Editing Kepler–Poinsot polyhedron (section)

== History ==

Most, if not all, of the Kepler–Poinsot polyhedra were known of in some form or other before Kepler. A small stellated dodecahedron appears in a marble tarsia (inlay panel) on the floor of [[St. Mark's Basilica]], [[Venice]], Italy. It dates from the 15th century and is sometimes attributed to [[Paolo Uccello]].<ref>{{cite book|contribution=Regular and semiregular polyhedra|first=H. S. M.|last=Coxeter|author-link= Harold Scott MacDonald Coxeter|pages=41–52|title=Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination|edition=2nd|editor-first=Marjorie|editor-last=Senechal|editor-link=Marjorie Senechal|publisher=Springer|year=2013|doi=10.1007/978-0-387-92714-5|isbn=978-0-387-92713-8 }} See in particular p. 42.</ref>

In his ''[[Perspectiva corporum regularium]]'' (''Perspectives of the regular solids''), a book of woodcuts published in 1568, [[Wenzel Jamnitzer]] depicts the [[great stellated dodecahedron]] and a [[great dodecahedron]] (both shown below). There is also a [[truncation (geometry)|truncated]] version of the [[small stellated dodecahedron]].<ref>[[:File:Perspectiva Corporum Regularium 27e.jpg]]</ref> It is clear from the general arrangement of the book that he regarded only the five Platonic solids as regular.

The small and great stellated dodecahedra, sometimes called the '''Kepler polyhedra''', were first recognized as regular by [[Johannes Kepler]] around 1619.<ref>H.S.M. Coxeter, P. Du Val, H.T. Flather and J.F. Petrie; ''The Fifty-Nine Icosahedra'', 3rd Edition, Tarquin, 1999. p.11</ref> He obtained them by [[stellation|stellating]] the regular convex dodecahedron, for the first time treating it as a surface rather than a solid. He noticed that by extending the edges or faces of the convex dodecahedron until they met again, he could obtain star pentagons. Further, he recognized that these star pentagons are also regular. In this way he constructed the two stellated dodecahedra. Each has the central convex region of each face "hidden" within the interior, with only the triangular arms visible. Kepler's final step was to recognize that these polyhedra fit the definition of regularity, even though they were not [[Convex polyhedron|convex]], as the traditional [[Platonic solid]]s were.

In 1809, [[Louis Poinsot]] rediscovered Kepler's figures, by assembling star pentagons around each vertex. He also assembled convex polygons around star vertices to discover two more regular stars, the great icosahedron and great dodecahedron. Some people call these two the '''Poinsot polyhedra'''. Poinsot did not know if he had discovered all the regular star polyhedra.

Three years later, [[Augustin Cauchy]] proved the list complete by [[stellation|stellating]] the [[Platonic solid]]s, and almost half a century after that, in 1858, [[Joseph Bertrand|Bertrand]] provided a more elegant proof by [[faceting]] them.

The following year, [[Arthur Cayley]] gave the Kepler–Poinsot polyhedra the names by which they are generally known today.

A hundred years later, [[John Horton Conway|John Conway]] developed a [[Stellation#Naming stellations|systematic terminology]] for stellations in up to four dimensions. Within this scheme the [[small stellated dodecahedron]] is just the ''stellated dodecahedron''.

{| style="width: 100%; text-align: center;"
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| [[File:Marble floor mosaic Basilica of St Mark Vencice.jpg|thumb|center|Floor [[mosaic]] in [[St Mark's Basilica|St Mark's]], [[Venice]] <small>(possibly by [[Paolo Uccello]])</small>]]
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{{multiple image
 | align = center | total_width = 440
 | image1 = Perspectiva Corporum Regularium 22c.jpg
 | image2 = Perspectiva Corporum Regularium MET DP239933, great stellated dodecahedron.jpg
 | footer = [[Great dodecahedron]] and [[great stellated dodecahedron]] in ''[[Perspectiva Corporum Regularium]]'' (1568)
}}
| [[File:Stellated dodecahedra Harmonices Mundi.jpg|thumb|center|Stellated dodecahedra, ''[[Harmonices Mundi]]'' by [[Johannes Kepler]] (1619)]]
| [[File:Sternpolyeder.jpg|thumb|center|Cardboard model of a [[great icosahedron]] from [[University of Tübingen|Tübingen University]] (around 1860)]]
|}