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Colvin, A; et al. (2018). "Peripatric speciation". WikiJournal of Science 2 (1): 8. doi:10.15347/wjs/2018.008. ISSN 24706345. https://en.wikiversity.org/wiki/WikiJournal_of_Science/Peripatric_speciation.
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The concept of peripatric speciation was first outlined by the evolutionary biologist Ernst Mayr in 1954.^{[3]} Since then, other alternative models have been developed such as centrifugal speciation, that posits that a species' population experiences periods of geographic range expansion followed by shrinking periods, leaving behind small isolated populations on the periphery of the main population. Other models have involved the effects of sexual selection on limited population sizes. Other related models of peripherally isolated populations based on chromosomal rearrangements have been developed such as budding speciation and quantum speciation.
The existence of peripatric speciation is supported by observational evidence and laboratory experiments.^{[1]}^{:106} Scientists observing the patterns of a species biogeographic distribution and its phylogenetic relationships are able to reconstruct the historical process by which they diverged. Further, oceanic islands are often the subject of peripatric speciation research due to their isolated habitats—with the Hawaiian Islands widely represented in much of the scientific literature.Peripatric speciation was originally proposed by Ernst Mayr in 1954,^{[3]} and fully theoretically modeled in 1982.^{[4]} It is related to the founder effect, where small living populations may undergo selection bottlenecks.^{[5]} The founder effect is based on models that suggest peripatric speciation can occur by the interaction of selection and genetic drift,^{[1]}^{:106} which may play a significant role.^{[6]} Mayr first conceived of the idea by his observations of kingfisher populations in New Guinea and its surrounding islands.^{[1]}^{:389} Tanysiptera galatea was largely uniform in morphology on the mainland, but the populations on the surrounding islands differed significantly—referring to this pattern as "peripatric".^{[1]}^{:389} This same pattern was observed by many of Mayr's contemporaries at the time such as by E. B. Ford's studies of Maniola jurtina.^{[7]}^{:522} Around the same time, the botanist Verne Grant developed a model of quantum speciation very similar to Mayr's model in the context of plants.^{[8]}
In what has been called Mayr's genetic revolutions, he postulated that genetic drift played the primary role that resulted in this pattern.^{[1]}^{:389} Seeing that a species cohesion is maintained by conservative forces such as epistasis and the slow pace of the spread of favorable alleles in a large population (based heavily on J. B. S. Haldane's calculations), he reasoned that speciation could only take place in which a population bottleneck occurred.^{[1]}^{:389} A small, isolated, founder population could be established on an island for example. Containing less genetic variation from the main population, shifts in allele frequencies may occur from different selection pressures.^{[1]}^{:390} This leads to further changes in the network of linked loci, driving a cascade of genetic change, or a "genetic revolution"—a largescale reorganization of the entire genome of the peripheral population.^{[1]}^{:391} Mayr did recognize that the chances of success were incredibly low and that extinction was likely; though noting that some examples of successful founder populations existed at the time.^{[7]}^{:522}
Shortly after Mayr, William Louis Brown, Jr. proposed an alternative model of peripatric speciation in 1957 called centrifugal speciation. In 1976 and 1980, the Kaneshiro model of peripatric speciation was developed by Kenneth Y. Kaneshiro which focused on sexual selection as a driver for speciation during population bottlenecks.^{[9]}^{[10]}^{[11]}
Figure 1  Diagrams representing the process of peripatric and centrifugal speciation. A) In peripatry, a small population becomes isolated on the periphery of the central population evolving reproductive isolation (blue) due to reduced gene flow. B) In centrifugal speciation, an original population (green) range expands and contracts, leaving an isolated fragment population behind. The central population (changed to blue) evolves reproductive isolation in contrast to peripatry.
Peripatric speciation models are identical to models of vicariance (allopatric speciation).^{[1]}^{:105} Requiring both geographic separation and time, speciation can result as a predictable byproduct.^{[12]} Peripatry can be distinguished from allopatric speciation by three key features:^{[1]}^{:105}
The size of a population is important because individuals colonizing a new habitat likely contain only a small sample of the genetic variation of the original population. This promotes divergence due to strong selective pressures, leading to the rapid fixation of an allele within the descendant population. This gives rise to the potential for genetic incompatibilities to evolve. These incompatibilities cause reproductive isolation, giving rise to—sometimes rapid—speciation events.^{[1]}^{:105} Furthermore, two important predictions are invoked, namely that geological or climactic changes cause populations to become locally fragmented (or regionally when considering allopatric speciation), and that an isolated population's reproductive traits evolve enough as to prevent interbreeding upon potential secondary contact.^{[13]} The role of genetic drift and founder effects in speciation remains controversial and unresolved.^{[1]}^{:394396}
The peripatric model results in, what have been called, progenitorderivative species pairs, whereby the derivative species (the peripherally isolated population)—geographically and genetically isolated from the progenitor species—diverges.^{[14]} A specific phylogenetic signature results from this mode of speciation: the geographically widespread progenitor species becomes paraphyletic (thereby becoming a paraspecies), with respect to the derivative species (the peripheral isolate).^{[1]}^{:470} The concept of a paraspecies is therefore a logical consequence of the evolutionary species concept, by which one species gives rise to a daughter species.^{[15]} It is thought that the character traits of the peripherally isolated species become apomorphic, while the central population remains pleisomorphic.^{[16]}
Modern cladistic methods have developed definitions that have incidentally removed derivative species by defining clades in a way that assumes that when a speciation event occurs, the original species no longer exists, while two new species arise; this is not the case in peripatric speciation.^{[8]} Mayr warned against this, as it causes a species to lose their classification status.^{[17]} Loren H. Rieseberg and Luc Brouillet recognized the same dilemma in plant classification.^{[18]}
The botanist Verne Grant proposed the term quantum speciation that combined the ideas of J. T. Gulick (his observation of the variation of species in semiisolation), Sewall Wright (his models of genetic drift), Mayr (both his peripatric and genetic revolution models), and George Gaylord Simpson (his development of the idea of quantum evolution).^{[19]}^{:114} Quantum speciation is a rapid process with large genotypic or phenotypic effects, whereby a new, crossfertilizing plant species buds off from a larger population as a semiisolated peripheral population.^{[20]}^{[19]}^{:114} Interbreeding and genetic drift takes place due to the reduced population size, driving changes to the genome that would most likely result in extinction (due to low adaptive value).^{[19]}^{:115} In rare instances, chromosomal traits with adaptive value may arise, resulting in the origin of a new, derivative species.^{[8]}^{[21]} Evidence for the occurrence of this type of speciation has been found in several plant species pairs: Layia discoidea and L. glandulosa, Clarkia lingulata and C. biloba, and Stephanomeria malheurensis and S. exigua ssp. coronaria.^{[8]}
A closely related model of peripatric speciation is called budding speciation—largely applied in the context of plant speciation.^{[22]} The budding process, where a new species originates at the margins of an ancestral range, is thought to be common in plants^{[22]}—especially in progenitorderivative species pairs.^{[23]}
William Louis Brown, Jr. proposed an alternative model of peripatric speciation in 1957 called centrifugal speciation. This model contrasts with peripatric speciation by virtue of the origin of the genetic novelty that leads to reproductive isolation.^{[24]} A population of a species experiences periods of geographic range expansion followed by periods of contraction. During the contraction phase, fragments of the population become isolated as small refugial populations on the periphery of the central population. Because of the large size and potentially greater genetic variation within the central population, mutations arise more readily. These mutations are left in the isolated peripheral populations, whereby, promoting reproductive isolation. Consequently, Brown suggested that during another expansion phase, the central population would overwhelm the peripheral populations, hindering speciation. However, if the species finds a specialized ecological niche, the two may coexist.^{[25]}^{[26]} The phylogenetic signature of this model is that the central population becomes derived, while the peripheral isolates (in this case, the progenitors) become paraphyletic^{[16]}—the reverse of the general model. In contrast to centrifugal speciation, peripatric speciation has sometimes been referred to as centripetal speciation (see figures 1 and 2 for a contrast).^{[27]} Centrifugal speciation has been largely ignored in the scientific literature, often dominated by the traditional model of peripatric speciation.^{[28]}^{[24]}^{[16]} Despite this, Brown cited a wealth of evidence to support his model, of which has not yet been refuted.^{[25]}
Peromyscus polionotus and P. melanotis (the peripherally isolated species from the central population of P. maniculatus) arose via the centrifugal speciation model.^{[29]} Centrifugal speciation may have taken place in tree kangaroos, South American frogs (Ceratophrys), shrews (Crocidura), and primates (Presbytis melalophos).^{[28]} John C. Briggs associates centrifugal speciation with centers of origin, contending that the centrifugal model is better supported by the data, citing species patterns from the proposed 'center of origin' within the IndoWest Pacific^{[28]}
Figure 2  In the Kaneshiro model, a sample of a larger population results in an isolated population with less males containing attractive traits. Over time, choosy females are selected against as the population increases. Sexual selection drives new traits to arise (green), reproductively isolating the new population from the old one (blue).^{[11]}
When a sexual species experiences a population bottleneck—that is, when the genetic variation is reduced due to small population size—mating discrimination among females may be altered by the decrease in courtship behaviors of males.^{[11]} Sexual selection pressures may become weakened by this in an isolated peripheral population, and as a byproduct of the altered mating recognition system, secondary sexual traits may appear.^{[9]} Eventually, a growth in population size paired with novel female mate preferences will give rise to reproductive isolation from the main populationthereby completing the peripatric speciation process.^{[10]} Support for this model comes from experiments and observation of species that exhibit asymmetric mating patterns such as the Hawaiian Drosophila species^{[30]}^{[31]} or the Hawaiian cricket Laupala.^{[32]} However, this model has not been entirely supported by experiments, and therefore, it may not represent a plausible process of peripatric speciation that takes place in nature.^{[11]}
Observational evidence from nature and laboratory experiments support the occurrence of peripatric speciation. Islands and archipelagos are often the subject of speciation studies in that they represent isolated populations of organisms. Island species provide direct evidence of speciation occurring peripatrically in such that, "the presence of endemic species on oceanic islands whose closest relatives inhabit a nearby continent" must have originated by a colonization event.^{[1]}^{:106107} Comparative phylogeography of oceanic archipelagos shows consistent patterns of sequential colonization and speciation along island chains, most notably on the Azores islands, Canary Islands, Society Islands, Marquesas Islands, Galápagos Islands, Austral Islands, and the Hawaiian Islands—all of which express geological patterns of spatial isolation and, in some cases, linear arrangement.^{[33]} Peripatric speciation also occurs on continents, as isolation of small populations can occur through various geographic and dispersion events. Laboratory studies have been conducted where populations of Drosophila, for example, are separated from one another and evolve in reproductive isolation.
Figure 3  A) Colonization events of species from the genus Cyanea (green) and species from the genus Drosophila (blue) on the Hawaiian island chain. Islands age from left to right, (Kauai being the oldest and Hawaii being the youngest). Speciation arises peripatrically as they spatiotemporally colonize new islands along the chain. Lighter blue and green indicate colonization in the reverse direction from youngtoold. B) A map of the Hawaiian archipelago showing the colonization routes of Theridion grallator superimposed. Purple lines indicate colonization occurring in conjunction with island age where light purple indicates backwards colonization. T. grallator is not present on Kauai or Niihau so colonization may have occurred from there, or the nearest continent. C) The sequential colonization and speciation of the ‘Elepaio subspecies along the Hawaiian island chain.
Drosophila species on the Hawaiian archipelago have helped researchers understand speciation processes in great detail. It is well established that Drosophila has undergone an adaptive radiation into hundreds of endemic species on the Hawaiian island chain;^{[1]}^{:107}^{[34]} originating from a single common ancestor (supported from molecular analysis).^{[35]} Studies consistently find that colonization of each island occurred from older to younger islands, and in Drosophila, speciating peripatrically at least fifty percent of the time.^{[1]}^{:108} In conjunction with Drosophila, Hawaiian lobeliads (Cyanea) have also undergone an adaptive radiation, with upwards of twentyseven percent of extant species arising after new island colonization—exemplifying peripatric speciation—once again, occurring in the oldtoyoung island direction.^{[36]}^{[37]}^{[38]}
Other endemic species in Hawaii also provide evidence of peripatric speciation such as the endemic flightless crickets (Laupala). It has been estimated that, "17 species out of 36 wellstudied cases of [Laupala] speciation were peripatric".^{[1]}^{:108}^{[39]} Plant species in genera's such as Dubautia, Wilkesia, and Argyroxiphium have also radiated along the archipelago.^{[40]} Other animals besides insects show this same pattern such as the Hawaiian amber snail (Succinea caduca),^{[41]} and ‘Elepaio flycatchers (Chasiempis).^{[42]}
Tetragnatha spiders have also speciated peripatrically on the Hawaiian islands,^{[43]}^{[44]} Numerous arthropods have been documented existing in patterns consistent with the geologic evolution of the island chain, in such that, phylogenetic reconstructions find younger species inhabiting the geologically younger islands and older species inhabiting the older islands^{[45]} (or in some cases, ancestors date back to when islands currently below sea level were exposed). Spiders such as those from the genus Orsonwelles exhibit patterns compatible with the oldtoyoung geology.^{[46]} Other endemic genera such as Argyrodes have been shown to have speciated along the island chain.^{[47]} Pagiopalus, Pedinopistha, and part of the Thomisidae family have adaptively radiated along the island chain,^{[48]} as well as the Lycosidae family of wolf spiders.^{[49]}
A host of other Hawaiian endemic arthropod species and genera have had their speciation and phylogeographical patterns studied: the Drosophila grimshawi species complex,^{[50]} damselflies (Megalagrion xanthomelas and Megalagrion pacificum),^{[51]} Doryonychus raptor, Littorophiloscia hawaiiensis, Anax strenuus, Nesogonia blackburni, Theridion grallator,^{[52]} Vanessa tameamea, Hyalopeplus pellucidus, Coleotichus blackburniae, Labula, Hawaiioscia, Banza (in the Tettigoniidae family), Caconemobius, Eupethicea, Ptycta, Megalagrion, Prognathogryllus, Nesosydne, Cephalops, Trupanea, and the tribe Platynini—all suggesting repeated radiations among the islands.^{[53]}
Phylogenetic studies of a species of crab spider (Misumenops rapaensis) in the genus Thomisidae located on the Austral Islands have established the, "sequential colonization of [the] lineage down the Austral archipelago toward younger islands". M. rapaensis has been traditionally thought of as a single species; whereas this particular study found distinct genetic differences corresponding to the sequential age of the islands.^{[54]} The figwart plant species Scrophularia lowei is thought to have arisen through a peripatric speciation event, with the more widespread mainland species, Scrophularia arguta dispersing to the Macaronesian islands.^{[55]}^{[56]} Other members of the same genus have also arisen by single colonization events between the islands.^{[57]}^{[58]}
The occurrence of peripatry on continents is more difficult to detect due to the possibility of vicariant explanations being equally likely.^{[1]}^{:110} However, studies concerning the Californian plant species Clarkia biloba and C. lingulata strongly suggest a peripatric origin.^{[59]} In addition, a great deal of research has been conducted on several species of land snails involving chirality that suggests peripatry (with some authors noting other possible interpretations).^{[1]}^{:111}
Figure 4 
A) The southern chestnuttailed antbird, Sciaphylax hemimelaena
B) Satellite image of the Noel Kempff Mercado National Park (outlined in green) in Bolivia, South America. The white arrow indicates the location of the isolated forest fragment.
