Nine Regular Polyhedra

Theorem

There exist $9$ regular polyhedra.

Proof

From Five Platonic Solids, there exist $5$ regular polyhedra which are convex:

the regular tetrahedron

the cube

the regular octahedron

the regular dodecahedron

the regular icosahedron.

From Four Kepler-Poinsot Polyhedra:

There exist exactly four Kepler-Poinsot polyhedra:

$(1): \quad$ the small stellated dodecahedron

$(2): \quad$ the great stellated dodecahedron

$(3): \quad$ the great dodecahedron

$(4): \quad$ the great icosahedron.

All $4$ of the above are regular polyhedra which are non-convex.

making the total $9$.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$