Editing Kepler–Poinsot polyhedron (section)

===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]]
|}