Editing Kepler–Poinsot polyhedron

{{Short description|Any of 4 regular star polyhedra}}
{{multiple image
 | align = right | perrow = 2 | total_width = 320
 | image1 = Great dodecahedron (gray with yellow face).svg | caption1 = [[Great dodecahedron]]
 | image2 = Small stellated dodecahedron (gray with yellow face).svg | caption2 = [[Small stellated dodecahedron]]
 | image3 = Great icosahedron (gray with yellow face).svg | caption3 = [[Great icosahedron]]
 | image4 = Great stellated dodecahedron (gray with yellow face).svg | caption4 = [[Great stellated dodecahedron]]
}}
In [[geometry]], a '''Kepler–Poinsot polyhedron''' is any of four [[Regular polyhedron|regular]] [[Star polyhedron|star polyhedra]].<ref>Coxeter, ''Star polytopes and the Schläfli function f(&alpha;,&beta;,&gamma;)'' p. 121 1. The Kepler–Poinsot polyhedra</ref>

They may be obtained by [[stellation|stellating]] the regular [[Convex polyhedron|convex]] [[dodecahedron]] and [[icosahedron]], and differ from these in having regular [[pentagram]]mic [[face (geometry)|face]]s or [[vertex figure]]s. They can all be seen as three-dimensional analogues of the pentagram in one way or another.

== Characteristics ==

=== Sizes ===
The great icosahedron edge length is <math>\phi^4 = \tfrac12\bigl(7+3\sqrt5\,\bigr)</math> times the original icosahedron edge length.
The small stellated dodecahedron, great dodecahedron, and great stellated dodecahedron edge lengths are respectively <math>\phi^3 = 2+\sqrt5,</math> <math>\phi^2 = \tfrac12\bigl(3+\sqrt5\,\bigr),</math> and <math>\phi^5 = \tfrac12\bigl(11+5\sqrt5\,\bigr)</math> times the original dodecahedron edge length.

=== Non-convexity ===
These figures have [[pentagram]]s (star pentagons) as faces or vertex figures. The [[small stellated dodecahedron|small]] and [[great stellated dodecahedron]] have [[star polygon|nonconvex regular]] [[pentagram]] faces. The [[great dodecahedron]] and [[great icosahedron]] have [[convex polygon|convex]] polygonal faces, but pentagrammic [[vertex figure]]s.

In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure. Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines intersect at a point that is not a corner of any face, these points are false vertices. The images below show spheres at the true vertices, and blue rods along the true edges.

For example, the [[small stellated dodecahedron]] has 12 [[pentagram]] faces with the central [[pentagon]]al part hidden inside the solid. The visible parts of each face comprise five [[isosceles triangle]]s which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical. Each edge would now be divided into three shorter edges (of two different kinds), and the 20 false vertices would become true ones, so that we have a total of 32 vertices (again of two kinds). The hidden inner pentagons are no longer part of the polyhedral surface, and can disappear. Now [[Planar graph#Euler's formula|Euler's formula]] holds: 60&nbsp;&minus;&nbsp;90&nbsp;+&nbsp;32&nbsp;=&nbsp;2. However, this polyhedron is no longer the one described by the [[Schläfli symbol]] {5/2,&nbsp;5}, and so can not be a Kepler–Poinsot solid even though it still looks like one from outside.

=== Euler characteristic χ ===
A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others.  Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the [[Euler characteristic|Euler relation]]

:<math>\chi=V-E+F=2\ </math>

does not always hold. Schläfli held that all polyhedra must have χ = 2, and he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never widely held.

A modified form of Euler's formula, using [[Density (polytope)|density]] (''D'') of the [[vertex figure]]s (<math>d_v</math>) and faces (<math>d_f</math>) was given by [[Arthur Cayley]], and holds both for convex polyhedra (where the correction factors are all 1), and the Kepler–Poinsot polyhedra:
:<math>d_v V - E + d_f F = 2D.</math>

=== Duality and Petrie polygons ===

The Kepler–Poinsot polyhedra exist in [[dual polyhedron|dual]] pairs. Duals have the same [[Petrie polygon]], or more precisely, Petrie polygons with the same two dimensional projection.

