Editing Kepler–Poinsot polyhedron (section)

== 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]]
|}