A) Hector Bottai, CCBYSA 3.0
The chestnuttailed antbird Sciaphylax hemimelaena (formerly called Myrmeciza hemimelaena) is located within the Noel Kempff Mercado National Park (Serrania de Huanchaca) in Bolivia. Within this region exists a forest fragment estimated to have been isolated for 1000–3000 years. The population of S. hemimelaena antbirds that reside in the isolated patch express significant song divergence; thought to be an "early step" in the process of peripatric speciation. Further, peripheral isolation "may partly explain the dramatic diversification of suboscines in Amazonia".^{[13]}
The montane spiny throated reed frog species complex (genus: Hyperolius) originated through occurrences of peripatric speciation events. Lucinda P. Lawson maintains that the species' geographic ranges within the Eastern Afromontane Biodiversity Hotspot support a peripatric model that is driving speciation; suggesting that this mode of speciation may play a significant role in "highly fragmented ecosystems".^{[2]}
In a study of the phylogeny and biogeography of the land snail genus Monacha, the species M. ciscaucasica is thought to have speciated peripatrically from a population of M. roseni. In addition, M. claussi consists of a small population located on the peripheral of the much larger range of M. subcarthusiana suggesting that it also arose by peripatric speciation.^{[60]}
Figure 5 
Foliage and cones of
A) Picea mariana, and
B) Picea rubens
A) MPF, CCBYSA 3.0, B) Keith Kanoti, CCBY 3.0
Red spruce (Picea rubens) has arisen from an isolated population of black spruce (Picea mariana). During the Pleistocene, a population of black spruce became geographically isolated, likely due to glaciation. The geographic range of the black spruce is much larger than the red spruce. The red spruce has significantly lower genetic diversity in both its nuclear DNA and its mitochondrial DNA than the black spruce.^{[61]}^{[62]} Furthermore, the genetic variation of the red spruce has no unique mitochondrial haplotypes, only subsets of those in the black spruce; suggesting that the red spruce speciated peripatrically from the black spruce population.^{[63]}^{[64]}^{[65]} It is thought that the entire genus Picea in North America has diversified by the process of peripatric speciation, as numerous pairs of closely related species in the genus have smaller southern population ranges; and those with overlapping ranges often exhibit weak reproductive isolation.^{[66]}^{[62]}
Using a phylogeographic approach paired with ecological niche models (i.e. prediction and identification of expansion and contraction species ranges into suitable habitats based on current ecological niches, correlated with fossil and molecular data), researchers found that the prairie dog species Cynomys mexicanus speciated peripatrically from Cynomys ludovicianus approximately 230,000 years ago. North American glacial cycles promoted range expansion and contraction of the prairie dogs, leading to the isolation of a relic population in a refugium located in the present day Coahuila, Mexico.^{[67]} This distribution and paleobiogeographic pattern correlates with other species expressing similar biographic range patterns^{[67]} such as with the Sorex cinereus complex.^{[68]}
Species  Replicates  Year 

Drosophila adiastola  1  1979^{[69]} 
Drosophila silvestris  1  1980^{[70]} 
Drosophila pseudoobscura  8  1985^{[71]} 
Drosophila simulans  8  1985^{[72]} 
Musca domestica  6  1991^{[73]} 
Drosophila pseudoobscura  42  1993^{[74]} 
Drosophila melanogaster  50  1998^{[75]} 
Drosophila melanogaster  19; 19  1999^{[76]} 
Drosophila grimshawi  1  N/A^{[11]} 
Table 1  A nonexhaustive table of laboratory experiments focused explicitly on peripatric speciation. Most of the studies also conducted experiments on vicariant speciation as well. The "replicates" column signifies the number of lines used in the experiment—that is, how many independent populations were used (not the population size or the number of generations performed).^{[11]} 
Laboratory studies have been conducted in an attempt to replicate peripatric speciation in a controlled setting. Jerry Coyne and H. Allen Orr in Speciation suggest that most laboratory studies of allopatric speciation are also examples of peripatric speciation due to their small population sizes and the inevitable divergent selection that they undergo.^{[1]}^{:106} Much of the laboratory research concerning peripatry is inextricably linked to founder effect research. Coyne and Orr conclude that selection's role in speciation is well established, whereas genetic drift's role is unsupported by experimental and field data—suggesting that foundereffect speciation does not occur.^{[1]}^{:410} Nevertheless, a great deal of research has been conducted on the matter, and one study conducted involving populations of Drosophila pseudoobscura found evidence of isolation after a single bottleneck.^{[77]}^{[78]}
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Author: Mikhail Boldyrev^{[a]}^{[i]}, et al.
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Boldyrev, M; et al. (2018). "Lead: properties, history, and applications". WikiJournal of Science 1 (1): 7. doi:10.15347/wjs/2018.007. ISSN 24706345. https://en.wikiversity.org/wiki/WikiJournal_of_Science/Lead:_properties,_history,_and_applications.
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Lead is a relatively unreactive posttransition metal. Its weak metallic character is illustrated by its amphoteric nature; lead and its oxides react with acids and bases, and it tends to form covalent bonds. Compounds of lead are usually found in the +2 oxidation state rather than the +4 state common with lighter members of the carbon group. Exceptions are mostly limited to organolead compounds. Like the lighter members of the group, lead tends to bond with itself; it can form chains, rings and polyhedral structures.
Lead is easily extracted from its ores; prehistoric people in Western Asia knew of it. Galena, a principal ore of lead, often bears silver, interest in which helped initiate widespread extraction and use of lead in ancient Rome. Lead production declined after the fall of Rome and did not reach comparable levels until the Industrial Revolution. In 2014, annual global production of lead was about ten million tonnes, over half of which was from recycling. Lead's high density, low melting point, ductility, and relative inertness to oxidation make it useful. These properties, combined with its relative abundance and low cost, resulted in its extensive use in construction, plumbing, batteries, bullets and shot, weights, solders, pewters, fusible alloys, white paints, leaded gasoline, and radiation shielding.
In the late 19th century, lead's toxicity was recognized, and its use has since been phased out of many applications. Lead is a toxin that accumulates in soft tissues and bones, it acts as a neurotoxin damaging the nervous system and interferences with the function of biological enzymes. It is particularly problematic in children: even if blood levels are promptly normalized with treatment, neurological disorders, such as brain damage and behavioral problems, may result.
Figure 1 
A sample of lead solidified from the molten state
Juergen Kummer, CCBY 3.0
A lead atom has 82 electrons, arranged in an electron configuration of [Xe]4f^{14}5d^{10}6s^{2}6p^{2}. The combined first and second ionization energies—the total energy required to remove the two 6p electrons—is close to that of tin, lead's upper neighbor in the carbon group. This is unusual; ionization energies generally fall going down a group, as an element's outer electrons become more distant from the nucleus, and more shielded by smaller orbitals. The similarity of ionization energies is caused by the lanthanide contraction—the decrease in element radii from lanthanum (atomic number 57) to lutetium (71), and the relatively small radii of the elements after hafnium (72). This is due to poor shielding of the nucleus by the lanthanide 4f electrons. The combined first four ionization energies of lead exceed those of tin,^{[1]} contrary to what periodic trends would predict. Relativistic effects, which become significant in heavier atoms, contribute to this behavior.^{[a]} One such effect is the inert pair effect: the 6s electrons of lead become reluctant to participate in bonding, which leads to elevated ionization energies and makes the distance between nearest atoms in crystalline lead unusually long.^{[3]}
Lead's lighter carbon group congeners form stable or metastable allotropes with the tetrahedrally coordinated and covalently bonded diamond cubic structure. The energy levels of their outer s and porbitals are close enough to allow mixing into four hybrid sp^{3} orbitals. In lead, the inert pair effect increases the separation between its s and porbitals, and the gap cannot be overcome by the energy that would be released by extra bonds following hybridization.^{[4]} Rather than having a diamond cubic structure, lead forms metallic bonds in which only the pelectrons are delocalized and shared between the Pb^{2+} ions. Lead consequently has a facecentered cubic structure^{[5]} like the similarly sized^{[6]} divalent metals calcium and strontium.^{[7]}^{[b]}^{[c]}^{[d]}
Pure lead has a bright, silvery appearance with a hint of blue.^{[12]} It tarnishes on contact with moist air, and takes on a dull appearance the hue of which depends on the prevailing conditions. Characteristic properties of lead include high density, malleability, ductility,^{[e]} and high resistance to corrosion due to passivation.^{[13]}
Isotopic abundances vary greatly by sample  
Standard atomic weight (A_{r, standard}) 


Lead's closepacked facecentered cubic structure and high atomic weight result in a density^{[15]} of 11.34 g/cm^{3}, which is greater than that of common metals such as iron (7.87 g/cm^{3}), copper (8.93 g/cm^{3}), and zinc (7.14 g/cm^{3}).^{[16]} This density is the origin of the idiom to go over like a lead balloon.^{[17]}^{[18]}^{[f]} Some rarer metals are denser: tungsten and gold are both at 19.3 g/cm^{3}, and osmium—the densest metal known—has a density of 22.59 g/cm^{3}, almost twice that of lead.^{[19]}
Lead is a very soft metal with a Mohs hardness of 1.5; it can be scratched with a fingernail.^{[20]} It is very malleable and quite ductile.^{[21]} The bulk modulus of lead—a measure of its ease of compressibility—is 45.8 GPa. In comparison, that of aluminium is 75.2 GPa; copper 137.8 GPa; and mild steel 160–169 GPa.^{[22]} Lead's tensile strength, at 12–17 MPa, is low (that of aluminium is 6 times higher, copper 10 times, and mild steel 15 times higher); it can be strengthened by adding small amounts of copper or antimony.^{[23]}
The melting point of lead—at 327.5 °C (621.5 °F)^{[24]}—is very low compared to most metals.^{[15]}^{[g]} Its boiling point of 1749 °C (3180 °F)^{[24]} is the lowest among the carbon group elements. The electrical resistivity of lead at 20 °C is 192 nanoohmmeters, almost an order of magnitude higher than those of other industrial metals (copper at 15.43 nΩ·m; gold 20.51 nΩ·m; and aluminium at 24.15 nΩ·m).^{[26]} Lead is a superconductor at temperatures lower than 7.19 K;^{[27]} this is the highest critical temperature of all typeI superconductors and the third highest of the elemental superconductors.^{[28]}
Natural lead consists of four stable isotopes with mass numbers of 204, 206, 207, and 208,^{[29]} and traces of five shortlived radioisotopes.^{[30]} The main isotopes of lead, with information on percent abundance, halflife, and decay mode and product, are listed in Table 1. The high number of isotopes is consistent with lead's atomic number being even.^{[h]} Lead has a magic number of protons (82), for which the nuclear shell model accurately predicts an especially stable nucleus.^{[31]} Lead208 has 126 neutrons, another magic number, which may explain why lead208 is extraordinarily stable.^{[31]}
With its high atomic number, lead is the heaviest element whose natural isotopes are regarded as stable; lead208 is the heaviest stable nucleus. This title was formerly held by bismuth, with an atomic number of 83, until its only primordial isotope, bismuth209, was found in 2003 to decay very slowly.^{[i]} The four stable isotopes of lead could theoretically undergo alpha decay to isotopes of mercury with a release of energy, but this has not been observed for any of them; their predicted halflives range from 10^{35} to 10^{189} years.^{[34]}
Figure 2 
The Holsinger meteorite, the largest piece of the Canyon Diablo meteorite. Uranium–lead dating and lead–lead dating on this meteorite allowed refinement of the age of the Earth to 4.55 billion ± 70 million years.