The following images show the two [[dual compound]]s with the same [[midsphere|edge radius]]. They also show that the Petrie polygons are [[skew polygon|skew]].
Two relationships described in the article below are also easily seen in the images: That the violet edges are the same, and that the green faces lie in the same planes.

{| class="wikitable"
! <span style="color: #a13870;">horizontal edge in front</span>
! <span style="color: #007400;">vertical edge in front</span>
! Petrie polygon
|-
| [[small stellated dodecahedron]] <math>\left\{\frac{5}{2}, 5\right\}</math>
| [[great dodecahedron]] <math>\left\{5, \frac{5}{2}\right\}</math>
| [[Regular hexagon|hexagon]] <math>\left\{\frac{6}{1,3}\right\}</math>
|-
| [[great icosahedron]] <math>\left\{3, \frac{5}{2}\right\}</math>
| [[great stellated dodecahedron]] <math>\left\{\frac{5}{2}, 3\right\}</math>
| [[Decagram (geometry)|decagram]] <math>\left\{\frac{10}{3,5}\right\}</math>
|}

{| style="width: 100%;"
|-
|
{{multiple image
 | align = center | width = 200
 | image1 = Skeleton pair Gr12 and dual, size s.png
 | image2 = Skeleton pair Gr12 and dual, Petrie, stick, size s.png
 | image3 = Skeleton pair Gr12 and dual, Petrie, stick, size s, 3-fold.png
 | footer = [[Compound of small stellated dodecahedron and great dodecahedron|Compound of sD and gD]] with Petrie hexagons
}}
|
{{multiple image
 | align = center | width = 200
 | image1 = Skeleton pair Gr20 and dual, size s.png
 | image2 = Skeleton pair Gr20 and dual, Petrie, stick, size s.png
 | image3 = Skeleton pair Gr20 and dual, Petrie, stick, size s, 5-fold.png
 | footer = [[Compound of great icosahedron and great stellated dodecahedron|Compound of gI and gsD]] with Petrie decagrams
}}
|}

