Nine Regular Polyhedra

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There exist $9$ regular polyhedra.


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 $9$.