Editing Kepler–Poinsot polyhedron (section)

=== 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
}}
|}