=== Summary ===

{| class="wikitable"
|-
!Name<br>(Conway's abbreviation)
!Picture
!Spherical<BR>tiling
![[Stellation]]<BR>diagram
![[Schläfli symbol|Schläfli]]<br />{p,&nbsp;q} and<br />[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!Faces<br />{p}
!Edges
!Vertices<br />{q}
![[Vertex figure|Vertex<BR>figure]]<BR>[[Vertex configuration|(config.)]]
![[Petrie polygon]]
![[Euler characteristic|χ]]
![[Density (polytope)|Density]]
![[Symmetry group|Symmetry]]
![[dual polyhedron|Dual]]
|- align=center
|[[great dodecahedron]]<br>(gD)
|[[Image:Great dodecahedron (gray with yellow face).svg|80px]]
|[[Image:Great dodecahedron tiling.svg|80px]]
|[[File:Second stellation of dodecahedron facets.svg|80px]]
|{5,&nbsp;5/2}<br />{{CDD|node_1|5|node|5|rat|d2|node}}
|12<br />{5}
|30||12<br />{5/2}||[[File:Great dodecahedron vertfig.png|80px]]<BR>(5<sup>5</sup>)/2
|[[File:Skeleton Gr12, Petrie, stick, size m, 3-fold.png|80px]]<br>{6}
| −6||3||I<sub>h</sub>||small stellated dodecahedron
|- align=center
|[[small stellated dodecahedron]]<br>(sD)
|[[Image:Small stellated dodecahedron (gray with yellow face).svg|80px]]
|[[Image:Small stellated dodecahedron tiling.png|80px]]
|[[File:First stellation of dodecahedron facets.svg|80px]]
|{5/2,&nbsp;5}<br />{{CDD|node|5|node|5|rat|d2|node_1}}
|12<br />{5/2}
|30||12<br />{5}||[[File:Small stellated dodecahedron vertfig.png|80px]]<BR>(5/2)<sup>5</sup>
|[[File:Skeleton St12, Petrie, stick, size m, 3-fold.png|80px]]<br>{6}
| −6||3||I<sub>h</sub>||great dodecahedron
|- align=center
|[[great icosahedron]]<br>(gI)
|[[Image:Great icosahedron (gray with yellow face).svg|80px]]
|[[Image:Great icosahedron tiling.svg|80px]]
|[[File:Great icosahedron stellation facets.svg|80px]]
|{3,&nbsp;5/2}<br />{{CDD|node_1|3|node|5|rat|d2|node}}
|20<br />{3}
|30||12<br />{5/2}||[[File:Great icosahedron vertfig.svg|80px]]<BR>(3<sup>5</sup>)/2
|[[File:Skeleton Gr20, Petrie, stick, size m, 5-fold.png|80px]]<br>{10/3}
|2||7||I<sub>h</sub>||great stellated dodecahedron
|- align=center
|[[great stellated dodecahedron]]<br>(sgD = gsD)
|[[Image:Great stellated dodecahedron (gray with yellow face).svg|80px]]
|[[Image:Great stellated dodecahedron tiling.svg|80px]]
|[[File:Third stellation of dodecahedron facets.svg|80px]]
|{5/2,&nbsp;3}<br />{{CDD|node|3|node|5|rat|d2|node_1}}
|12<br />{5/2}
|30||20<br />{3}||[[File:Great stellated dodecahedron vertfig.png|80px]]<BR>(5/2)<sup>3</sup>
|[[File:Skeleton GrSt12, Petrie, stick, size m, 5-fold.png|80px]]<br>{10/3}
|2||7||I<sub>h</sub>||great icosahedron
|}

== Relationships among the regular polyhedra ==

[[File:Relationship among regular star polyhedra (direction colors).png|thumb|400px|Conway's system of relations between the six polyhedra (ordered vertically by [[Density (polytope)|density]])<ref>Conway et al. (2008), p.405 Figure 26.1 Relationships among the three-dimensional star-polytopes</ref>]]

===Conway's operational terminology===

[[John Horton Conway|John Conway]] defines the Kepler–Poinsot polyhedra as ''greatenings'' and ''stellations'' of the convex solids.<br>
In his [[Stellation#Naming stellations|naming convention]], the [[small stellated dodecahedron]] is just the ''stellated dodecahedron''.

{| class="wikitable"
| icosahedron (I)
| dodecahedron (D)
|-
| great dodecahedron (gD)
| stellated dodecahedron (sD)
|-
| great icosahedron (gI)
| great stellated dodecahedron (sgD = gsD)
|}

''Stellation'' changes pentagonal faces into pentagrams. (In this sense stellation is a unique operation, and not to be confused with the more general [[stellation]] described below.)

''Greatening'' maintains the type of faces, shifting and resizing them into parallel planes.

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="7"| Conway relations illustrated
|-
|-
! diagram
| 
[[File:Relationship among regular star polyhedra (green and violet).png|450px]]
<small><br>The polyhedra in this section are shown with the same [[midsphere|midradius]].</small>
|-
!style="color:#e57500"| stellation
|
{|
|
{{multiple image
 | align = right  | width = 200
 | image1 = Skeleton 12, size s.png | caption1 = <span style="color: #a13870;">'''D'''</span>
 | image2 = Skeleton St12, size s.png | caption2 = <span style="color: #a13870;">'''sD'''</span>
}}
|
{{multiple image
 | align = right  | width = 200
 | image1 = Skeleton Gr12, size s.png | caption1 = <span style="color: #007400;">'''gD'''</span>
 | image2 = Skeleton GrSt12, size s.png | caption2 = <span style="color: #007400;">sgD = '''gsD'''</span>
}}
|}

