Kepler–Poinsot polyhedron

These figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures.

In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure. Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines intersect at a point that is not a corner of any face, these points are false vertices. The images below show spheres at the true vertices, and blue rods along the true edges.

For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid. The visible parts of each face comprise five isosceles triangles which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical. Each edge would now be divided into three shorter edges (of two different kinds), and the 20 false vertices would become true ones, so that we have a total of 32 vertices (again of two kinds). The hidden inner pentagons are no longer part of the polyhedral surface, and can disappear. Now Euler's formula holds: 60 − 90 + 32 = 2. However, this polyhedron is no longer the one described by the Schläfli symbol {5/2, 5}, and so can not be a Kepler–Poinsot solid even though it still looks like one from outside.

A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others. Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation

${\displaystyle \chi =V-E+F=2\ }$

does not always hold. Schläfli held that all polyhedra must have χ = 2, and he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never widely held.

A modified form of Euler's formula, using density (D) of the vertex figures (${\displaystyle d_{v}}$) and faces (${\displaystyle d_{f}}$) was given by Arthur Cayley, and holds both for convex polyhedra (where the correction factors are all 1), and the Kepler–Poinsot polyhedra:

${\displaystyle d_{v}V-E+d_{f}F=2D.}$

The Kepler–Poinsot polyhedra exist in 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 compounds with the same edge radius. They also show that the Petrie polygons are 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.

Compound of sD and gD with Petrie hexagons   (sD and gD alone)

Compound of gI and gsD with Petrie decagrams   (gI and gsD alone)

John Conway defines the Kepler–Poinsot polyhedra as greatenings and stellations of the convex solids.
In his naming convention the small stellated dodecahedron is just the stellated dodecahedron.

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.

Conway relations illustrated
diagram	The polyhedra in this section are shown with the same midradius.
stellation	D sD gD sgD = gsD
greatening	D and gD I and gI sD and gsD
duality	D and I gD and sD gI and gsD

The great icosahedron is one of the stellations of the icosahedron. (See The Fifty-Nine Icosahedra)
The three others are all the stellations of the dodecahedron.

The great stellated dodecahedron is a faceting of the dodecahedron.
The three others are facetings of the icosahedron.

Stellations and facetings
Convex	icosahedron			dodecahedron
Stellations	gI (the one with yellow faces)			gD	sD	gsD
Facetings	gI	gD	sD	gsD (the one with yellow vertices)

If the intersections are treated as new edges and vertices, the figures obtained will not be regular, but they can still be considered stellations.

(See also List of Wenninger polyhedron models)

The great stellated dodecahedron shares its vertices with the dodecahedron. The other three Kepler–Poinsot polyhedra share theirs with the icosahedron. The skeletons of the solids sharing vertices are topologically equivalent.

icosahedron	great dodecahedron	great icosahedron	small stellated dodecahedron	dodecahedron	great stellated dodecahedron
share vertices and edges		share vertices and edges		share vertices, skeletons form dodecahedral graph
share vertices, skeletons form icosahedral graph

The small and great stellated dodecahedron can be seen as a regular and a great dodecahedron with their edges and faces extended until they intersect.
The pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces.
For the small stellated dodecahedron the hull is ${\displaystyle \varphi }$ times bigger than the core, and for the great it is ${\displaystyle \varphi +1=\varphi ^{2}}$ times bigger. (See Golden ratio)
(The midradius is a common measure to compare the size of different polyhedra.)

Hull and core of the stellated dodecahedra
Hull	Star polyhedron	Core	${\displaystyle {\frac {\text{hull midradius}}{\text{core midradius}}}}$	${\displaystyle {\frac {\text{core midradius}}{\text{hull midradius}}}}$
			${\displaystyle {\frac {{\sqrt {5}}+1}{2}}=1.61803...}$	${\displaystyle {\frac {{\sqrt {5}}-1}{2}}=0.61803...}$
			${\displaystyle {\frac {3+{\sqrt {5}}}{2}}=2.61803...}$	${\displaystyle {\frac {3-{\sqrt {5}}}{2}}=0.38196...}$
The platonic hulls in these images have the same midradius. This implies that the pentagrams have the same size, and that the cores have the same edge length. (Compare the 5-fold orthographic projections below.)

Traditionally the two star polyhedra have been defined as augmentations (or cumulations), i.e. as dodecahedron and icosahedron with pyramids added to their faces.

Kepler calls the small stellation an augmented dodecahedron (then nicknaming it hedgehog).[3]

In his view the great stellation is related to the icosahedron as the small one is to the dodecahedron.[4]

These 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.[5] [6]

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. If they were, the two star polyhedra would be topologically equivalent to the pentakis dodecahedron and the triakis icosahedron.

Stellated dodecahedra as augmentations
Core	Star polyhedron	Catalan solid

• J. Bertrand, Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences, 46 (1858), pp. 79–82, 117.
• Augustin-Louis Cauchy, Recherches sur les polyèdres. J. de l'École Polytechnique 9, 68-86, 1813.
• Arthur Cayley, On Poinsot's Four New Regular Solids. Phil. Mag. 17, pp. 123–127 and 209, 1859.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetry of Things 2008, ISBN 978-1-56881-220-5 (Chapter 24, Regular Star-polytopes, pp. 404–408)
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
  • (Paper 1) H.S.M. Coxeter, The Nine Regular Solids [Proc. Can. Math. Congress 1 (1947), 252–264, MR 8, 482]
  • (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) [Elemente der Mathematik 44 (2) (1989) 25–36]
• Theoni Pappas, (The Kepler–Poinsot Solids) The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 113, 1989.
• Louis Poinsot, Memoire sur les polygones et polyèdres. J. de l'École Polytechnique 9, pp. 16–48, 1810.
• Lakatos, Imre; Proofs and Refutations, Cambridge University Press (1976) - discussion of proof of Euler characteristic
• Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 0-521-54325-8., pp. 39–41.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 3)
• Anthony Pugh (1976). Polyhedra: A Visual Approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 8: Kepler Poisot polyhedra