Editing Kepler–Poinsot polyhedron (section)

==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]]
|}