|-
!style="color:#00a7e1"| greatening
|
{| style="width: 100%;"
| [[File:Skeleton pair 12 and greatening, size s.png|thumb|center|200px|<span style="color: #a13870;">'''D'''</span> and <span style="color: #007400;">'''gD'''</span>]]
| [[File:Skeleton pair 20 and greatening, size s.png|thumb|center|200px|<span style="color: #007400;">'''I'''</span> and <span style="color: #a13870;">'''gI'''</span>]]
| [[File:Skeleton pair St12 and greatening, size s.png|thumb|center|200px|<span style="color: #a13870;">'''sD'''</span> and <span style="color: #007400;">'''gsD'''</span>]]
|}

|-
!style="color:#00cb00"| duality
|
{| style="width: 100%;"
| [[File:Skeleton pair 12-20, size s.png|thumb|center|200px|<span style="color: #a13870;">'''D'''</span> and <span style="color: #007400;">'''I'''</span>]]
| [[File:Skeleton pair Gr12 and dual, size s.png|thumb|center|200px|<span style="color: #007400;">'''gD'''</span> and <span style="color: #a13870;">'''sD'''</span>]]
| [[File:Skeleton pair Gr20 and dual, size s.png|thumb|center|200px|<span style="color: #a13870;">'''gI'''</span> and <span style="color: #007400;">'''gsD'''</span>]]
|}

|}

===Stellations and facetings===

The [[great icosahedron]] is one of the [[stellation]]s of the [[regular icosahedron|icosahedron]]. (See ''[[The Fifty-Nine Icosahedra]]'')<br>
The three others are all the stellations of the [[regular dodecahedron|dodecahedron]].

The [[great stellated dodecahedron]] is a [[faceting]] of the dodecahedron.<br>
The three others are facetings of the icosahedron.

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="7"| Stellations and facetings
|-
! Convex
|colspan="3"| [[File:Polyhedron 20 big.png|160px]]<br>[[regular icosahedron|icosahedron]]
|colspan="3"| [[File:Polyhedron 12 big.png|160px]]<br>[[regular dodecahedron|dodecahedron]]
|-
! Stellations
|colspan="3"| [[File:Polyhedron great 20.png|90px]]<br>[[great icosahedron|gI]] <small>(the one with yellow faces)</small>
| [[File:Polyhedron great 12.png|90px]]<br>[[great dodecahedron|gD]]
| [[File:Polyhedron great 12 dual.png|90px]]<br>[[small stellated dodecahedron|sD]]
| [[File:Polyhedron great 20 dual.png|90px]]<br>[[great stellated dodecahedron|gsD]]
|-
! Facetings
| [[File:Polyhedron great 20.png|90px]]<br>[[great icosahedron|gI]]
| [[File:Polyhedron great 12.png|90px]]<br>[[great dodecahedron|gD]]
| [[File:Polyhedron great 12 dual.png|90px]]<br>[[small stellated dodecahedron|sD]]
|colspan="3"| [[File:Polyhedron great 20 dual.png|90px]]<br>[[great stellated dodecahedron|gsD]] <small>(the one with yellow vertices)</small>
|}

If the intersections are treated as new edges and vertices, the figures obtained will not be [[regular polyhedron|regular]], but they can still be considered [[stellation]]s.{{Example needed|s|date=December 2018}}

(See also [[List of Wenninger polyhedron models#Stellations of dodecahedron|List of Wenninger polyhedron models]])

===Shared vertices and edges===

The great stellated dodecahedron shares its vertices with the dodecahedron. The other three Kepler–Poinsot polyhedra share theirs with the icosahedron.
{{awrap|The [[n-skeleton|skeletons]] of the solids sharing vertices are [[topology|topologically]] equivalent.}}

