---
tag: CKQRZVBGTS
kind: object
name: Kepler–Poinsot polyhedron
msc: [51M20, 52B15]
instance_of: [0325SGSJAR]
uses: [F60PD912D5, MAE6BF4DFK, RNDDWDAFJ5]
sources:
  - url: https://en.wikipedia.org/wiki/Kepler%E2%80%93Poinsot_polyhedron
same_as:
  wikidata: Q1023978
  wikipedia: Kepler–Poinsot polyhedron
---

## Description

A *Kepler–Poinsot polyhedron* is a regular star [[polyhedron|0325SGSJAR]].
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|MAE6BF4DFK]], where $5/2$ denotes the pentagram:

1. the small stellated dodecahedron $\set{5/2, 5}$;
2. the great dodecahedron $\set{5, 5/2}$;
3. the great stellated dodecahedron $\set{5/2, 3}$;
4. 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|F60PD912D5]], but not convex.
Kepler described the two stellated dodecahedra, Poinsot the other two, and Cauchy showed that there are no other regular star polyhedra.

## Explanations

### L1 human

Extend the faces of a dodecahedron or an icosahedron until they meet again, and spiky star-shaped solids appear.
Four of these are as regular as the Platonic solids: every face, edge and corner looks like every other.
Their faces pass through each other, which is why they are not convex.