Marcin Wichary, CCBY 2.0
Three of the stable isotopes are found in three of the four major decay chains: lead206, lead207, and lead208 are the final decay products of uranium238, uranium235, and thorium232, respectively. These decay chains are called the uranium chain, the actinium chain, and the thorium chain.^{[35]} Their isotopic concentrations in a natural rock sample depends greatly on the presence of these three parent uranium and thorium isotopes. For example, the relative abundance of lead208 can range from 52% in normal samples to 90% in thorium ores;^{[36]} for this reason, the standard atomic weight of lead is given to only one decimal place.^{[37]} As time passes, the ratios of lead206 and lead207 to lead204 increase, since the former two are supplemented by radioactive decay of heavier elements while the latter is not; analysis of lead207/lead204 ratios versus lead206/lead204 ratios constitutes lead–lead dating. As uranium decays into lead, their relative amounts change; analysis of uranium238/lead206 ratios versus uranium235/lead207 ratios constitutes uranium–lead dating.^{[38]} Lead207 exhibits nuclear magnetic resonance, a property that has been used to study its compounds in solution and solid state,^{[39]}^{[40]} including in human body.^{[41]}
Apart from the stable isotopes, which make up almost all lead that exists naturally, there are trace quantities of a few radioactive isotopes. One of them is lead210; although it has a halflife of only 22.3 years,^{[29]} small quantities occur in nature because lead210 is produced by a long decay series that starts with uranium238 (which has been present for billions of years on Earth). Lead211, 212, and 214 are present in the decay chains of uranium235, thorium232, and uranium238, respectively, so traces of all three of these lead isotopes are found naturally. Minute traces of lead209 arise from the very rare cluster decay of radium223, one of the daughter products of natural uranium235. Lead210 is particularly useful for helping to identify the ages of samples by measuring its ratio to lead206 (both isotopes are present in a single decay chain).^{[42]}
In total, 43 lead isotopes have been synthesized, with mass numbers 178–220.^{[29]} Lead205 is the most stable radioisotope, with a halflife of around 1.5×10^{7} years.^{[j]} The secondmost stable is lead202, which has a halflife of about 53,000 years, longer than any of the natural trace radioisotopes.^{[29]}
Figure 3 
Flame test: lead colors flame pale blue
Herge, public domain
Bulk lead exposed to moist air forms a protective layer of varying composition. Lead(II) carbonate is a common constituent;^{[44]}^{[45]}^{[46]} the sulfate or chloride may also be present in urban or maritime settings.^{[47]} This layer makes bulk lead effectively chemically inert in the air.^{[47]} Finely powdered lead, as with many metals, is pyrophoric,^{[48]} and burns with a bluishwhite flame.^{[49]}
Fluorine reacts with lead at room temperature, forming lead(II) fluoride. The reaction with chlorine is similar but requires heating, as the resulting chloride layer diminishes the reactivity of the elements.^{[47]} Molten lead reacts with the chalcogens to give lead(II) chalcogenides.^{[50]}
Lead metal resists sulfuric and phosphoric acid but not hydrochloric or nitric acid; the outcome depends on solubility and subsequent passivation of the product salt.^{[51]} Organic acids, such as acetic acid, dissolve lead in the presence of oxygen.^{[47]} Concentrated alkalis will dissolve lead and form plumbites.^{[52]}
Lead shows two main oxidation states: +4 and +2. The tetravalent state is common for the carbon group. The divalent state is rare for carbon and silicon, minor for germanium, important (but not prevailing) for tin, and is the more important of the two oxidation states for lead.^{[47]} This is attributable to relativistic effects, specifically the inert pair effect, which manifests itself when there is a large difference in electronegativity between lead and oxide, halide, or nitride anions, leading to a significant partial positive charge on lead. The result is a stronger contraction of the lead 6s orbital than is the case for the 6p orbital, making it rather inert in ionic compounds. The inert pair effect is less applicable to compounds in which lead forms covalent bonds with elements of similar electronegativity, such as carbon in organolead compounds. In these, the 6s and 6p orbitals remain similarly sized and sp^{3} hybridization is still energetically favorable. Lead, like carbon, is predominantly tetravalent in such compounds.^{[53]}
There is a relatively large difference in the electronegativity of lead(II) at 1.87 and lead(IV) at 2.33. This difference marks the reversal in the trend of increasing stability of the +4 oxidation state going down carbon group; tin, by comparison, has values of 1.80 in the +2 oxidation state and 1.96 in the +4 state.^{[54]}
Figure 4 
Polymorphs of lead(II) oxide, litharge (αPbO, left) and massicot (βPbO, right)
Hubertus Giefers, public domain
Lead(II) compounds are characteristic of the inorganic chemistry of lead. Even strong oxidizing agents like fluorine and chlorine react with lead to give only PbF_{2} and PbCl_{2}.^{[47]} Lead(II) ions are usually colorless in solution.^{[55]} No simple hydroxide is found; in aqueous solution, the lead(II) ion undergoes a series of pHdependent hydrolysis and condensation reactions, including Pb(OH)+ and the most common hydrolysis product, [Pb4(OH)4]4+.b_{4}(OH)_{4}]^{4+} (in which the hydroxyl ions act as bridging ligands).^{[56]}^{[57]} Lead(II) ions are not reducing agents as tin(II) ions are. Techniques for identifying the presence of the Pb^{2+} ion in water generally rely on the precipitation of lead(II) chloride using dilute hydrochloric acid. As the chloride salt is sparingly soluble in water, in very dilute solutions the precipitation of lead(II) sulfide is achieved by bubbling hydrogen sulfide through the solution..^{[58]}
Lead monoxide exists in two polymorphs, litharge αPbO (red) and massicot βPbO (yellow), the latter being stable only above around 488 °C. It is the most commonly used inorganic compound of lead.^{[59]} There is no lead(II) hydroxide; increasing the pH of solutions of lead(II) salts leads to hydrolysis and condensation.^{[60]} Lead commonly reacts with heavier chalcogens. Lead sulfide is a semiconductor, a photoconductor, and an extremely sensitive infrared radiation detector. The other two chalcogenides, lead selenide and lead telluride, are likewise photoconducting. They are unusual in that their color becomes lighter going down the group.^{[61]}
Figure 5 
Lead and oxygen in a tetragonal unit cell of lead(II,IV) oxide
Ben Mills, public domain
Lead dihalides are wellcharacterized; this includes the diastatide,^{[62]} and mixed halides, such as PbFCl. The relative insolubility of the latter forms a useful basis for the gravimetric determination of fluorine. The difluoride was the first solid ionically conducting compound to be discovered (in 1834, by Michael Faraday).^{[63]} The other dihalides decompose on exposure to ultraviolet or visible light, especially the diiodide.^{[64]} Many lead(II) pseudohalides are known, such as the cyanide, cyanate, and thiocyanate.^{[61]}^{[65]} Lead(II) forms an extensive variety of halide coordination complexes, such as [PbCl_{4}]^{2−}, [PbCl_{6}]^{4−}, and the [Pb_{2}Cl_{9}]_{n}^{5n−} chain anion.^{[64]}
Lead(II) sulfate is insoluble in water, like the sulfates of other heavy divalent cations. Lead(II) nitrate and lead(II) acetate are very soluble, and this is exploited in the synthesis of other lead compounds.^{[66]}
Few inorganic lead(IV) compounds are known. They are only formed in highly oxidizing solutions and do not normally exist under standard conditions.^{[67]} Lead(II) oxide gives a mixed oxide on further oxidation, Pb_{3}O_{4}. It is described as lead(II,IV) oxide, or structurally 2PbO·PbO_{2}, and is the bestknown mixed valence lead compound. Lead dioxide is a strong oxidizing agent, capable of oxidizing hydrochloric acid to chlorine gas.^{[68]} This is because the expected PbCl_{4} that would be produced is unstable and spontaneously decomposes to PbCl_{2} and Cl_{2}.^{[69]} Analogously to lead monoxide, lead dioxide is capable of forming plumbate anions. Lead disulfide^{[70]} and lead diselenide^{[71]} are only stable at high pressures. Lead tetrafluoride, a yellow crystalline powder, is stable, but less so than the difluoride. Lead tetrachloride (a yellow oil) decomposes at room temperature, lead tetrabromide is less stable still, and the existence of lead tetraiodide is questionable.^{[72]}
Figure 6 
The capped square antiprismatic anion [Pb_{9}]^{4−}^{[73]}
Ben Mills, public domain
Some lead compounds exist in formal oxidation states other than +4 or +2. Lead(III) may be obtained, as an intermediate between lead(II) and lead(IV), in larger organolead complexes; this oxidation state is not stable, as both the lead(III) ion and the larger complexes containing it are radicals.^{[74]}^{[75]}^{[76]} The same applies for lead(I), which can be found in such radical species.^{[77]}
Numerous mixed lead(II,IV) oxides are known. When PbO_{2} is heated in air, it becomes Pb_{12}O_{19} at 293 °C, Pb_{12}O_{17} at 351 °C, Pb_{3}O_{4} at 374 °C, and finally PbO at 605 °C. A further sesquioxide, Pb_{2}O_{3}, can be obtained at high pressure, along with several nonstoichiometric phases. Many of them show defective fluorite structures in which some oxygen atoms are replaced by vacancies: PbO can be considered as having such a structure, with every alternate layer of oxygen atoms absent.^{[78]}
Negative oxidation states can occur as Zintl phases, as either free lead anions, as in Ba_{2}Pb, with lead formally being lead(−IV),^{[79]} or in oxygensensitive ringshaped or polyhedral cluster ions such as the trigonal bipyramidal Pb_{5}^{2−} ion, where two lead atoms are lead(−I) and three are lead(0).^{[80]} In such anions, each atom is at a polyhedral vertex and contributes two electrons to each covalent bond along an edge from their sp^{3} hybrid orbitals, the other two being an external lone pair.^{[56]} They may be made in liquid ammonia via the reduction of lead by sodium.^{[81]}
Figure 7 
Structure of a tetraethyllead molecule:
Carbon
Hydrogen
Lead
Jynto, public domain
Lead can form multiplybonded chains, a property it shares with its lighter homologs in the carbon group. The capacity for catenation decreases going down the group due to decreasing bond energy. The Pb–Pb bond energy is over three and a half times lower than that of the C–C bond.^{[50]} With itself lead can build metal–metal bonds of an order up to three.^{[82]} With carbon, lead forms organolead compounds similar to, but generally less stable than, typical organic compounds^{[83]} (due to the Pb–C bond being rather weak).^{[56]} This makes the organometallic chemistry of lead far less wideranging than that of tin.^{[84]} Lead predominantly forms organolead(IV) compounds, even when starting with inorganic lead(II) reactants; very few organolead(II) compounds are known. The most wellcharacterized exceptions are Pb[CH(SiMe_{3})_{2}]_{2} and Pb(η^{5}C_{5}H_{5})_{2}.^{[84]}
The lead analog of the simplest organic compound, methane, is plumbane. Plumbane may be obtained in a reaction between metallic lead and atomic hydrogen.^{[85]} It is thermodynamically unstable and its chemistry has not been determined.^{[86]} Two simple derivatives, tetramethyllead and tetraethyllead, are the bestknown organolead compounds. These compounds are relatively stable: tetraethyllead only starts to decompose if heated^{[87]} or if exposed to sunlight or ultraviolet light.^{[88]} (Tetraphenyllead is even more thermally stable, decomposing at 270 °C.^{[84]}) With sodium metal, lead readily forms an equimolar alloy that reacts with alkyl halides to form organometallic compounds such as tetraethyllead.^{[89]} The oxidizing nature of many organolead compounds is usefully exploited: lead tetraacetate is an important laboratory reagent for oxidation in organic synthesis,^{[90]} and tetraethyllead was once produced in larger quantities than any other organometallic compound.^{[84]} Other organolead compounds are less chemically stable.^{[83]} For many organic compounds, a lead analog does not exist.^{[85]}
Atomic number 
Element  Relative amount 

42  Molybdenum  0.798 
46  Palladium  0.440 
50  Tin  1.146 
78  Platinum  0.417 
80  Mercury  0.127 
82  Lead  1 
90  Thorium  0.011 
92  Uranium  0.003 
Lead's perparticle abundance in the Solar System is 0.121 ppb (parts per billion).^{[91]}^{[k]} This figure is two and a half times higher than that of platinum, eight times more than mercury, and seventeen times more than gold.^{[91]} The amount of lead in the universe is slowly increasing^{[92]} as most heavier atoms (all of which are unstable) gradually decay to lead.^{[93]} The abundance of lead in the Solar System since its formation 4.5 billion years ago has increased by about 0.75%.^{[94]} The solar system abundances table shows that lead, despite its relatively high atomic number, is more prevalent than most other elements with atomic numbers greater than 40.^{[91]}
Primordial lead—which comprises the isotopes lead204, lead206, lead207, and lead208—was mostly created as a result of repetitive neutron capture processes occurring in stars. The two main modes of capture are the s and rprocesses.^{[95]}
In the sprocess (s is for "slow"), captures are separated by years or decades, allowing less stable nuclei to undergo beta decay.^{[96]} A stable thallium203 nucleus can capture a neutron and become thallium204; this undergoes beta decay to give stable lead204; on capturing another neutron, it becomes lead205, which has a halflife of around 15 million years. Further captures result in lead206, lead207, and lead208. On capturing another neutron, lead208 becomes lead209, which quickly decays into bismuth209. On capturing another neutron, bismuth209 becomes bismuth210, and this beta decays to polonium210, which alpha decays to lead206. The cycle hence ends at lead206, lead207, lead208, and bismuth209.^{[97]}
Figure 8  Chart of the final part of the sprocess, from mercury to polonium. Red lines and circles represent neutron captures; blue arrows represent beta decays; the green arrow represents an alpha decay; cyan arrows represent electron captures.
In the rprocess (r is for "rapid"), captures happen faster than nuclei can decay.^{[98]} This occurs in environments with a high neutron density, such as a supernova or the merger of two neutron stars. The neutron flux involved may be on the order of 10^{22} neutrons per square centimeter per second.^{[99]} The rprocess does not form as much lead as the sprocess.^{[100]} It tends to stop once neutronrich nuclei reach 126 neutrons.^{[101]} At this point, the neutrons are arranged in complete shells in the atomic nucleus, and it becomes harder to energetically accommodate more of them.^{[102]} When the neutron flux subsides, these nuclei beta decay into stable isotopes of osmium, iridium, and platinum.^{[103]}
Figure 9 
Lead is a fairly common element in the Earth's crust for its high atomic number (82). Most elements of atomic number greater than 40 are less abundant.
Gordon Haxel, Sara Boore, Susan Mayfield, public domain
Lead is classified as a chalcophile under the Goldschmidt classification, meaning it is generally found combined with sulfur.^{[104]} It rarely occurs in its native, metallic form.^{[105]} Many lead minerals are relatively light and, over the course of the Earth's history, have remained in the crust instead of sinking deeper into the Earth's interior. This accounts for lead's relatively high crustal abundance of 14 ppm; it is the 38th most abundant element in the crust.^{[106]}^{[l]}
The main leadbearing mineral is galena (PbS), which is mostly found with zinc ores.^{[108]} Most other lead minerals are related to galena in some way; boulangerite, Pb_{5}Sb_{4}S_{11}, is a mixed sulfide derived from galena; anglesite, PbSO_{4}, is a product of galena oxidation; and cerussite or white lead ore, PbCO_{3}, is a decomposition product of galena. Arsenic, tin, antimony, silver, gold, copper, and bismuth are common impurities in lead minerals.^{[108]}
World lead resources exceed 2 billion tons. Significant deposits are located in Australia, China, Ireland, Mexico, Peru, Portugal, Russia, and the United States. Global reserves—resources that are economically feasible to extract—totaled 88 million tons in 2016, of which Australia had 35 million, China 17 million, and Russia 6.4 million.^{[109]}
Typical background concentrations of lead do not exceed 0.1 μg/m^{3} in the atmosphere; 100 mg/kg in soil; and 5 μg/L in freshwater and seawater.^{[110]}
The modern English word "lead" is of Germanic origin; it comes from the Middle English leed and Old English lēad (with the macron above the "e" signifying that the vowel sound of that letter is long).^{[111]} The Old English word is derived from the hypothetical reconstructed ProtoGermanic *lauda ("lead").^{[112]} According to linguistic theory, this word bore descendants in multiple Germanic languages of exactly the same meaning.^{[112]}
The origin of the ProtoGermanic *lauda is not agreed in the linguistic community. One hypothesis suggests it is derived from ProtoIndoEuropean *lAudh ("lead"; capitalization of the vowel is equivalent to the macron).^{[113]} Another hypothesis suggests it is borrowed from ProtoCeltic *φloudio ("lead"). This word is related to the Latin plumbum, which gave the element its chemical symbol Pb. The word *φloudio is thought to be the origin of ProtoGermanic *bliwa (which also means "lead"), from which stemmed the German Blei.^{[114]}
The name of the chemical element is not related to the verb of the same spelling, which is derived from ProtoGermanic *laidijan ("to lead").^{[115]}
Figure 10 
World lead production peaking in the Roman period and the Industrial Revolution.^{[116]}
Fox 52, CCBYSA 4.0
Metallic lead beads dating back to 7000–6500 BCE have been found in Asia Minor and may represent the first example of metal smelting.^{[117]} At that time lead had few (if any) applications due to its softness and dull appearance.^{[117]} The major reason for the spread of lead production was its association with silver, which may be obtained by burning galena (a common lead mineral).^{[118]} The Ancient Egyptians were the first to use lead minerals in cosmetics, an application that spread to Ancient Greece and beyond;^{[119]} the Egyptians may have used lead for sinkers in fishing nets, glazes, glasses, enamels, and for ornaments.^{[118]} Various civilizations of the Fertile Crescent used lead as a writing material, as currency, and for construction.^{[118]} Lead was used in the Ancient Chinese royal court as a stimulant,^{[118]} as currency,^{[120]} and as a contraceptive;^{[121]} the Indus Valley civilization and the Mesoamericans^{[118]} used it for making amulets; and the eastern and southern African peoples used lead in wire drawing.^{[122]}
Because silver was extensively used as a decorative material and an exchange medium, lead deposits came to be worked in Asia Minor since 3000 BCE; later, lead deposits were developed throughout Mediterranea.^{[124]}^{[125]}
Rome's territorial expansion in Europe and across the Mediterranean, and its development of mining, led to it becoming the greatest producer of lead during the classical era, with an estimated annual output peaking at 80,000 tonnes. Like their predecessors, the Romans obtained lead mostly as a byproduct of silver smelting.^{[116]}^{[126]}
Box 1
This metal was by far the most used material in classical antiquity, and it is appropriate to refer to the (Roman) Lead Age. Lead was to the Romans what plastic is to us.