{| class="wikitable" style="width: 100%; text-align: center;"
|-
| [[File:Polyhedron 20 big.png|160px]]<br>[[regular icosahedron|icosahedron]]
| [[File:Polyhedron great 12.png|160px]]<br>[[great dodecahedron]]
| [[File:Polyhedron great 20.png|160px]]<br>[[great icosahedron]]
| [[File:Polyhedron great 12 dual.png|160px]]<br>[[small stellated dodecahedron]]
| [[File:Polyhedron 12 big.png|160px]]<br>[[regular dodecahedron|dodecahedron]]
| [[File:Polyhedron great 20 dual.png|160px]]<br>[[great stellated dodecahedron]]
|-
|colspan="2"| share vertices and edges
|colspan="2"| share vertices and edges
|colspan="2" rowspan="2"| share vertices, {{awrap|skeletons form [[dodecahedral graph]]}}
|-
|colspan="4"| share vertices, skeletons form [[icosahedral graph]]
|}

==The stellated dodecahedra==

===Hull and core===

The [[small stellated dodecahedron|small]] and [[great stellated dodecahedron|great]] stellated dodecahedron
can be seen as a [[regular dodecahedron|regular]] and a [[great dodecahedron]] with their edges and faces extended until they intersect.<br>
The pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces.<br>
For the small stellated dodecahedron the hull is <math>\varphi</math> times bigger than the core, and for the great it is <math>\varphi + 1 = \varphi^2</math> times bigger.
{{awrap|(See [[Golden ratio]])}}<br>
<small>(The [[midsphere|midradius]] is a common measure to compare the size of different polyhedra.)</small>

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="5"| Hull and core of the stellated dodecahedra
|-
! Hull
! Star polyhedron
! Core
! <math>\frac{\text{hull midradius}}{\text{core midradius}}</math>
! <math>\frac{\text{core midradius}}{\text{hull midradius}}</math>
|-
| [[File:Polyhedron 20 big.png|160px]]
| [[File:Polyhedron great 12 dual.png|160px]]
| [[File:Polyhedron 12 (core of great 12 dual).png|160px]]
| <math>\frac{\sqrt{5} + 1}{2} = 1.61803...</math>
| <math>\frac{\sqrt{5} - 1}{2} = 0.61803...</math>
|-
| [[File:Polyhedron 12 big.png|160px]]
| [[File:Polyhedron great 20 dual.png|160px]]
| [[File:Polyhedron great 12 (core of great 20 dual).png|160px]]
| <math>\frac{3 + \sqrt{5}}{2} = 2.61803...</math>
| <math>\frac{3 - \sqrt{5}}{2} = 0.38196...</math>
|- style="text-align: left; font-size: small;"
|colspan="5"|
The platonic hulls in these images have the same [[midsphere|midradius]].<br>
This implies that the pentagrams have the same size, and that the cores have the same edge length.<br>
(Compare the 5-fold orthographic projections below.)
|}

===Augmentations===

Traditionally the two star polyhedra have been defined as ''augmentations'' (or ''cumulations''),
{{awrap|i.e. as dodecahedron and icosahedron with pyramids added to their faces.}}

Kepler calls the small stellation an ''augmented dodecahedron'' (then nicknaming it ''hedgehog'').<ref>"augmented dodecahedron to which I have given the name of ''Echinus''"
(''[[Harmonices Mundi]]'', Book V, Chapter III — p. 407 in the translation by E. J. Aiton)</ref>

{{awrap|In his view the great stellation is related to the icosahedron as the small one is to the dodecahedron.<ref>"These figures are so closely related the one to the dodecahedron the other to the icosahedron that the latter two figures, particularly the dodecahedron, seem somehow truncated or maimed when compared to the figures with spikes."
(''[[Harmonices Mundi]]'', Book II, Proposition XXVI — p. 117 in the translation by E. J. Aiton)</ref>}}

