Editing Regular polyhedron (section)

== Regular polyhedra in nature ==
Each of the Platonic solids occurs naturally in one form or another.

The tetrahedron, cube, and octahedron all occur as [[crystal]]s. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither the [[regular icosahedron]] nor the [[regular dodecahedron]] are amongst them, but crystals can have the shape of a [[pyritohedron]], which is visually almost indistinguishable from a regular dodecahedron. Truly icosahedral crystals may be formed by [[quasicrystal|quasicrystalline materials]] which are very rare in nature but can be produced in a laboratory.

A more recent discovery is of a series of new types of [[carbon]] molecule, known as the [[fullerene]]s (see Curl, 1991). Although C<sub>60</sub>, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C<sub>240</sub>, C<sub>480</sub> and C<sub>960</sub>) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.

Regular polyhedra appear in biology as well. The [[coccolithophore]] ''[[Braarudosphaera bigelowii]]'' has a regular dodecahedral structure, about 10 [[micrometre]]s across.<ref name=Hagino2013>Hagino, K., Onuma, R., Kawachi, M. and Horiguchi, T. (2013) "Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in ''Braarudosphaera bigelowii'' (Prymnesiophyceae)". ''PLoS One'', '''8'''(12): e81749. {{doi|10.1371/journal.pone.0081749}}.</ref> In the early 20th century, [[Ernst Haeckel]] described a number of species of [[radiolarians]], some of whose shells are shaped like various regular polyhedra.<ref name=Haeckel1904>Haeckel, E. (1904). ''[[Kunstformen der Natur]]''. Available as Haeckel, E. ''Art forms in nature'', Prestel USA (1998), {{isbn|3-7913-1990-6}}. Online version at [http://www.biolib.de/haeckel/kunstformen/index.html Kurt Stüber's Biolib] (in german)</ref> Examples include ''Circoporus octahedrus'', ''Circogonia icosahedra'', ''Lithocubus geometricus'' and ''Circorrhegma dodecahedra''; the shapes of these creatures are indicated by their names.<ref name=Haeckel1904 /> The outer protein shells of many [[virus]]es form regular polyhedra. For example, [[HIV]] is enclosed in a regular icosahedron, as is the head of a typical [[myovirus]].<ref>{{cite book | title=Virus Taxonomy | chapter=Myoviridae | publisher=Elsevier | year=2012 | pages=46–62 | doi=10.1016/b978-0-12-384684-6.00002-1 | isbn=9780123846846 | ref={{sfnref | Elsevier | 2012}}}}</ref><ref>{{cite book | last1=STRAUSS | first1=JAMES H. | last2=STRAUSS | first2=ELLEN G. | title=Viruses and Human Disease | chapter=The Structure of Viruses | publisher=Elsevier | year=2008 | pages=35–62 | doi=10.1016/b978-0-12-373741-0.50005-2| pmc=7173534 | isbn=9780123737410 | s2cid=80803624 }}</ref>

<gallery mode=packed style=float:left heights=180px>
File:Braarudosphaera bigelowii.jpg| The [[coccolithophore]] ''[[Braarudosphaera bigelowii]]'' has a regular dodecahedral structure
File:Circogonia icosahedra.jpg| The [[radiolarian]] ''[[Circoporidae|Circogonia icosahedra]]'' has a regular icosahedral structure
File:Structure of a Myoviridae bacteriophage 2.jpg| A [[myovirus]] typically has a regular icosahedral [[capsid]] (head) about 100 [[nanometer]]s across.
</gallery>
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In ancient times the [[Pythagoreanism|Pythagoreans]] believed that there was a harmony between the regular polyhedra and the orbits of the [[planet]]s. In the 17th century, [[Johannes Kepler]] studied data on planetary motion compiled by [[Tycho Brahe]] and for a decade tried to establish the Pythagorean ideal by finding a match between the sizes of the polyhedra and the sizes of the planets' orbits. His search failed in its original objective, but out of this research came Kepler's discoveries of the Kepler solids as regular polytopes, the realisation that the orbits of planets are not circles, and [[Kepler's laws of planetary motion|the laws of planetary motion]] for which he is now famous. In Kepler's time only five planets (excluding the earth) were known, nicely matching the number of Platonic solids. Kepler's work, and the discovery since that time of [[Uranus]] and [[Neptune]], have invalidated the Pythagorean idea.

Around the same time as the Pythagoreans, Plato described a theory of matter in which the five elements (earth, air, fire, water and spirit) each comprised tiny copies of one of the five regular solids. Matter was built up from a mixture of these polyhedra, with each substance having different proportions in the mix. Two thousand years later [[Dalton's atomic theory]] would show this idea to be along the right lines, though not related directly to the regular solids.