Editing Kepler–Poinsot polyhedron (section)

== 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
|}