Editing Kepler–Poinsot polyhedron (section)

==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]]}}
|}