These [[Informal mathematics|naïve]] definitions are still used.
E.g. [[MathWorld]] states that the two star polyhedra can be constructed by adding pyramids to the faces of the Platonic solids.<ref>"A small stellated dodecahedron can be constructed by cumulation of a dodecahedron,
i.e., building twelve pentagonal pyramids and attaching them to the faces of the original dodecahedron."
{{MathWorld |id=SmallStellatedDodecahedron |title=Small Stellated Dodecahedron |access-date=2018-09-21}}</ref>
<ref>"Another way to construct a great stellated dodecahedron via cumulation is to make 20 triangular pyramids [...] and attach them to the sides of an icosahedron."
{{MathWorld |id=GreatStellatedDodecahedron |title=Great Stellated Dodecahedron |access-date=2018-09-21}}</ref>

{{awrap|This is just a help to visualize the shape of these solids, and not actually a claim that the edge intersections (false vertices) are vertices.}}
{{awrap|If they were, the two star polyhedra would be [[Topology|topologically]] equivalent to the [[pentakis dodecahedron]] and the [[triakis icosahedron]].}}

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="5"| Stellated dodecahedra as augmentations
|-
! Core
! Star polyhedron
! [[Catalan solid]]
|-
| [[File:Polyhedron 12 (core of great 12 dual).png|160px]]
| [[File:Polyhedron great 12 dual (as pentakis 12).png|160px]]
| [[File:Polyhedron truncated 20 dual big.png|160px]]
|-
| [[File:Polyhedron 20 (core of great 20 dual).png|160px]]
| [[File:Polyhedron great 20 dual (as triakis 20).png|160px]]
| [[File:Polyhedron truncated 12 dual big.png|160px]]
|}

==Symmetry==

All Kepler–Poinsot polyhedra have full [[icosahedral symmetry]], just like their convex hulls.

The [[great icosahedron]] and [[great stellated dodecahedron|its dual]] resemble the icosahedron and its dual in that they have faces and vertices on the 3-fold (yellow) and 5-fold (red) symmetry axes.<br>
In the [[great dodecahedron]] and [[small stellated dodecahedron|its dual]] all faces and vertices are on 5-fold symmetry axes (so there are no yellow elements in these images).

The following table shows the solids in pairs of duals. In the top row they are shown with [[pyritohedral symmetry]], in the bottom row with icosahedral symmetry (to which the mentioned colors refer).

The table below shows [[orthographic projection]]s from the 5-fold (red), 3-fold (yellow) and 2-fold (blue) symmetry axes.