Lead was used for making water pipes; the Latin word for the metal, plumbum, is the origin of the English word "plumbing". Its ease of working and resistance to corrosion^{[128]} ensured its widespread use in other applications including pharmaceuticals, roofing, currency, and warfare.^{[129]}^{[130]}^{[131]} Writers of the time recommended lead (or leadcoated) vessels for the preparation of sweeteners and preservatives added to wine and food. The lead conferred an agreeable taste due to the formation of "sugar of lead" (lead(II) acetate), whereas copper or bronze vessels could impart a bitter flavor through verdigris formation.^{[132]}
The Roman author Vitruvius reported the health dangers of lead^{[133]} and modern writers have suggested that lead poisoning played a major role in the decline of the Roman Empire.^{[134]}^{[135]}^{[m]} Other researchers have criticized such claims, pointing out, for instance, that not all abdominal pain is caused by lead poisoning.^{[137]}^{[138]} According to archaeological research, Roman lead pipes increased lead levels in tap water but such an effect was "unlikely to have been truly harmful".^{[139]}^{[140]} When lead poisoning did occur, victims were called "saturnine", dark and cynical, after the ghoulish father of the gods, Saturn. By association, lead was considered the father of all metals.^{[141]} Its status in Roman society was low as it was readily available^{[142]} and cheap.^{[143]}
Figure 12 
Roman lead pipes^{[n]}
Wolfgang Sauber, CCBYSA 3.0
During the classical era (and even up to the 17th century), tin was often not distinguished from lead: Romans called lead plumbum nigrum ("black lead"), and tin plumbum candidum ("bright lead"). The association of lead and tin can be seen in other languages: the word olovo in Czech translates to "lead", but in Russian the cognate олово (olovo) means "tin".^{[144]} To add to the confusion, lead bore a close relation to antimony: both elements commonly occur as sulfides (galena and stibnite), often together. Pliny incorrectly wrote that stibnite would give lead on heating, instead of antimony.^{[145]} In countries such as Turkey and India, the originally Persian name surma came to refer to either antimony sulfide or lead sulfide,^{[146]} and in some languages, such as Russian, gave its name to antimony (сурьма).^{[147]}
Lead mining in Western Europe declined after the fall of the Western Roman Empire, with Arabian Iberia being the only region having a significant output.^{[148]}^{[149]} The largest production of lead occurred in South and East Asia, especially China and India, where lead mining grew strongly.^{[149]}
Figure 13 
Elizabeth I of England was commonly depicted with a whitened face. Lead in face whiteners is thought to have contributed to her death.^{[150]}
Nicholas Hilliard, public domain
In Europe, lead production only began to revive in the 11th and 12th centuries, when it was again used for roofing and piping. Starting in the 13th century, lead was used to create stained glass.^{[151]} In the European and Arabian traditions of alchemy, lead was considered an impure base metal which, by the separation, purification and balancing of its constituent essences, could be transformed to pure and incorruptible gold.^{[152]} During the period, lead was used increasingly for adulterating wine. The use of such wine was forbidden for use in Christian rites by a papal bull in 1498, but it continued to be imbibed and resulted in mass poisonings up to the late 18th century.^{[148]}^{[153]} Lead was a key material in parts of the printing press, which was invented around 1440; lead dust was commonly inhaled by print workers, causing lead poisoning.^{[154]} Firearms were invented at around the same time, and lead, despite being more expensive than iron, became the chief material for making bullets. It was less damaging to iron gun barrels, had a higher density (which allowed for better retention of velocity), and its lower melting point made the production of bullets easier as they could be made using a wood fire.^{[155]} Lead, in the form of Venetian ceruse, was extensively used in cosmetics by Western European aristocracy as whitened faces were regarded as a sign of modesty.^{[156]}^{[157]} This practice later expanded to white wigs and eyeliners, and only faded out with the French Revolution in the late 18th century. A similar fashion appeared in Japan in the 18th century with the emergence of the geishas, a practice that continued long into the 20th century. The white faces of women "came to represent their feminine virtue as Japanese women",^{[158]} with lead commonly used in the whitener.^{[159]}
In the New World, lead was produced soon after the arrival of European settlers. The earliest recorded lead production dates to 1621 in the English Colony of Virginia, fourteen years after its foundation.^{[160]} In Australia, the first mine opened by colonists on the continent was a lead mine, in 1841.^{[161]} In Africa, lead mining and smelting were known in the Benue Trough^{[162]} and the lower Congo Basin, where lead was used for trade with Europeans and as a currency by the 17th century,^{[163]} well before the scramble for Africa.
Figure 14 
Lead mining in the upper Mississippi River region in the United States in 1865
John Barber and Henry Howe, public domain
In the second half of the 18th century, Britain, and later continental Europe and the United States, experienced the Industrial Revolution. This was the first time during which production rates exceeded those of Rome.^{[116]} Britain was the leading producer, losing this status by the mid19th century with the depletion of its mines and the development of lead mining in Germany, Spain, and the United States.^{[164]} By 1900, the United States was the leader in global lead production, and other nonEuropean nations—Canada, Mexico, and Australia—had begun significant production; production outside Europe exceeded that within.^{[165]} A great share of the demand for lead came from plumbing and painting—lead paints were in regular use.^{[166]} At this time, more (working class) people were exposed to the metal and lead poisoning cases escalated. This led to research into the effects of lead intake. Lead was proven to be more dangerous in its fume form than as a solid metal. Lead poisoning and gout were linked. The effects of chronic ingestion of lead, including mental disorders, were also studied in the 19th century. The first laws aimed at decreasing lead poisoning in factories were enacted during the 1870s and 1880s in the United Kingdom.^{[166]}
Figure 15 
Promotional poster for Dutch Boy lead paint, United States, 1912
Dutch Boy paints, public domain
Further evidence of the threat that lead posed to humans was discovered in the late 19th and early 20th centuries. Mechanisms of harm were better understood, lead blindness was documented, and the element was phased out of public use in the United States and Europe. The United Kingdom introduced mandatory factory inspections in 1878 and appointed the first Medical Inspector of Factories in 1898; as a result, a 25fold decrease in lead poisoning incidents from 1900 to 1944 was reported.^{[167]} The last major human exposure to lead was the addition of tetraethyllead to gasoline as an antiknock agent, a practice that originated in the United States in 1921. It was phased out in the United States and the European Union by 2000.^{[166]} Most European countries banned lead paint—commonly used because of its opacity and water resistance^{[168]}—for interiors by 1930.^{[169]}
In the 1970s, the United States and Western European countries introduced legislation to reduce lead air pollution.^{[170]}^{[171]} The impact was significant: the share of people in the United States with elevated blood lead levels fell from 77.8% in 1976–1980 to 2.2% in 1991–1994.^{[172]} The main product made of lead by the end of the 20th century was the lead–acid battery,^{[173]} which posed no direct threat to humans. From 1960 to 1990, lead output in the Western Bloc grew by a third.^{[174]} The share of the world's lead production by the Eastern Bloc increased from 10% to 30%, from 1950 to 1990, with the Soviet Union being the world's largest producer during the mid1970s and the 1980s, and China starting major lead production in the late 20th century.^{[175]} Unlike the European communist countries, China was largely unindustrialized by the mid20th century; in 2004, China surpassed Australia as the largest producer of lead.^{[176]} As was the case during European industrialization, lead has had a negative effect on health in China.^{[177]}
Figure 16 
Primary production of lead since 1840
Borvan53, CCBYSA 4.0
Production of lead is increasing worldwide due to its use in lead–acid batteries.^{[178]} There are two major categories of production: primary from mined ores, and secondary from scrap. In 2014, 4.58 million metric tons came from primary production and 5.64 million from secondary production. The top three producers of mined lead concentrate in that year were China, Australia, and the United States. The top three producers of refined lead were China, the United States, and South Korea.^{[179]} According to the International Resource Panel's Metal Stocks in Society report of 2010, the total amount of lead in use, stockpiled, discarded, or dissipated into the environment, on a global basis, is 8 kg per capita. Much of this is in more developed countries (20–150 kg per capita) rather than less developed ones (1–4 kg per capita).^{[180]}
The primary and secondary lead production processes are similar. Some primary production plants now supplement their operations with scrap lead, and this trend is likely to increase in the future. Given adequate techniques, lead obtained via secondary processes is indistinguishable from lead obtained via primary processes. Scrap lead from the building trade is usually fairly clean and is remelted without the need for smelting, though refining is sometimes needed. Secondary lead production is therefore cheaper, in terms of energy requirements, than is primary production, often by 50% or more.^{[181]}
Most lead ores contain a low percentage of lead (rich ores have a typical content of 3–8%) which must be concentrated for extraction.^{[182]} During initial processing, ores typically undergo crushing, densemedium separation, grinding, froth flotation, and drying. The resulting concentrate, which has a lead content of 30–80% by mass (regularly 50–60%),^{[182]} is then turned into (impure) lead metal.
There are two main ways of doing this: a twostage process involving roasting followed by blast furnace extraction, carried out in separate vessels; or a direct process in which the extraction of the concentrate occurs in a single vessel. The latter has become the most common route, though the former is still significant.^{[183]}
Country  Output (thousand tons) 

China  2,400 
Australia  500 
United States  335 
Peru  310 
Mexico  250 
Russia  225 
India  135 
Bolivia  80 
Sweden  76 
Turkey  75 
Iran  41 
Kazakhstan  41 
Poland  40 
South Africa  40 
North Korea  35 
Ireland  33 
Macedonia  33 
Other countries  170 
First, the sulfide concentrate is roasted in air to oxidize the lead sulfide:^{[184]}
As the original concentrate was not pure lead sulfide, roasting yields not only the desired lead(II) oxide, but a mixture of oxides, sulfates and silicates of lead and other metals contained in the ore.^{[185]} This impure lead oxide is reduced in a cokefired blast furnace to the (again, impure) metal:^{[186]}
Impurities are mostly arsenic, antimony, bismuth, zinc, copper, silver, and gold. The melt is treated in a reverberatory furnace with air, steam, and sulfur, which oxidizes the impurities except for silver, gold, and bismuth. Oxidized contaminants float to the top of the melt and are skimmed off.^{[187]}^{[188]} Metallic silver and gold are removed and recovered economically by means of the Parkes process, in which zinc is added to lead. The zinc, which is immiscible in lead, dissolves in silver and gold. The zinc solution can be separated from the lead, and the silver and gold retrieved.^{[189]}^{[188]} Desilvered lead is freed of bismuth by the Betterton–Kroll process, treating it with metallic calcium and magnesium. The resulting bismuth dross can be skimmed off.^{[188]}
Very pure lead can be obtained by processing smelted lead electrolytically using the Betts process. Anodes of impure lead and cathodes of pure lead are placed in an electrolyte of lead fluorosilicate (PbSiF_{6}). Once electrical potential is applied, impure lead at the anode dissolves and plates onto the cathode, leaving the majority of the impurities in solution.^{[188]}^{[190]} This is a highcost process and thus mostly reserved for refining bullion containing high percentages of impurities.^{[191]}
In this process, lead bullion and slag is obtained directly from lead concentrates. The lead sulfide concentrate is melted in a furnace and oxidized, forming lead monoxide. Carbon (as coke or coal gas^{[o]}) is added to the molten charge along with fluxing agents. The lead monoxide is thereby reduced to metallic lead, in the midst of a slag rich in lead monoxide.^{[183]}
As much as 80% of the lead in very highcontent initial concentrates can be obtained as bullion; the remaining 20% forms a slag rich in lead monoxide. For a lowgrade feed, all of the lead can be oxidized to a highlead slag.^{[183]} Metallic lead is further obtained from the highlead (25–40%) slags via submerged fuel combustion or injection, reduction assisted by an electric furnace, or a combination of both.^{[183]}
Research on a cleaner, less energyintensive lead extraction process continues; a major drawback is that either too much lead is lost as waste, or the alternatives result in a high sulfur content in the resulting lead metal. Hydrometallurgical extraction, in which anodes of impure lead are immersed into an electrolyte and pure lead is deposited onto a cathode, is a technique that may have potential.^{[192]}
Smelting, which is an essential part of the primary production, is often skipped during secondary production. It is only performed when metallic lead had undergone significant oxidation.^{[181]} The process is similar to that of primary production in either a blast furnace or a rotary furnace; blast furnaces produce hard lead (10% antimony) while reverberatory and rotary kiln furnaces produced semisoft lead (3–4% antimony).^{[193]} The Isasmelt process is a more recent method that may act as an extension to primary production; battery paste from spent lead–acid batteries has sulfur removed by treating it with alkali, and is then treated in a coalfueled furnace in the presence of oxygen, which yields impure lead, with antimony the most common impurity.^{[194]} Refining of secondary lead is similar to that of primary lead. Some refining processes may be skipped depending on the material recycled and its potential contamination. ^{[195]}
Of the sources of lead for recycling, lead–acid batteries are the most important; lead pipe, sheet, and cable sheathing are also significant.^{[181]}
Contrary to popular belief, pencil leads in wooden pencils have never been made from lead. When the pencil originated as a wrapped graphite writing tool, the particular type of graphite used was named plumbago (literally, act for lead or lead mockup).^{[197]}
Lead metal has several useful mechanical properties, including high density, low melting point, ductility, and relative inertness. Many metals are superior to lead in some of these aspects but are generally less common and more difficult to extract from parent ores. Lead's toxicity has led to its phasing out for some uses.^{[198]}
Lead has been used for bullets since their invention in the Middle Ages. It is inexpensive; its low melting point means small arms ammunition and shotgun pellets can be cast with minimal technical equipment; and it is denser than other common metals, which allows for better retention of velocity. Concerns have been raised that lead bullets used for hunting can damage the environment.^{[p]}
Lead's high density and resistance to corrosion have been exploited in a number of related applications. It is used as ballast in sailboat keels;^{[200]} its density allows it to take up a small volume and minimize water resistance, thus counterbalancing the heeling effect of wind on the sails. It is used in scuba diving weight belts to counteract the diver's buoyancy.^{[201]} In 1993, the base of the Leaning Tower of Pisa was stabilized with 600 tonnes of lead.^{[202]} Because of its corrosion resistance, lead is used as a protective sheath for underwater cables.^{[203]}
Figure 18 
A 17thcentury goldcoated lead sculpture
Coyau, CCBYSA 3.0
Lead has many uses in the construction industry; lead sheets are used as architectural metals in roofing material, cladding, flashing, gutters and gutter joints, and on roof parapets.^{[204]}^{[205]} Detailed lead moldings are used as decorative motifs to fix lead sheet. Lead is still used in statues and sculptures,^{[q]} including for armatures.^{[207]} In the past it was often used to balance the wheels of cars; for environmental reasons this use is being phased out in favor of other materials.^{[109]}
