Somewhat Regular Polyhedra

Platonic Solids

The platonic solids are also known as the regular polyhedra. Note that the duals are also all platonic solids.

Polyhedra	Vertex Config	Vertices	Edges	Faces	Dual
tetrahedron	3,3,3	4	6	4 triangles	tetrahedron
octahedron	3,3,3,3	6	12	8 triangles	cube
cube	4,4,4	8	12	6 squares	octahedron
icosahedron	3,3,3,3,3	12	30	20 triangles	dodecahedron
dodecahedron	5,5,5	20	30	12 pentagons	icosahedron

The archimedean solids are also known as the semi-regular polyhedra. The duals are known as the catalan solids.

Polyhedra	Vertex Config	Vertices	Edges	Faces	Dual
truncated tetrahedron	3,6,6	12	18	8: 4t+4h	triakis tetrahedron
cuboctahedron	3,4,3,4	12	24	14: 8t+6s	rhombic dodecahedron
truncated cube	3,8,8	24	36	14: 8t+6o	small triakis octahedron
truncated octahedron	4,6,6	24	36	14: 6s+8h	tetrakis hexahedron
rhombicuboctahedron	3,4,4,4	24	48	26: 8t+18s	deltoidal icositetrahedron
snub cube	3,3,3,3,4	24	60	38: 32t+6s	pentagonal icositetrahedron
icosidodecahedron	3,5,3,5	30	60	32: 20t+12p	rhombic triacontahedron
rhombitruncated cuboctahedron	4,6,8	48	72	26: 12s+8h+6o	disdyakis dodecahedron
truncated dodecahedron	3,10,10	60	90	32: 20t+12d	triakis icosahedron
truncated icosahedron	5,6,6	60	90	32: 12p+20h	pentakis dodecahedron
rhombicosidodecahedron	3,4,5,4	60	120	62: 20t+30s+12p	deltoidal hexecontahedron
snub dodecahedron	3,3,3,3,5	60	150	92: 80t+12p	pentagonal hexecontahedron
rhombitruncated icosidodecahedron	4,6,10	120	180	62: 30s+20h+12d	disdyakis triacontahedron

The Golden Rhomb has an acute angle of 70° 32' and an obtuse angle of 109° 28'.

Polyhedra	Vertex Config	Vertices	Edges	Faces
Oblate Rhombic Hexahedron	2ooo + 6aao	8	12	6
Prolate Rhombic Hexahedron	2aaa + 6aoo	8	12	6
Oblate Rhombic Dodecahedron	4ooo + 4ooa + 4aaao + 2aaaa	14	24	12
Prolate Rhombic Dodecahedron	6aaaa + 8ooo	14	24	12
Rhombic Icosahedron	2aaaaa + 10ooo + 10aaao	22	40	20
Rhombic Triacontahedron	12aaaaa + 20ooo	32	60	30

Polyhedra	Class	Angle Deficiency
tetrahedron	P	180°
octahedron	P	120°
cube	P	90°
truncated tetrahedron	A	60°
cuboctahedron	A	60°
icosahedron	P	60°
dodecahedron	P	36°
truncated cube	A	30°
truncated octahedron	A	30°
rhombicuboctahedron	A	30°
snub cube	A	30°
icosidodecahedron	A	24°
rhombitruncated cuboctahedron	A	15°
truncated dodecahedron	A	12°
truncated icosihedron	A	12°
rhombicosidodecahedron	A	12°
snub dodecahedron	A	12°
rhombitruncated icosidodecahedron	A	6°

Angle Deficiency = 360° minus the sum of the polygon angles at a vertex.

Polyhedron	Category	Angle Deficiency	SA Base	Total SA	Stability
Triangular Prism	Prism	120	1	3.866	25.9
Cube	Platonic	90	1	6	16.7
Hexagonal Prism	Prism	60	2.598	11.196	23.2
Rhombic Dodecahedron	Catalan	31.59 or 77.88	0.943	11.314	8.3
Truncated Octahedron	Archimedean	30	2.598	26.785	9.7

Stability = Percentage ratio of surface area of base over total surface area.

Polyhedron	Number of Vertices	Cartesian Coordinates
Tetrahedron	4	(±1,±1,±1) but only odd octants
Octahedron	6	(±1,0,0), (0,±1,0), (0,0,±1)
Cube	8	(±1,±1,±1)
Icosahedron	12	(±Phi,±1,0), (±1,0,±Phi), (0,±Phi,±1)
Dodecahedron	20	(±Phi,±phi,0), (±phi,0,±Phi), (0,±Phi,±phi), (±1,±1,±1)
Truncated Tetrahedron	12	(±3,±1,±1) but only odd octants
Cuboctahedron	12	(±1,±1,0), (±1,0,±1), (0,±1,±1)
Truncated Cube	24	(±(√2-1),±1,±1), (±1,±(√2-1),±1), (±1,±1,±(√2-1))
Truncated Octahedron	24	(0,±1,±2), (0,±2,±1), (±1,0,±2), (±1,±2,0), (±2,0,±1), (±2,±1,0)
Rhombicuboctahedron	24	(±(√2+1),±1,±1), (±1,±(√2+1),±1), (±1,±1,±(√2+1))
Snub Cube	24	?
Icosidodecahedron	30	?
Rhombitruncated Cuboctahedron	48	?
Truncated Dodecahedron	60	?
Truncated Icosahedron	60	?
Rhombicosidodecahedron	60	?
Snub Dodecahedron	60	?
Rhombitruncated Icosidodecahedron	120	?
Rhombic Dodecahedron	8	(±2,0,0), (0,±2,0), (0,0,±2), (±1,±1,±1)

Phi = (√5 + 1) / 2
phi = (√5 − 1) / 2