Kepler–Poinsot polyhedron

Builds on Regular polytope Schläfli symbol Euler characteristic

Description

A Kepler–Poinsot polyhedron is a regular star polyhedron. It is non-convex, its faces are congruent regular polygons or regular star polygons, and the same number of faces meet at each vertex. Its symmetries act transitively on its flags. There are exactly four, given here with their Schläfli symbols, where $5/2$ denotes the pentagram:

the small stellated dodecahedron $\set{5/2, 5}$ ;
the great dodecahedron $\set{5, 5/2}$ ;
the great stellated dodecahedron $\set{5/2, 3}$ ;
the great icosahedron $\set{3, 5/2}$ .

Remarks

Their vertices, edges and faces number $(V, E, F) = (12, 30, 12)$ , $(12, 30, 12)$ , $(20, 30, 12)$ and $(12, 30, 20)$ in this order. So $V - E + F$ equals $-6$ for the first two and $2$ for the last two. The first two do not satisfy Euler's formula $V - E + F = 2$ , since their pentagram or crossing pentagon faces do not fit together into a surface shaped like a sphere. All four share the icosahedral symmetry group of the icosahedron and dodecahedron. They are regular in the flag sense of a regular polytope, but not convex. Kepler described the two stellated dodecahedra, Poinsot the other two, and Cauchy showed that there are no other regular star polyhedra.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`CKQRZVBGTS`
URI	`m2://CKQRZVBGTS`
Kind	object
Writing	W0 stub
MSC	51M20, 52B15
Instance of	Polyhedron
Uses	Regular polytope, Schläfli symbol, Euler characteristic
Same as	Wikidata Q1023978 Wikipedia Kepler–Poinsot polyhedron

Kepler–Poinsot polyhedron

Description

Remarks

Reading path

Further reading

What this page needs for W1

What links here