Lead is added to copper alloys such as brass and bronze, to improve machinability and for its lubricating qualities. Being practically insoluble in copper, lead forms solid globules in imperfections throughout the alloy, such as grain boundaries. In low concentrations, as well as acting as a lubricant, the globules hinder the formation of swarf as the alloy is worked, thereby improving machinability. Copper alloys with larger concentrations of lead are used in bearings. The lead provides lubrication, and the copper provides the loadbearing support.^{[208]}
Lead's high density, atomic number, and formability form the basis for use of lead as a barrier that absorbs sound, vibration, and radiation.^{[209]} Lead has no natural resonance frequencies;^{[209]} as a result, sheetlead is used as a sound deadening layer in the walls, floors, and ceilings of sound studios.^{[210]} Organ pipes are often made from a lead alloy, mixed with various amounts of tin to control the tone of each pipe.^{[211]}^{[212]} Lead is an established shielding material from radiation in nuclear science and in Xray rooms^{[213]} due to its denseness and high attenuation coefficient.^{[214]} Molten lead has been used as a coolant for leadcooled fast reactors.^{[215]}
Figure 19 
Lead glass
Paolo Neo, public domain
The largest use of lead in the early 21st century is in lead–acid batteries. The reactions in the battery between lead, lead dioxide, and sulfuric acid provide a reliable source of voltage.^{[r]} The lead in batteries undergoes no direct contact with humans, so there are fewer toxicity concerns. Supercapacitors incorporating lead–acid batteries have been installed in kilowatt and megawatt scale applications in Australia, Japan, and the United States in frequency regulation, solar smoothing and shifting, wind smoothing, and other applications.^{[217]} These batteries have lower energy density and chargedischarge efficiency than lithiumion batteries, but are significantly cheaper.^{[218]}
Lead is used in high voltage power cables as sheathing material to prevent water diffusion into insulation; this use is decreasing as lead is being phased out.^{[219]} Its use in solder for electronics is also being phased out by some countries to reduce the amount of environmentally hazardous waste.^{[220]} Lead is one of three metals used in the Oddy test for museum materials, helping detect organic acids, aldehydes, and acidic gases.^{[221]}^{[222]}
Lead compounds are used as, or in, coloring agents, oxidants, plastic, candles, glass, and semiconductors. While lead paints are phased out in Europe and North America, they remain in use in less developed countries such as China^{[223]} or India.^{[224]} Leadbased coloring agents are used in ceramic glazes and glass, especially for red and yellow shades.^{[225]} Lead tetraacetate and lead dioxide are used as oxidizing agents in organic chemistry. Lead is frequently used in the polyvinyl chloride coating of electrical cords.^{[226]}^{[227]} It can be used to treat candle wicks to ensure a longer, more even burn. Because of its toxicity, European and North American manufacturers use alternatives such as zinc.^{[228]}^{[229]} Lead glass is composed of 12–28% lead oxide, changing its optical characteristics and reducing the transmission of ionizing radiation.^{[230]} Leadbased semiconductors such as lead telluride and lead selenide are used in photovoltaic cells and infrared detectors.^{[231]}
Lead has no confirmed biological role.^{[232]} Its prevalence in the human body—at an adult average of 120 mg^{[s]}—is nevertheless exceeded only by zinc (2500 mg) and iron (4000 mg) among the heavy metals.^{[234]} Lead salts are very efficiently absorbed by the body.^{[235]} A small amount of lead (1%) is stored in bones; the rest is excreted in urine and feces within a few weeks of exposure. Only about a third of lead is excreted by a child.^{[236]} Continual exposure may result in the bioaccumulation of lead.^{[236]}
Lead is a highly poisonous metal (whether inhaled or swallowed), affecting almost every organ and system in the human body.^{[237]} At airborne levels of 100 mg/m^{3}, it is immediately dangerous to life and health.^{[238]} Most ingested lead is absorbed into the bloodstream.^{[239]} The primary cause of its toxicity is its predilection for interfering with the proper functioning of enzymes. It does so by binding to the sulfhydryl groups found on many enzymes,^{[240]} or mimicking and displacing other metals which act as cofactors in many enzymatic reactions.^{[241]} Among the essential metals that lead interacts with are calcium, iron, and zinc.^{[242]} High levels of calcium and iron tend to provide some protection from lead poisoning; low levels cause increased susceptibility.^{[235]}
Lead can cause severe damage to the brain and kidneys and, ultimately, death. By mimicking calcium, lead can cross the bloodbrain barrier. It degrades the myelin sheaths of neurons, reduces their numbers, interferes with neurotransmission routes, and decreases neuronal growth.^{[240]} In the human body, lead inhibits porphobilinogen synthase and ferrochelatase, preventing both porphobilinogen formation and the incorporation of iron into protoporphyrin IX, the final step in heme synthesis. This causes ineffective heme synthesis and microcytic anemia.^{[243]}
Symptoms of lead poisoning include nephropathy, coliclike abdominal pains, and possibly weakness in the fingers, wrists, or ankles. Small blood pressure increases, particularly in middleaged and older people, may be apparent and can cause anemia. Several studies, mostly crosssectional, found an association between increased lead exposure and decreased heart rate variability.^{[244]} In pregnant women, high levels of exposure to lead may cause miscarriage. Chronic, highlevel exposure has been shown to reduce fertility in males.^{[245]}
In a child's developing brain, lead interferes with synapse formation in the cerebral cortex, neurochemical development (including that of neurotransmitters), and the organization of ion channels.^{[246]} Early childhood exposure has been linked with an increased risk of sleep disturbances and excessive daytime sleepiness in later childhood.^{[247]} High blood levels are associated with delayed puberty in girls.^{[248]} The rise and fall in exposure to airborne lead from the combustion of tetraethyl lead in gasoline during the 20th century has been linked with historical increases and decreases in crime levels, a hypothesis which is not universally accepted.^{[249]}
Lead exposure is a global issue since lead mining and smelting, and battery manufacturing/disposal/recycling, are common in many countries. Lead enters the body via inhalation, ingestion, or skin absorption. Almost all inhaled lead is absorbed into the body; for ingestion, the rate is 20–70%, with children absorbing a higher percentage than adults.^{[250]}
Figure 21 
Battery collection site in Dakar, Senegal, where at least 18 children died of lead poisoning in 2008
J Caravanos, CCBYSA 4.0
Poisoning typically results from ingestion of food or water contaminated with lead, and less commonly after accidental ingestion of contaminated soil, dust, or leadbased paint.^{[251]} Seawater products can contain lead if affected by nearby industrial waters.^{[252]} Fruit and vegetables can be contaminated by high levels of lead in the soils they were grown in. Soil can be contaminated through particulate accumulation from lead in pipes, lead paint, and residual emissions from leaded gasoline.^{[253]}
The use of lead for water pipes is problematic in areas with soft or acidic water.^{[254]} Hard water forms insoluble layers in the pipes whereas soft and acidic water dissolves the lead pipes.^{[255]} Dissolved carbon dioxide in the carried water may result in the formation of soluble lead bicarbonate; oxygenated water may similarly dissolve lead as lead(II) hydroxide.^{[256]} Drinking such water, over time, can cause health problems due to the toxicity of the dissolved lead. The harder the water the more calcium bicarbonate and sulfate it will contain, and the more the inside of the pipes will be coated with a protective layer of lead carbonate or lead sulfate.^{[257]}
Ingestion of applied leadbased paint is the major source of exposure for children: a direct source is chewing on old painted window sills. Alternatively, as the applied dry paint deteriorates, it peels, is pulverized into dust, and then enters the body through handtomouth contact or contaminated food, water, or alcohol. Ingesting certain home remedies may result in exposure to lead or its compounds.^{[258]}
Inhalation is the second major exposure pathway, affecting smokers and especially workers in leadrelated occupations.^{[239]} Cigarette smoke contains, among other toxic substances, radioactive lead210.^{[259]}
Skin exposure may be significant for people working with organic lead compounds. The rate of skin absorption is lower for inorganic lead.^{[260]}
Treatment for lead poisoning normally involves the administration of dimercaprol and succimer.^{[261]} Acute cases may require the use of disodium calcium edetate, the calcium chelate of the disodium salt of ethylenediaminetetraacetic acid (EDTA). It has a greater affinity for lead than calcium, with the result that lead chelate is formed by exchange and excreted in the urine, leaving behind harmless calcium.^{[262]}
The extraction, production, use, and disposal of lead and its products have caused significant contamination of the Earth's soils and waters. Atmospheric emissions of lead were at their peak during the Industrial Revolution, and the leaded gasoline period in the second half of the twentieth century. Lead releases originate from natural sources (i.e., concentration of the naturally occurring lead), industrial production, incineration and recycling, and mobilization of previously buried lead.^{[263]} Elevated concentrations of lead persist in soils and sediments in postindustrial and urban areas; industrial emissions, including those arising from coal burning,^{[264]} continue in many parts of the world, particularly in the developing countries.^{[265]}
Lead can accumulate in soils, especially those with a high organic content, where it remains for hundreds to thousands of years. Environmental lead can compete with other metals found in and on plants surfaces potentially inhibiting photosynthesis and at high enough concentrations, negatively affecting plant growth and survival. Contamination of soils and plants can allow lead to ascend the food chain affecting microorganisms and animals. In animals, lead exhibits toxicity in many organs, damaging the nervous, renal, reproductive, hematopoietic, and cardiovascular systems after ingestion, inhalation, or skin absorption.^{[266]} Fish uptake lead from both water and sediment;^{[267]} bioaccumulation in the food chain poses a hazard to fish, birds, and sea mammals.^{[268]}
Figure 23 
Radiography of a swan found dead in Condesurl'Escaut (northern France), highlighting lead shot. There are hundreds of lead pellets; a dozen is enough to kill an adult swan within a few days. Such bodies are sources of environmental contamination by lead.
Rafael Mateo, CCBYSA 3.0
Antropogenic lead includes lead from shot and sinkers. These are among the most potent sources of lead contamination along with lead production sites.^{[269]} Lead was banned for shot and sinkers in the United States in 2017,^{[270]} although that ban was only effective for a month,^{[271]} and a similar ban is being considered in the European Union.^{[272]}
Analytical methods for the determination of lead in the environment include spectrophotometry, Xray fluorescence, atomic spectroscopy and electrochemical methods. A specific ionselective electrode has been developed based on the ionophore S,S'methylenebis(N,Ndiisobutyldithiocarbamate).^{[273]} An important biomarker assay for lead poisoning is δaminolevulinic acid levels in plasma, serum, and urine.^{[274]}
By the mid1980s, there was significant decline in the use of lead in industry. In the United States, environmental regulations reduced or eliminated the use of lead in nonbattery products, including gasoline, paints, solders, and water systems. Particulate control devices were installed in coalfired power plants to capture lead emissions.^{[264]} Lead use was further curtailed by the European Union's 2003 Restriction of Hazardous Substances Directive.^{[275]} A large drop in lead deposition occurred in the Netherlands after the 1993 national ban on use of lead shot for hunting and sport shooting: from 230 tonnes in 1990 to 47.5 tonnes in 1995.^{[276]}
In the United States, the permissible exposure limit for lead in the workplace, comprising metallic lead, inorganic lead compounds, and lead soaps, was set at 50 μg/m^{3} over an 8hour workday, and the blood lead level limit at 5 μg per 100 g of blood in 2012.^{[277]} Lead may still be found in harmful quantities in stoneware,^{[278]} vinyl^{[279]} (such as that used for tubing and the insulation of electrical cords), and Chinese brass.^{[t]} Old houses may still contain lead paint.^{[279]} White lead paint has been withdrawn from sale in industrialized countries, but specialized uses of other pigments such as yellow lead chromate remain.^{[168]} Stripping old paint by sanding produces dust which can be inhaled.^{[281]} Lead abatement programs have been mandated by some authorities in properties where young children live.^{[282]}
Lead waste, depending of the jurisdiction and the nature of the waste, may be treated as household waste (in order to facilitate lead abatement activities),^{[283]} or potentially hazardous waste requiring specialized treatment or storage.^{[284]} Lead is released to the wildlife in shooting places and a number of lead management practices, such as stewardship of the environment and reduced public scrutiny, have been developed to counter the lead contamination.^{[285]} Lead migration can be enhanced in acidic soils; to counter that, it is advised soils be treated with lime to neutralize the soils and prevent leaching of lead.^{[286]}
Research has been conducted on how to remove lead from biosystems by biological means: Fish bones are being researched for their ability to bioremediate lead in contaminated soil.^{[287]}^{[288]} The fungus Aspergillus versicolor is effective at removing lead ions.^{[289]} Several bacteria have been researched for their ability to remove lead from the environment, including the sulfatereducing bacteria Desulfovibrio and Desulfotomaculum, both of which are highly effective in aqueous solutions.^{[290]}
I would like to thank the many Wikipedia editors who commented on, copyedited, contributed to, or otherwise helped to improve this article, including those who assisted at the Wikipedia featured article candidacy: Graeme Bartlett, Jimfbleak, Sandbh, Nergaal, DePiep, John, Cas Liber, Axl, Double sharp, Smurrayinchester, edwininlondon, and Parcly Taxel. I would also like to thank the reviewers who provided journal peer reviews: Marshall Sumter, Jaclyn Catalano, an anonymous reviewer, and Robert M. Gogal Jr.
Global demand for lead has more than doubled since the early 1990s and almost 90% of use is now in leadacid batteries
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WikiJSci Editorial Board (2018). "The aims and scope of WikiJournal of Science". WikiJournal of Science 1 (1): 1. doi:10.15347/wjs/2018.001. ISSN 24706345. https://en.wikiversity.org/wiki/WikiJournal_of_Science/The_aims_and_scope_of_WikiJournal_of_Science.
Wikipedia plays an increasingly important role in scientific research and teaching, by providing a universally accessible compilation of wellknown ideas and information. As a result, a typical Wikipedia article on a scientific subject has about as many viewers every day as a typical scientific article has readers over its lifetime.^{[1]} However, relatively few academics write in Wikipedia, and the coverage of many topics could greatly benefit from more attention by specialists.^{[2]}^{[3]}^{[4]} Indeed most articles are 12 paragraph stubs.^{[5]}
Researchers currently have little incentive to contribute to Wikipedia, largely because such contributions do not count as academic publications, the main drivers of academics' careers.^{[6]} Further, Wikipedia is not an ideal source to cite in scholarly publications, since it is an evolving, tertiary source written by large numbers of mostly pseudonymous or anonymous authors.^{[7]} A main aim of the WikiJournals is to address this issue, by allowing academics to write Wikipedia articles that also count as bona fide scholarly publications. This builds on experimentation in dualpublishing academic articles into Wikipedia by several journals, such as PLOS Computational biology,^{[8]} Gene,^{[9]} RNA Biology,^{[10]} and Open Medicine.^{[11]} Conversely, expertwritten online encyclopedias such as Scholarpedia^{[12]} and Citizendium^{[13]} have remained entirely separate from Wikipedia, and this has severely limited their reach and scope.