{| class="wikitable" style="width: 100%; text-align: center;"
|-
! {3, 5} ([[regular icosahedron|I]]) &nbsp; and &nbsp; {5, 3} ([[regular dodecahedron|D]])
! {5, 5/2} ([[great dodecahedron|gD]]) &nbsp; and &nbsp; {5/2, 5} ([[small stellated dodecahedron|sD]])
! {3, 5/2} ([[great icosahedron|gI]]) &nbsp; and &nbsp; {5/2, 3} ([[great stellated dodecahedron|gsD]])
|-
|{{awrap|[[File:Polyhedron 20 pyritohedral big.png|160px]][[File:Polyhedron 12 pyritohedral big.png|160px]]}}<br>
<small>([[c:Animations of Kepler-Poinsot solids with direction colors#20pyr|animations]])</small>
|{{awrap|[[File:Polyhedron great 12 pyritohedral.png|160px]][[File:Polyhedron great 12 dual pyritohedral.png|160px]]}}<br>
<small>([[c:Animations of Kepler-Poinsot solids with direction colors#great12pyr|animations]])</small>
|{{awrap|[[File:Polyhedron great 20 pyritohedral.png|160px]][[File:Polyhedron great 20 dual pyritohedral.png|160px]]}}<br>
<small>([[c:Animations of Kepler-Poinsot solids with direction colors#great20pyr|animations]])</small>
|-
|{{awrap|[[File:Polyhedron 20 big.png|160px]][[File:Polyhedron 12 big.png|160px]]}}<br>
<small>([[c:Animations of Kepler-Poinsot solids with direction colors#20ico|animations]])</small>
|{{awrap|[[File:Polyhedron great 12.png|160px]][[File:Polyhedron great 12 dual.png|160px]]}}<br>
<small>([[c:Animations of Kepler-Poinsot solids with direction colors#great12ico|animations]])</small>
|{{awrap|[[File:Polyhedron great 20.png|160px]][[File:Polyhedron great 20 dual.png|160px]]}}<br>
<small>([[c:Animations of Kepler-Poinsot solids with direction colors#great20ico|animations]])</small>
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
!colspan="3"| orthographic projections
|- style="text-align: left; font-size: small;"
|colspan="3"|
The platonic hulls in these images have the same [[midsphere|midradius]], so all the 5-fold projections below are in a [[decagon]] of the same size.
{{awrap|(Compare [[:File:Polyhedron pair 12-20 big from red.png|projection of the compound]].)}}
{{awrap|This implies that [[small stellated dodecahedron|sD]], [[great stellated dodecahedron|gsD]] and [[great icosahedron|gI]] have the same edge length,
namely the side length of a pentagram in the surrounding decagon.}}
|-
|{{awrap|[[File:Polyhedron 20 big from red.png|160px]][[File:Polyhedron 12 big from red.png|160px]]}}
|{{awrap|[[File:Polyhedron great 12 from red.png|160px]][[File:Polyhedron great 12 dual from red.png|160px]]}}
|{{awrap|[[File:Polyhedron great 20 from red.png|160px]][[File:Polyhedron great 20 dual from red.png|160px]]}}
|-
|{{awrap|[[File:Polyhedron 20 big from yellow.png|160px]][[File:Polyhedron 12 big from yellow.png|160px]]}}
|{{awrap|[[File:Polyhedron great 12 from yellow.png|160px]][[File:Polyhedron great 12 dual from yellow.png|160px]]}}
|{{awrap|[[File:Polyhedron great 20 from yellow.png|160px]][[File:Polyhedron great 20 dual from yellow.png|160px]]}}
|-
|{{awrap|[[File:Polyhedron 20 big from blue.png|160px]][[File:Polyhedron 12 big from blue.png|160px]]}}
|{{awrap|[[File:Polyhedron great 12 from blue.png|160px]][[File:Polyhedron great 12 dual from blue.png|160px]]}}
|{{awrap|[[File:Polyhedron great 20 from blue.png|160px]][[File:Polyhedron great 20 dual from blue.png|160px]]}}
|}

== 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;"
|- style="vertical-align: top;"
| [[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>]]
|
{{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)]]
|}

== Regular star polyhedra in art and culture ==
[[Image:Alexander's Star.jpg|thumb|Alexander's Star]]
Regular star polyhedra first appear in Renaissance art. A small stellated dodecahedron is depicted in a marble tarsia on the floor of [[St Mark's Basilica|St. Mark's Basilica]], Venice, Italy, dating from ca. 1430 and sometimes attributed to [[Paolo Uccello|Paulo Uccello]].

In the 20th century, artist [[M. C. Escher]]'s interest in geometric forms often led to works based on or including regular solids; ''[[Gravitation (M. C. Escher)|Gravitation]]'' is based on a small stellated dodecahedron.

A [[Dissection (geometry)|dissection]] of the great dodecahedron was used for the 1980s puzzle [[Alexander's Star]].

Norwegian artist [[Vebjørn Sand]]'s sculpture ''The Kepler Star'' is displayed near [[Oslo Airport, Gardermoen]]. The star spans 14 meters, and consists of an [[icosahedron]] and a [[dodecahedron]] inside a great stellated dodecahedron.

