Quartic Johnson Solids | M.E. Irizarry-Gelpí

Sun 09 April 2017
Maths
#polyhedra

I am searching for polyhedra that have four edges at every vertex. The simplest ones I have found so far are the $n$-antiprisms, with $2n+2$ faces, $2n$ vertices, and $4n$ edges. Each convex polyhedron must satisfy the polyhedral formula:

$$V - E + F = 2$$

The triangular antiprism is the octahedron, which is a Platonic solid. There are four Archimedean solids that follow this rule too:

cuboctahedron (12 vertices)
rhombicuboctahedron (24 vertices)
icosidodecahedron (30 vertices)
rhombicosidodecahedron (60 vertices)

Besides the Platonic and Archimedean solids, you also have the Johnson solids. Here is a list of relevant Johnson solids:

$J_{15}$: Elongated square bipyramid (10 vertices)
$J_{27}$: Triangular orthobicupola (12 vertices)
$J_{57}$: Triaugmented hexagonal prism (15 vertices)
$J_{28}$: Square orthobicupola (16 vertices)
$J_{29}$: Square gyrobicupola (16 vertices)
$J_{35}$: Elongated triangular orthobicupola (18 vertices)
$J_{36}$: Elongated triangular gyrobicupola (18 vertices)
$J_{92}$: Triangular hebesphenorotunda (18 vertices)
$J_{30}$: Pentagonal orthobicupola (20 vertices)
$J_{31}$: Pentagonal gyrobicupola (20 vertices)
$J_{37}$: Elongated square gyrobicupola (24 vertices)
$J_{32}$: Pentagonal orthocupolarotunda (25 vertices)
$J_{33}$: Pentagonal gyrocupolarotunda (25 vertices)
$J_{34}$: Pentagonal orthobirotunda (30 vertices)
$J_{38}$: Elongated pentagonal orthobicupola (30 vertices)
$J_{39}$: Elongated pentagonal gyrobicupola (30 vertices)
$J_{40}$: Elongated pentagonal orthocupolarotunda (35 vertices)
$J_{41}$: Elongated pentagonal gyrocupolarotunda (35 vertices)
$J_{42}$: Elongated pentagonal orthobirotunda (40 vertices)
$J_{43}$: Elongated pentagonal gyrobirotunda (40 vertices)
$J_{72}$: Gyrate rhombicosidodecahedron (60 vertices)
$J_{73}$: Parabigyrate rhombicosidodecahedron (60 vertices)
$J_{74}$: Metabigyrate rhombicosidodecahedron (60 vertices)
$J_{75}$: Trigyrate rhombicosidodecahedron (60 vertices)

The number of vertices is always a multiple of 4 or 5. According to WolframAlpha, the polyhedral graph for the polyhedra with odd number of vertices has edge chromatic number equal to 5. So $J_{57}$, $J_{32}$, $J_{33}$, $J_{40}$, and $J_{41}$ do not work for the ABCD model.