Figure 1  Outline of publication process for Wiki.J.Sci. Articles may be written from scratch or adapted from existing Wikipedia material. Submission via the preprint server is recommended. External, academic peer review is organised by the journal editors. If published, suitable material is integrated back into Wikipedia to improve the encyclopedia.
A primary feature of WikiJournals is their focus on dualpublication of review articles in the journal and in Wikipedia. In particular, WikiJournals publish a stable, citeable version of record as well as seeing a copy in Wikipedia that serves as a highly visible, living version (Figure 1).^{[1]} WikiJournals are formatted and published using the same MediaWiki software as Wikipedia. This enables suitable material to be integrated from WikiJournal articles into Wikipedia, including Lysine and ShK toxin.^{[14]}^{[15]} Similarly, Wikipedia articles can be submitted to WikiJournals for peer review, for example Radiocarbon dating and Spaces in mathematics.^{[16]}^{[17]} In both cases, these broad, reviewstyle articles need to conform to both scholarly best practices and Wikipedia's rules. WikiJournals can also publish original research articles that are not destined to be integrated into Wikipedia, such as A card game for Bell's theorem and its loopholes but can nevertheless benefit from the collaborative editing that MediaWiki enables.^{[18]}
As academic journals, WikiJournals strive to follow recognised best practices in scholarly publishing, especially features advocated by the open access movement.^{[19]}^{[20]}^{[21]} In particular, we want to encourage open and collaborative reviews,^{[22]} the use of preprints,^{[23]}^{[24]} and reviews based on scientific merit, rather than perceived novelty or importance.^{[25]} In order to achieve this,
The format was pioneered by WikiJournal of Medicine,^{[26]} which itself took inspiration from earlier experiences in Wikipediaintegrated publishing on medical and biomedicine topics.^{[1]}^{[11]} The model is now being extended by WikiJournal of Science to cover STEM topics more broadly.
Wiki.J.Sci. aims to achieve a broad scope across the STEM fields, and to become the main journal of its kind in these fields, including biology, physics, chemistry, technology, engineering, and mathematics. In addition to publishing on topics that were previously absent or underdeveloped in Wikipedia, it also allows the less common format of Wikipediafirst publication, treating Wikipedia as a preprint server.
In addition to review articles, Wiki.J.Sci. also publishes other formats that are not able to be integrated directly into Wikipedia, but still benefit from the libre open access model. Running the journal on Wikimedia software is well suited to collaborative, transparent, versioned, open writing. It also enables the journal to run on a very low budget, allowing articles to be processed without asking authors to pay fees. This contrasts with the "predatory journal" model that has proliferated to take advantage of the openaccess movement.^{[27]}
The unique features of the WikiJournal format have prompted us to codify them in a statement of publication ethics.^{[28]} This statement lays out principles for handling Wikipediaintegrated content, in addition to standard publication ethics. The editorial board encompasses a combination of academic research and publishing experience, as well as knowledge of Wikipedia's workings. We are therefore wellpositioned to help set best practices for future Wikipediaintegrated academic publishing. We will be happy to support other publishing groups in introducing their own Wikipediaintegrated article formats.
The versatility of the format also gives wide scope for development of new features. Depending on author demand, we may implement interactive figures,^{[29]} live data,^{[30]} repeated versioned reviews,^{[31]} or the option of doubleblind peer review.^{[32]} Indeed, Wiki.J.Med. has already experimented with interactive figures to a limited extent.^{[33]} We also intend to implement greater integration with the structured knowledge database WikiData. As demand increases, one major decision will be whether to split off specialised journals, or stay a large, unified journal with specialised subjournals. As part of the journal's MediaWiki heritage, these decisions will be made in open discussions with the editor, author, reviewer, and reader communities.
Our thanks to the editorial board of Wiki.J.Med. for their help and advice in setting up their sister journal.
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Tsirelson, B; et al. (2018). "Spaces in mathematics". WikiJournal of Science 1 (1): 2. doi:10.15347/wjs/2018.002. ISSN 24706345. https://en.wikiversity.org/wiki/WikiJournal_of_Science/Spaces_in_mathematics.
Wikipedia: This work is adapted from the Wikipedia article Space (mathematics). Content has also subsequently been used to update that same Wikipedia article (Page views).
License: This is an open access article distributed under the Creative Commons Attribution ShareAlike License, which permits unrestricted use, distribution, and reproduction, provided the original author and source are credited.
Figure 1 
Overview of types of abstract spaces. An arrow from space A to space B implies that space A is also a kind of space B. That means, for instance, that a normed vector space is also a metric space.
Stefan Eckert, CCBYSA 3.0
While modern mathematics use many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.^{[1]}^{[details 1]}
A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can be elements of a set, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a onetoone correspondence between their points that preserves the relationships. For example, the relationships between the points of a threedimensional Euclidean space are uniquely determined by Euclid's axioms,^{[details 2]} and all threedimensional Euclidean spaces are considered identical.
Topological notions such as continuity have natural definitions in every Euclidean space. However, topology does not distinguish straight lines from curved lines, and the relation between Euclidean and topological spaces is thus "forgetful". Relations of this kind are sketched in Figure 1, and treated in more detail in the Section "Types of spaces".
It is not always clear whether a given mathematical object should be considered as a geometric "space", or an algebraic "structure". A general definition of "structure", proposed by Bourbaki^{[2]}, embraces all common types of spaces, provides a general definition of isomorphism, and justifies the transfer of properties between isomorphic structures.Classic  Modern 

axioms are obvious implications of definitions  axioms are conventional 
theorems are absolute objective truth  theorems are implications of the corresponding axioms 
relationships between points, lines etc. are determined by their nature  relationships between points, lines etc. are essential; their nature is not 
mathematical objects are given to us with their structure  each mathematical theory describes its objects by some of their properties 
geometry corresponds to an experimental reality  geometry is a mathematical truth 
all geometric properties of the space follow from the axioms  axioms of a space need not determine all geometric properties 
geometry is an autonomous and living science  classical geometry is a universal language of mathematics 
space is threedimensional  different concepts of dimension apply to different kind of spaces 
space is the universe of geometry  spaces are just mathematical structures, they occur in various branches of mathematics 
In ancient Greek mathematics, "space" was a geometric abstraction of the threedimensional reality observed in everyday life. About 300 BC, Euclid gave axioms for the properties of space. Euclid built all of mathematics on these geometric foundations, going so far as to define numbers by comparing the lengths of line segments to the length of a chosen reference segment.
The method of coordinates (analytic geometry) was adopted by René Descartes in 1637.^{[3]} At that time, geometric theorems were treated as absolute objective truths knowable through intuition and reason, similar to objects of natural science;^{[4]}^{:11} and axioms were treated as obvious implications of definitions.^{[4]}^{:15}
Two equivalence relations between geometric figures were used: congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures; homotheties — into similar figures. For example, all circles are mutually similar, but ellipses are not similar to circles. A third equivalence relation, introduced by Gaspard Monge in 1795, occurs in projective geometry: not only ellipses, but also parabolas and hyperbolas, turn into circles under appropriate projective transformations; they all are projectively equivalent figures.
The relation between the two geometries, Euclidean and projective,^{[4]}^{:133} shows that mathematical objects are not given to us with their structure.^{[4]}^{:21} Rather, each mathematical theory describes its objects by some of their properties, precisely those that are put as axioms at the foundations of the theory.^{[4]}^{:20}
Distances and angles cannot appear in theorems of projective geometry, since these notions are neither mentioned in the axioms of projective geometry nor defined from the notions mentioned there. The question "what is the sum of the three angles of a triangle" is meaningful in Euclidean geometry but meaningless in projective geometry.
A different situation appeared in the 19th century: in some geometries the sum of the three angles of a triangle is welldefined but different from the classical value (180 degrees). NonEuclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829 and János Bolyai in 1832 (and Carl Gauss in 1816, unpublished)^{[4]}^{:133} stated that the sum depends on the triangle and is always less than 180 degrees. Eugenio Beltrami in 1868 and Felix Klein in 1871 obtained Euclidean "models" of the nonEuclidean hyperbolic geometry, and thereby completely justified this theory as a logical possibility.^{[4]}^{:24}^{[5]}
This discovery forced the abandonment of the pretensions to the absolute truth of Euclidean geometry. It showed that axioms are not "obvious", nor "implications of definitions". Rather, they are hypotheses. To what extent do they correspond to an experimental reality? This important physical problem no longer has anything to do with mathematics. Even if a "geometry" does not correspond to an experimental reality, its theorems remain no less "mathematical truths".^{[4]}^{:15}
A Euclidean model of a nonEuclidean geometry is a choice of some objects existing in Euclidean space and some relations between these objects that satisfy all axioms (and therefore, all theorems) of the nonEuclidean geometry. These Euclidean objects and relations "play" the nonEuclidean geometry like contemporary actors playing an ancient performance. Actors can imitate a situation that never occurred in reality. Relations between the actors on the stage imitate relations between the characters in the play. Likewise, the chosen relations between the chosen objects of the Euclidean model imitate the nonEuclidean relations. It shows that relations between objects are essential in mathematics, while the nature of the objects is not.
The word "geometry" (from Ancient Greek: geo "earth", metron "measurement") initially meant a practical way of processing lengths, regions and volumes in the space in which we live, but was then extended widely (as well as the notion of space in question here).
According to Bourbaki,^{[4]}^{:131} the period between 1795 (Géométrie descriptive of Monge) and 1872 (the "Erlangen programme" of Klein) can be called the golden age of geometry. The original space investigated by Euclid is now called threedimensional Euclidean space. Its axiomatization, started by Euclid 23 centuries ago, was reformed with Hilbert's axioms, Tarski's axioms and Birkhoff's axioms. These axiom systems describe the space via primitive notions (such as "point", "between", "congruent") constrained by a number of axioms.
Analytic geometry made great progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups.^{[4]}^{:134,5} Since that time, new theorems of classical geometry have been of more interest to amateurs than to professional mathematicians.^{[4]}^{:136} However, the heritage of classical geometry was not lost. According to Bourbaki,^{[4]}^{:138} "passed over in its role as an autonomous and living science, classical geometry is thus transfigured into a universal language of contemporary mathematics".
Simultaneously, numbers began to displace geometry as the foundation of mathematics. For instance, in Richard Dedekind's 1872 essay Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), he asserts that points on a line ought to have the properties of Dedekind cuts, and that therefore a line was the same thing as the set of real numbers. Dedekind is careful to note that this is an assumption that is incapable of being proven. In modern treatments, Dedekind's assertion is often taken to be the definition of a line, thereby reducing geometry to arithmetic. Threedimensional Euclidean space is defined to be an affine space whose associated vector space of differences of its elements is equipped with an inner product.^{[6]} A definition "from scratch", as in Euclid, is now not often used, since it does not reveal the relation of this space to other spaces. Also, a threedimensional projective space is now defined as the space of all onedimensional subspaces (that is, straight lines through the origin) of a fourdimensional vector space. This shift in foundations requires a new set of axioms, and if these axioms are adopted, the classical axioms of geometry become theorems.
A space now consists of selected mathematical objects (for instance, functions on another space, or subspaces of another space, or just elements of a set) treated as points, and selected relationships between these points. Therefore, spaces are just mathematical structures of convenience. One may expect that the structures called "spaces" are perceived more geometrically than other mathematical objects, but this is not always true.
According to the famous inaugural lecture given by Bernhard Riemann in 1854, every mathematical object parametrized by n real numbers may be treated as a point of the ndimensional space of all such objects.^{[4]}^{:140} Contemporary mathematicians follow this idea routinely and find it extremely suggestive to use the terminology of classical geometry nearly everywhere.^{[4]}^{:138}
Functions are important mathematical objects. Usually they form infinitedimensional function spaces, as noted already by Riemann^{[4]}^{:141} and elaborated in the 20th century by functional analysis.
While each type of spaces has its own definition, the general idea of "space" evades formalization. Some structures are called spaces, other are not, without a formal criterion. Moreover, there is no consensus on the general idea of "structure". According to Pudlák,^{[7]} "Mathematics [...] cannot be explained completely by a single concept such as the mathematical structure. Nevertheless, Bourbaki's structuralist approach is the best that we have." We will return to Bourbaki's structuralist approach in the last section "Spaces and structures", while we now outline a possible classification of spaces (and structures) in the spirit of Bourbaki.
We classify spaces on three levels. Given that each mathematical theory describes its objects by some of their properties, the first question to ask is: which properties? This leads to the first (upper) classification level. On the second level, one takes into account answers to especially important questions (among the questions that make sense according to the first level). On the third level of classification, one takes into account answers to all possible questions.
For example, the upperlevel classification distinguishes between Euclidean and projective spaces, since the distance between two points is defined in Euclidean spaces but undefined in projective spaces.
Another example. The question "what is the sum of the three angles of a triangle" makes sense in a Euclidean space but not in a projective space. In a nonEuclidean space the question makes sense but is answered differently, which is not an upperlevel distinction.
Also, the distinction between a Euclidean plane and a Euclidean 3dimensional space is not an upperlevel distinction; the question "what is the dimension" makes sense in both cases.
The secondlevel classification distinguishes, for example, between Euclidean and nonEuclidean spaces; between finitedimensional and infinitedimensional spaces; between compact and noncompact spaces, etc.
In Bourbaki's terms,^{[2]} the secondlevel classification is the classification by "species". Unlike biological taxonomy, a space may belong to several species.
The thirdlevel classification distinguishes, for example, between spaces of different dimension, but does not distinguish between a plane of a threedimensional Euclidean space, treated as a twodimensional Euclidean space, and the set of all pairs of real numbers, also treated as a twodimensional Euclidean space. Likewise it does not distinguish between different Euclidean models of the same nonEuclidean space.
More formally, the third level classifies spaces up to isomorphism. An isomorphism between two spaces is defined as a onetoone correspondence between the points of the first space and the points of the second space, that preserves all relations stipulated according to the first level. Mutually isomorphic spaces are thought of as copies of a single space. If one of them belongs to a given species then they all do.
The notion of isomorphism sheds light on the upperlevel classification. Given a onetoone correspondence between two spaces of the same upperlevel class, one may ask whether it is an isomorphism or not. This question makes no sense for two spaces of different classes.
An isomorphism to itself is called an automorphism. Automorphisms of a Euclidean space are shifts, rotations, reflections and compositions of these. Euclidean space is homogeneous in the sense that every point can be transformed into every other point by some automorphism.
Euclidean axioms^{[details 2]} leave no freedom; they determine uniquely all geometric properties of the space. More exactly: all threedimensional Euclidean spaces are mutually isomorphic. In this sense we have "the" threedimensional Euclidean space. In Bourbaki's terms, the corresponding theory is univalent. In contrast, topological spaces are generally nonisomorphic; their theory is multivalent. A similar idea occurs in mathematical logic: a theory is called categorical if all its models of the same cardinality are mutually isomorphic. According to Bourbaki,^{[8]} the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.