== See also ==
* [[Regular polytope]]
* [[Regular polyhedron]]
* [[List of regular polytopes#Finite Non-Convex Polytopes - star-polytopes|List of regular polytopes]]
* [[Uniform polyhedron]]
* [[Uniform star polyhedron]]
* [[Polyhedral compound]]
* [[Regular star 4-polytope]] – the ten regular star [[4-polytope]]s, 4-dimensional analogues of the Kepler–Poinsot polyhedra

== References ==
===Notes===
{{Reflist}}

===Bibliography===
* [[Joseph Bertrand|J. Bertrand]], Note sur la théorie des polyèdres réguliers, ''Comptes rendus des séances de l'Académie des Sciences'', '''46''' (1858), pp.&nbsp;79–82, 117.
* [[Augustin-Louis Cauchy]], ''Recherches sur les polyèdres.'' J. de l'École Polytechnique 9, 68–86, 1813.
* [[Arthur Cayley]], On Poinsot's Four New Regular Solids. ''Phil. Mag.'' '''17''', pp.&nbsp;123–127 and 209, 1859.
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, [[Chaim Goodman-Strauss]], ''The Symmetry of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 24, Regular Star-polytopes, pp.&nbsp;404–408)
* ''Kaleidoscopes: Selected Writings of [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]]'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] {{Webarchive|url=https://web.archive.org/web/20160711140441/http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html |date=2016-07-11 }}
** (Paper 1) H.S.M. Coxeter, ''The Nine Regular Solids'' [Proc. Can. Math. Congress 1 (1947), 252–264, MR 8, 482]
** (Paper 10) H.S.M. Coxeter, ''Star Polytopes and the Schlafli Function f(α,β,γ)'' [Elemente der Mathematik 44 (2) (1989) 25–36]
* [[Theoni Pappas]], (The Kepler–Poinsot Solids) ''The Joy of Mathematics''. San Carlos, CA: Wide World Publ./Tetra, p.&nbsp;113, 1989.
* [[Louis Poinsot]], Memoire sur les polygones et polyèdres. ''J. de l'École Polytechnique'' '''9''', pp.&nbsp;16–48, 1810.
* Lakatos, Imre; ''Proofs and Refutations'', Cambridge University Press (1976) - discussion of proof of Euler characteristic
* {{cite book | first=Magnus | last=Wenninger | author-link=Magnus Wenninger  | title=Dual Models | publisher=Cambridge University Press | date=1983 | isbn=0-521-54325-8 }}, pp.&nbsp;39–41.
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, [[Chaim Goodman-Strauss]], ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;404: Regular star-polytopes Dimension 3)
* {{cite book | author= Anthony Pugh | date= 1976 | title= Polyhedra: A Visual Approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7  }} Chapter 8: Kepler Poisot polyhedra

==External links==
{{Commons category|Kepler-Poinsot solids}}
*{{Mathworld | urlname=Kepler-PoinsotSolid | title=Kepler–Poinsot solid }}
*[http://www.software3d.com/Kepler.php Paper models of Kepler–Poinsot polyhedra]
*[http://www.korthalsaltes.com/cuadros.php?type=k Free paper models (nets) of Kepler–Poinsot polyhedra]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://dmccooey.com/polyhedra/KeplerPoinsot.html Kepler-Poinsot Solids] in Visual Polyhedra
*[http://www.georgehart.com/virtual-polyhedra/kepler-poinsot-info.html VRML models of the Kepler–Poinsot polyhedra]
*[http://www.steelpillow.com/polyhedra/StelFacet/history.html Stellation and facetting - a brief history]
*[http://www.software3d.com/Stella.php Stella: Polyhedron Navigator]: Software used to create many of the images on this page.
{{Nonconvex polyhedron navigator}}

{{DEFAULTSORT:Kepler-Poinsot Polyhedron}}
[[Category:Kepler–Poinsot polyhedra| ]]
[[Category:Johannes Kepler]]
[[Category:Nonconvex polyhedra]]