Kepler-Poinsot Polyhedron — Definition, Formula & Examples
A Kepler-Poinsot polyhedron is one of the four regular star polyhedra — solids whose faces or vertex figures are star polygons, yet which still satisfy the strict definition of regularity (identical faces, identical vertices).
The Kepler-Poinsot polyhedra are the four non-convex regular polyhedra: the small stellated dodecahedron, the great stellated dodecahedron, the great dodecahedron, and the great icosahedron. Each is composed of congruent regular star-polygon faces or has star-polygon vertex figures, with full flag-transitive symmetry.
How It Works
Start with a Platonic solid such as the dodecahedron or icosahedron. By extending (stellating) the faces until they intersect again in a symmetric pattern, or by choosing star polygons as faces, you can build new solids that are still perfectly regular. Kepler discovered the first two in 1619 by stellating the dodecahedron. Poinsot found the remaining two in 1809 by using pentagrams as faces or vertex figures. Despite their self-intersecting surfaces, each one has a well-defined number of faces, edges, and vertices, and you can check them against Euler's formula for polyhedra (with an appropriate adjustment for the genus of the surface).
Worked Example
Problem: The small stellated dodecahedron has 12 pentagrammic faces, 30 edges, and 12 vertices. Verify whether the standard Euler formula V − E + F = 2 holds.
Identify counts: Faces F = 12, Edges E = 30, Vertices V = 12.
Apply Euler's formula: Compute V − E + F.
Interpret the result: The result is −6, not 2. This is expected because the small stellated dodecahedron is non-convex and self-intersecting. Its Euler characteristic corresponds to a surface of genus 4, giving χ = 2 − 2(4) = −6.
Answer: Euler's standard formula does not hold: V − E + F = −6, which is consistent with the topology of this star polyhedron (genus 4).
Why It Matters
The Kepler-Poinsot solids show that regularity in geometry extends beyond convex shapes, deepening your understanding of symmetry. They appear in advanced geometry courses and crystallography, and they illustrate why Euler's formula needs careful handling on non-convex surfaces.
Common Mistakes
Mistake: Assuming all regular polyhedra are Platonic solids.
Correction: The five Platonic solids are the convex regular polyhedra. The four Kepler-Poinsot polyhedra are also regular but non-convex, bringing the total count of regular polyhedra to nine.