Topological notions (continuity, convergence, open sets, closed sets etc.) are defined naturally in every Euclidean space. In other words, every Euclidean space is also a topological space. Every isomorphism between two Euclidean spaces is also an isomorphism between the corresponding topological spaces (called "homeomorphism"), but the converse is wrong: a homeomorphism may distort distances. In Bourbaki's terms,^{[2]} "topological space" is an underlying structure of the "Euclidean space" structure. Similar ideas occur in category theory: the category of Euclidean spaces is a concrete category over the category of topological spaces; the forgetful (or "stripping") functor maps the former category to the latter category.
A threedimensional Euclidean space is a special case of a Euclidean space. In Bourbaki's terms,^{[2]} the species of threedimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space.
Such relations between species of spaces may be expressed diagrammatically as shown in Fig. 3. An arrow from A to B means that every Aspace is also a Bspace, or may be treated as a Bspace, or provides a Bspace, etc. Treating A and B as classes of spaces one may interpret the arrow as a transition from A to B. (In Bourbaki's terms,^{[9]} "procedure of deduction" of a Bspace from a Aspace. Not quite a function unless the classes A,B are sets; this nuance does not invalidate the following.) The two arrows on Fig. 3 are not invertible, but for different reasons.
The transition from "Euclidean" to "topological" is forgetful. Topology distinguishes continuous from discontinuous, but does not distinguish rectilinear from curvilinear. Intuition tells us that the Euclidean structure cannot be restored from the topology. A proof uses an automorphism of the topological space (that is, selfhomeomorphism) that is not an automorphism of the Euclidean space (that is, not a composition of shifts, rotations and reflections). Such transformation turns the given Euclidean structure into a (isomorphic but) different Euclidean structure; both Euclidean structures correspond to a single topological structure.
In contrast, the transition from "3dim Euclidean" to "Euclidean" is not forgetful; a Euclidean space need not be 3dimensional, but if it happens to be 3dimensional, it is fullfledged, no structure is lost. In other words, the latter transition is injective (onetoone), while the former transition is not injective (manytoone). We denote injective transitions by an arrow with a barbed tail, "↣" rather than "→".
Both transitions are not surjective, that is, not every Bspace results from some Aspace. First, a 3dim Euclidean space is a special (not general) case of a Euclidean space. Second, a topology of a Euclidean space is a special case of topology (for instance, it must be noncompact, and connected, etc). We denote surjective transitions by a twoheaded arrow, "↠" rather than "→". See for example Fig. 4; there, the arrow from "real linear topological" to "real linear" is twoheaded, since every real linear space admits some (at least one) topology compatible with its linear structure.
Such topology is nonunique in general, but unique when the real linear space is finitedimensional. For these spaces the transition is both injective and surjective, that is, bijective; see the arrow from "finitedim real linear topological" to "finitedim real linear" on Fig. 4. The inverse transition exists (and could be shown by a second, backward arrow). The two species of structures are thus equivalent. In practice, one makes no distinction between equivalent species of structures.^{[10]} Equivalent structures may be treated as a single structure, as shown by a large box on Fig. 4.
The transitions denoted by the arrows obey isomorphisms. That is, two isomorphic Aspaces lead to two isomorphic Bspaces.
The diagram on Fig. 4 is commutative. That is, all directed paths in the diagram with the same start and endpoints lead to the same result. Other diagrams below are also commutative, except for dashed arrows on Fig. 9. The arrow from "topological" to "measurable" is dashed for the reason explained there: "In order to turn a topological space into a measurable space one endows it with a σalgebra. The σalgebra of Borel sets is the most popular, but not the only choice." A solid arrow denotes a prevalent, socalled "canonical" transition that suggests itself naturally and is widely used, often implicitly, by default. For example, speaking about a continuous function on a Euclidean space, one need not specify its topology explicitly. In fact, alternative topologies exist and are used sometimes, for example, the fine topology; but these are always specified explicitly, since they are much less notable that the prevalent topology. A dashed arrow indicates that several transitions are in use and no one is quite prevalent.
Two basic spaces are linear spaces (also called vector spaces) and topological spaces.
Linear spaces are of algebraic nature; there are real linear spaces (over the field of real numbers), complex linear spaces (over the field of complex numbers), and more generally, linear spaces over any field. Every complex linear space is also a real linear space (the latter underlies the former), since each real number is also a complex number.^{[details 3]} More generally, a vector space over a field also has the structure of a vector space over a subfield of that field. Linear operations, given in a linear space by definition, lead to such notions as straight lines (and planes, and other linear subspaces); parallel lines; ellipses (and ellipsoids). However, it is impossible to define orthogonal (perpendicular) lines, or to single out circles among ellipses, because in a linear space there is no structure like a scalar product that could be used for measuring angles. The dimension of a linear space is defined as the maximal number of linearly independent vectors or, equivalently, as the minimal number of vectors that span the space; it may be finite or infinite. Two linear spaces over the same field are isomorphic if and only if they are of the same dimension. A ndimensional complex linear space is also a 2ndimensional real linear space.
Topological spaces are of analytic nature. Open sets, given in a topological space by definition, lead to such notions as continuous functions, paths, maps; convergent sequences, limits; interior, boundary, exterior. However, uniform continuity, bounded sets, Cauchy sequences, differentiable functions (paths, maps) remain undefined. Isomorphisms between topological spaces are traditionally called homeomorphisms; these are onetoone correspondences continuous in both directions. The open interval (0,1) is homeomorphic to the whole real line (∞,∞) but not homeomorphic to the closed interval [0,1], nor to a circle. The surface of a cube is homeomorphic to a sphere (the surface of a ball) but not homeomorphic to a torus. Euclidean spaces of different dimensions are not homeomorphic, which seems evident, but is not easy to prove. The dimension of a topological space is difficult to define; inductive dimension (based on the observation that the dimension of the boundary of a geometric figure is usually one less than the dimension of the figure itself) and Lebesgue covering dimension can be used. In the case of a ndimensional Euclidean space, both topological dimensions are equal to n.
Every subset of a topological space is itself a topological space (in contrast, only linear subsets of a linear space are linear spaces). Arbitrary topological spaces, investigated by general topology (called also pointset topology) are too diverse for a complete classification up to homeomorphism. Compact topological spaces are an important class of topological spaces ("species" of this "type"). Every continuous function is bounded on such space. The closed interval [0,1] and the extended real line [∞,∞] are compact; the open interval (0,1) and the line (∞,∞) are not. Geometric topology investigates manifolds (another "species" of this "type"); these are topological spaces locally homeomorphic to Euclidean spaces (and satisfying a few extra conditions). Lowdimensional manifolds are completely classified up to homeomorphism.
Both the linear and topological structures underly the linear topological space (in other words, topological vector space) structure. A linear topological space is both a real or complex linear space and a topological space, such that the linear operations are continuous. So a linear space that is also topological is not in general a linear topological space.
Every finitedimensional real or complex linear space is a linear topological space in the sense that it carries one and only one topology that makes it a linear topological space. The two structures, "finitedimensional real or complex linear space" and "finitedimensional linear topological space", are thus equivalent, that is, mutually underlying. Accordingly, every invertible linear transformation of a finitedimensional linear topological space is a homeomorphism. The three notions of dimension (one algebraic and two topological) agree for finitedimensional real linear spaces. In infinitedimensional spaces, however, different topologies can conform to a given linear structure, and invertible linear transformations are generally not homeomorphisms.
It is convenient to introduce affine and projective spaces by means of linear spaces, as follows. A ndimensional linear subspace of a (n+1)dimensional linear space, being itself a ndimensional linear space, is not homogeneous; it contains a special point, the origin. Shifting it by a vector external to it, one obtains a ndimensional affine subspace. It is homogeneous. An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All ndimensional affine spaces are mutually isomorphic. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". In particular, every linear space is also an affine space.
Given an ndimensional affine subspace A in a (n+1)dimensional linear space L, a straight line in A may be defined as the intersection of A with a twodimensional linear subspace of L that intersects A: in other words, with a plane through the origin that is not parallel to A. More generally, a kdimensional affine subspace of A is the intersection of A with a (k+1)dimensional linear subspace of L that intersects A.
Every point of the affine subspace A is the intersection of A with a onedimensional linear subspace of L. However, some onedimensional subspaces of L are parallel to A; in some sense, they intersect A at infinity. The set of all onedimensional linear subspaces of a (n+1)dimensional linear space is, by definition, a ndimensional projective space. And the affine subspace A is embedded into the projective space as a proper subset. However, the projective space itself is homogeneous. A straight line in the projective space corresponds to a twodimensional linear subspace of the (n+1)dimensional linear space. More generally, a kdimensional projective subspace of the projective space corresponds to a (k+1)dimensional linear subspace of the (n+1)dimensional linear space, and is isomorphic to the kdimensional projective space.
Defined this way, affine and projective spaces are of algebraic nature; they can be real, complex, and more generally, over any field.
Every real or complex affine or projective space is also a topological space. An affine space is a noncompact manifold; a projective space is a compact manifold. In a real projective space a straight line is homeomorphic to a circle, therefore compact, in contrast to a straight line in a linear of affine space.
Distances between points are defined in a metric space. Isomorphisms between metric spaces are called isometries. Every metric space is also a topological space. A topological space is called metrizable, if it underlies a metric space. All manifolds are metrizable.
In a metric space, we can define bounded sets and Cauchy sequences. A metric space is called complete if all Cauchy sequences converge. Every incomplete space is isometrically embedded, as a dense subset, into a complete space (the completion). Every compact metric space is complete; the real line is noncompact but complete; the open interval (0,1) is incomplete.
Every Euclidean space is also a complete metric space. Moreover, all geometric notions immanent to a Euclidean space can be characterized in terms of its metric. For example, the straight segment connecting two given points A and C consists of all points B such that the distance between A and C is equal to the sum of two distances, between A and B and between B and C.
The Hausdorff dimension (related to the number of small balls that cover the given set) applies to metric spaces, and can be noninteger (especially for fractals). For a ndimensional Euclidean space, the Hausdorff dimension is equal to n.
Uniform spaces do not introduce distances, but still allow one to use uniform continuity, Cauchy sequences (or filters or nets), completeness and completion. Every uniform space is also a topological space. Every linear topological space (metrizable or not) is also a uniform space, and is complete in finite dimension but generally incomplete in infinite dimension. More generally, every commutative topological group is also a uniform space. A noncommutative topological group, however, carries two uniform structures, one leftinvariant, the other rightinvariant.
Vectors in a Euclidean space form a linear space, but each vector has also a length, in other words, norm, . A real or complex linear space endowed with a norm is a normed space. Every normed space is both a linear topological space and a metric space. A Banach space is a complete normed space. Many spaces of sequences or functions are infinitedimensional Banach spaces.
The set of all vectors of norm less than one is called the unit ball of a normed space. It is a convex, centrally symmetric set, generally not an ellipsoid; for example, it may be a polygon (in the plane) or. more generally, a polytope (in arbitrary finite dimension). The parallelogram law (called also parallelogram identity)
generally fails in normed spaces, but holds for vectors in Euclidean spaces, which follows from the fact that the squared Euclidean norm of a vector is its inner product with itself, .
An inner product space is a real or complex linear space, endowed with a bilinear or respectively sesquilinear form, satisfying some conditions and called an inner product. Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid. Angles between vectors are defined in inner product spaces. A Hilbert space is defined as a complete inner product space. (Some authors insist that it must be complex, others admit also real Hilbert spaces.) Many spaces of sequences or functions are infinitedimensional Hilbert spaces. Hilbert spaces are very important for quantum theory.^{[11]}
All ndimensional real inner product spaces are mutually isomorphic. One may say that the ndimensional Euclidean space is the ndimensional real inner product space that forgot its origin.
Smooth manifolds are not called "spaces", but could be. Every smooth manifold is a topological manifold, and can be embedded into a finitedimensional linear space. Smooth surfaces in a finitedimensional linear space are smooth manifolds: for example, the surface of an ellipsoid is a smooth manifold, a polytope is not. Real or complex finitedimensional linear, affine and projective spaces are also smooth manifolds.
At each one of its points, a smooth path in a smooth manifold has a tangent vector that belongs to the manifold's tangent space at this point. Tangent spaces to an ndimensional smooth manifold are ndimensional linear spaces. The differential of a smooth function on a smooth manifold provides a linear functional on the tangent space at each point.
A Riemannian manifold, or Riemann space, is a smooth manifold whose tangent spaces are endowed with inner products satisfying some conditions. Euclidean spaces are also Riemann spaces. Smooth surfaces in Euclidean spaces are Riemann spaces. A hyperbolic nonEuclidean space is also a Riemann space. A curve in a Riemann space has a length, and the length of the shortest curve between two points defines a distance, such that the Riemann space is a metric space. The angle between two curves intersecting at a point is the angle between their tangent lines.
Waiving positivity of inner products on tangent spaces, one obtains pseudoRiemann spaces, including the Lorentzian spaces that are very important for general relativity.
Waiving distances and angles while retaining volumes (of geometric bodies) one reaches measure theory. Besides the volume, a measure generalizes the notions of area, length, mass (or charge) distribution, and also probability distribution, according to Andrey Kolmogorov's approach to probability theory.
A "geometric body" of classical mathematics is much more regular than just a set of points. The boundary of the body is of zero volume. Thus, the volume of the body is the volume of its interior, and the interior can be exhausted by an infinite sequence of cubes. In contrast, the boundary of an arbitrary set of points can be of nonzero volume (an example: the set of all rational points inside a given cube). Measure theory succeeded in extending the notion of volume to a vast class of sets, the socalled measurable sets. Indeed, nonmeasurable sets almost never occur in applications.
Measurable sets, given in a measurable space by definition, lead to measurable functions and maps. In order to turn a topological space into a measurable space one endows it with a σalgebra. The σalgebra of Borel sets is the most popular, but not the only choice. (Baire sets, universally measurable sets, etc, are also used sometimes.) The topology is not uniquely determined by the Borel σalgebra; for example, the norm topology and the weak topology on a separable Hilbert space lead to the same Borel σalgebra. Not every σalgebra is the Borel σalgebra of some topology.^{[details 4]} Actually, a σalgebra can be generated by a given collection of sets (or functions) irrespective of any topology. Every subset of a measurable space is itself a measurable space.
Standard measurable spaces (also called standard Borel spaces) are especially useful due to some similarity to compact spaces (see EoM). Every bijective measurable mapping between standard measurable spaces is an isomorphism; that is, the inverse mapping is also measurable. And a mapping between such spaces is measurable if and only if its graph is measurable in the product space. Similarly, every bijective continuous mapping between compact metric spaces is a homeomorphism; that is, the inverse mapping is also continuous. And a mapping between such spaces is continuous if and only if its graph is closed in the product space.
Every Borel set in a Euclidean space (and more generally, in a complete separable metric space), endowed with the Borel σalgebra, is a standard measurable space. All uncountable standard measurable spaces are mutually isomorphic.
A measure space is a measurable space endowed with a measure. A Euclidean space with the Lebesgue measure is a measure space. Integration theory defines integrability and integrals of measurable functions on a measure space.
Sets of measure 0, called null sets, are negligible. Accordingly, a "mod 0 isomorphism" is defined as isomorphism between subsets of full measure (that is, with negligible complement).
A probability space is a measure space such that the measure of the whole space is equal to 1. The product of any family (finite or not) of probability spaces is a probability space. In contrast, for measure spaces in general, only the product of finitely many spaces is defined. Accordingly, there are many infinitedimensional probability measures (especially, Gaussian measures), but no infinitedimensional Lebesgue measures.
Standard probability spaces are especially useful. On a standard probability space a conditional expectation may be treated as the integral over the conditional measure (regular conditional probabilities, see also disintegration of measure). Given two standard probability spaces, every homomorphism of their measure algebras is induced by some measure preserving map. Every probability measure on a standard measurable space leads to a standard probability space. The product of a sequence (finite or not) of standard probability spaces is a standard probability space. All nonatomic standard probability spaces are mutually isomorphic mod 0; one of them is the interval (0,1) with the Lebesgue measure.
These spaces are less geometric. In particular, the idea of dimension, applicable (in one form or another) to all other spaces, does not apply to measurable, measure and probability spaces.
The theoretical study of calculus, known as mathematical analysis, led in the early 20th century to the consideration of linear spaces of realvalued or complexvalued functions. The earliest examples of these were function spaces, each one adapted to its own class of problems. These examples shared many common features, and these features were soon abstracted into Hilbert spaces, Banach spaces, and more general topological vector spaces. These were a powerful toolkit for the solution of a wide range of mathematical problems.
The most detailed information was carried by a class of spaces called Banach algebras. These are Banach spaces together with a continuous multiplication operation. An important early example was the Banach algebra of essentially bounded measurable functions on a measure space X. This set of functions is a Banach space under pointwise addition and scalar multiplication. With the operation of pointwise multiplication, it becomes a special type of Banach space, one now called a commutative von Neumann algebra. Pointwise multiplication determines a representation of this algebra on the Hilbert space of square integrable functions on X. An early observation of John von Neumann was that this correspondence also worked in reverse: Given some mild technical hypotheses, a commutative von Neumann algebra together with a representation on a Hilbert space determines a measure space, and these two constructions (of a von Neumann algebra plus a representation and of a measure space) are mutually inverse.
Von Neumann then proposed that noncommutative von Neumann algebras should have geometric meaning, just as commutative von Neumann algebras do. Together with Francis Murray, he produced a classification of von Neumann algebras. The direct integral construction shows how to break any von Neumann algebra into a collection of simpler algebras called factors. Von Neumann and Murray classified factors into three types. Type I was nearly identical to the commutative case. Types II and III exhibited new phenomena. A type II von Neumann algebra determined a geometry with the peculiar feature that the dimension could be any nonnegative real number, not just an integer. Type III algebras were those that were neither types I nor II, and after several decades of effort, these were proven to be closely related to type II factors.
A slightly different approach to the geometry of function spaces developed at the same time as von Neumann and Murray's work on the classification of factors. This approach is the theory of C*algebras. Here, the motivating example is the C*algebra , where X is a locally compact Hausdorff topological space. By definition, this is the algebra of continuous complexvalued functions on X that vanish at infinity (which loosely means that the farther you go from a chosen point, the closer the function gets to zero) with the operations of pointwise addition and multiplication. The Gelfand–Naimark theorem implied that there is a correspondence between commutative C*algebras and geometric objects: Every commutative C*algebra is of the form for some locally compact Hausdorff space X. Consequently it is possible to study locally compact Hausdorff spaces purely in terms of commutative C*algebras. Noncommutative geometry takes this as inspiration for the study of noncommutative C*algebras: If there were such a thing as a "noncommutative space X," then its would be a noncommutative C*algebra; if in addition the Gelfand–Naimark theorem applied to these nonexistent objects, then spaces (commutative or not) would be the same as C*algebras; so, for lack of a direct approach to the definition of a noncommutative space, a noncommutative space is defined to be a noncommutative C*algebra. Many standard geometric tools can be restated in terms of C*algebras, and this gives geometricallyinspired techniques for studying noncommutative C*algebras.
Both of these examples are now cases of a field called noncommutative geometry. The specific examples of von Neumann algebras and C*algebras are known as noncommutative measure theory and noncommutative topology, respectively. Noncommutative geometry is not merely a pursuit of generality for its own sake and is not just a curiosity. Noncommutative spaces arise naturally, even inevitably, from some constructions. For example, consider the nonperiodic Penrose tilings of the plane by kites and darts. It is a theorem that, in such a tiling, every finite patch of kites and darts appears infinitely often. As a consequence, there is no way to distinguish two Penrose tilings by looking at a finite portion. This makes it impossible to assign the set of all tilings a topology in the traditional sense. Despite this, the Penrose tilings determine a noncommutative C*algebra, and consequently they can be studied by the techniques of noncommutative geometry. Another example, and one of great interest within differential geometry, comes from foliations of manifolds. These are ways of splitting the manifold up into smallerdimensional submanifolds called leaves, each of which is locally parallel to others nearby. The set of all leaves can be made into a topological space. However, the example of an irrational rotation shows that this topological space can be inacessible to the techniques of classical measure theory. However, there is a noncommutative von Neumann algebra associated to the leaf space of a foliation, and once again, this gives an otherwise unintelligible space a good geometric structure.
Algebraic geometry studies the geometric properties of polynomial equations. Polynomials are a type of function defined from the basic arithmetic operations of addition and multiplication. Because of this, they are closely tied to algebra. Algebraic geometry offers a way to apply geometric techniques to questions of pure algebra, and vice versa.
Prior to the 1940s, algebraic geometry worked exclusively over the complex numbers, and the most fundamental variety was projective space. The geometry of projective space is closely related to the theory of perspective, and its algebra is described by homogeneous polynomials. All other varieties were defined as subsets of projective space. Projective varieties were subsets defined by a set of homogeneous polynomials. At each point of the projective variety, all the polynomials in the set were required to equal zero. The complement of the zero set of a linear polynomial is an affine space, and an affine variety was the intersection of a projective variety with an affine space.
André Weil saw that geometric reasoning could sometimes be applied in numbertheoretic situations where the spaces in question might be discrete or even finite. In pursuit of this idea, Weil rewrote the foundations of algebraic geometry, both freeing algebraic geometry from its reliance on complex numbers and introducing abstract algebraic varieties which were not embedded in projective space. These are now simply called varieties.
The type of space that underlies most modern algebraic geometry is even more general than Weil's abstract algebraic varieties. It was introduced by Alexander Grothendieck and is called a scheme. One of the motivations for scheme theory is that polynomials are unusually structured among functions, and algebraic varieties are consequently rigid. This presents problems when attempting to study degenerate situations. For example, almost any pair of points on a circle determines a unique line called the secant line, and as the two points move around the circle, the secant line varies continuously. However, when the two points collide, the secant line degenerates to a tangent line. The tangent line is unique, but the geometry of this configuration—a single point on a circle—is not expressive enough to determine a unique line. Studying situations like this requires a theory capable of assigning extra data to degenerate situations.
One of the building blocks of a scheme is a topological space. Topological spaces have continuous functions, but continuous functions are too general to reflect the underlying algebraic structure of interest. The other ingredient in a scheme, therefore, is a sheaf on the topological space, called the "structure sheaf". On each open subset of the topological space, the sheaf specifies a collection of functions, called "regular functions". The topological space and the structure sheaf together are required to satisfy conditions that mean the functions come from algebraic operations.
Like manifolds, schemes are defined as spaces that are locally modeled on a familiar space. In the case of manifolds, the familiar space is Euclidean space. For a scheme, the local models are called affine schemes. Affine schemes provide a direct link between algebraic geometry and commutative algebra. The fundamental objects of study in commutative algebra are commutative rings. If is a commutative ring, then there is a corresponding affine scheme which translates the algebraic structure of into geometry. Conversely, every affine scheme determines a commutative ring, namely, the ring of global sections of its structure sheaf. These two operations are mutually inverse, so affine schemes provide a new language with which to study questions in commutative algebra. By definition, every point in a scheme has an open neighborhood which is an affine scheme.
There are many schemes that are not affine. In particular, projective spaces satisfy a condition called properness which is analogous to compactness. Affine schemes cannot be proper (except in trivial situations like when the scheme has only a single point), and hence no projective space is an affine scheme (except for zerodimensional projective spaces). Projective schemes, meaning those that arise as closed subschemes of a projective space, are the single most important family of schemes.^{[12]}
Several generalizations of schemes have been introduced. Michael Artin defined an algebraic space as the quotient of a scheme by the equivalence relations that define étale morphisms. Algebraic spaces retain many of the useful properties of schemes while simultaneously being more flexible. For instance, the Keel–Mori theorem can be used to show that many moduli spaces are algebraic spaces.
More general than an algebraic space is a Deligne–Mumford stack. DM stacks are similar to schemes, but they permit singularities that cannot be described solely in terms of polynomials. They play the same role for schemes that orbifolds do for manifolds. For example, the quotient of the affine plane by a finite group of rotations around the origin yields a Deligne–Mumford stack that is not a scheme or an algebraic space. Away from the origin, the quotient by the group action identifies finite sets of equally spaced points on a circle. But at the origin, the circle consists of only a single point, the origin itself, and the group action fixes this point. In the quotient DM stack, however, this point comes with the extra data of being a quotient. This kind of refined structure is useful in the theory of moduli spaces, and in fact, it was originally introduced to describe moduli of algebraic curves.
A further generalization are the algebraic stacks, also called Artin stacks. DM stacks are limited to quotients by finite group actions. While this suffices for many problems in moduli theory, it is too restrictive for others, and Artin stacks permit more general quotients.
In Grothendieck's work on the Weil conjectures, he introduced a new type of topology now called a Grothendieck topology. A topological space (in the ordinary sense) axiomatizes the notion of "nearness," making two points be nearby if and only if they lie in many of the same open sets. By contrast, a Grothendieck topology axiomatizes the notion of "covering". A covering of a space is a collection of subspaces that jointly contain all the information of the ambient space. Since sheaves are defined in terms of coverings, a Grothendieck topology can also be seen as an axiomatization of the theory of sheaves.
Grothendieck's work on his topologies led him to the theory of topoi. In his memoir Récoltes et Semailles, he called them his "most vast conception".^{[13]} A sheaf (either on a topological space or with respect to a Grothendieck topology) is used to express local data. The category of all sheaves carries all possible ways of expressing local data. Since topological spaces are constructed from points, which are themselves a kind of local data, the category of sheaves can therefore be used as a replacement for the original space. Grothendieck consequently defined a topos to be a category of sheaves and studied topoi as objects of interest in their own right. These are now called Grothendieck topoi.
Every topological space determines a topos, and vice versa. There are topological spaces where taking the associated topos loses information, but these are generally considered pathological. (A necessary and sufficient condition is that the topological space be a sober space.) Conversely, there are topoi whose associated topological spaces do not capture the original topos. But, far from being pathological, these topoi can be of great mathematical interest. For instance, Grothendieck's theory of étale cohomology (which eventually led to the proof of the Weil conjectures) can be phrased as cohomology in the étale topos of a scheme, and this topos does not come from a topological space.
Topological spaces in fact lead to very special topoi called locales. The set of open subsets of a topological space determines a lattice. The axioms for a topological space cause these lattices to be complete Heyting algebras. The theory of locales takes this as its starting point. A locale is defined to be a complete Heyting algebra, and the elementary properties of topological spaces are reexpressed and reproved in these terms. The concept of a locale turns out to be more general than a topological space, in that every sober topological space determines a unique locale, but many interesting locales do not come from topological spaces. Because locales need not have points, the study of locales is somewhat jokingly called pointless topology.
Topoi also display deep connections to mathematical logic. Every Grothendieck topos has a special sheaf called a subobject classifier. This subobject classifier functions like the set of all possible truth values. In the topos of sets, the subobject classifier is the set , corresponding to "False" and "True". But in other topoi, the subobject classifier can be much more complicated. Lawvere and Tierney recognized that axiomatizing the subobject classifier yielded a more general kind of topos, now known as an elementary topos, and that elementary topoi were models of intuitionistic logic. In addition to providing a powerful way to apply tools from logic to geometry, this made possible the use of geometric methods in logic.
According to Kevin Carlson,
Nevertheless, a general definition of "structure" was proposed by Bourbaki^{[2]}; it embraces all types of spaces mentioned above, (nearly?) all types of mathematical structures used till now, and more. It provides a general definition of isomorphism, and justifies transfer of properties between isomorphic structures. However, it was never used actively in mathematical practice (not even in the mathematical treatises written by Bourbaki himself). Here are the last phrases from a review by Robert Reed^{[14]} of a book by Leo Corry:
For more information on mathematical structures see Wikipedia: mathematical structure, equivalent definitions of mathematical structures, and transport of structure.
The distinction between geometric "spaces" and algebraic "structures" is sometimes clear, sometimes elusive. Clearly, groups are algebraic, while Euclidean spaces are geometric. Modules over rings are as algebraic as groups. In particular, when the ring appears to be a field, the module appears to be a linear space; is it algebraic or geometric? In particular, when it is finitedimensional, over real numbers, and endowed with inner product, it becomes Euclidean space; now geometric. The (algebraic?) field of real numbers is the same as the (geometric?) real line. Its algebraic closure, the (algebraic?) field of complex numbers, is the same as the (geometric?) complex plane. It is first of all "a place we do analysis" (rather than algebra or geometry).
Every space treated in Section "Types of spaces" above, except for "Noncommutative geometry", "Schemes" and "Topoi" subsections, is a set (the "principal base set" of the structure, according to Bourbaki) endowed with some additional structure; elements of the base set are usually called "points" of this space. In contrast, elements of (the base set of) an algebraic structure usually are not called "points".
However, sometimes one uses more than one principal base set. For example, twodimensional projective geometry may be formalized via two base sets, the set of points and the set of lines. Moreover, a striking feature of projective planes is the symmetry of the roles played by points and lines. A less geometric example: a graph may be formalized via two base sets, the set of vertices (called also nodes or points) and the set of edges (called also arcs or lines). Generally, finitely many principal base sets and finitely many auxiliary base sets are stipulated by Bourbaki.
Many mathematical structures of geometric flavor treated in the "Noncommutative geometry", "Schemes" and "Topoi" subsections above do not stipulate a base set of points. For example, "pointless topology" (in other words, pointfree topology, or locale theory) starts with a single base set whose elements imitate open sets in a topological space (but are not sets of points); see also mereotopology and pointfree geometry.
The author thanks Wikipedian Ozob for their contributions.
Subsections "Noncommutative geometry", "Schemes" and "Topoi" are created in 20172018 on Wikipedia by Ozob. Otherwise, large part of this article was created in 2009 on Citizendium, ported in 2009 to Wikipedia, and therefrom in 2017 hereto.
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