User:Fropuff/Tables/Platonic solids

This is a table of properties of the five convex regular polyhedra (also known as Platonic solids).

Name	Tetrahedron	Hexahedron	Octahedron	Dodecahedron	Icosahedron
Picture
Schläfli symbol	{3, 3}	{4, 3}	{3, 4}	{5, 3}	{3, 5}	$\{p,q\}$
Face polygon	triangle	square	triangle	pentagon	triangle	$\{p\}$
Vertex figure	triangle	triangle	square	triangle	pentagon	$\{q\}$
Vertices ($N_{0}$)	4	8	6	20	12	${\frac {4p}{4-(p-2)(q-2)}}$
Edges ($N_{1}$)	6	12		30		${\frac {2pq}{4-(p-2)(q-2)}}$
Faces ($N_{2}$)	4	6	8	12	20	${\frac {4q}{4-(p-2)(q-2)}}$
Euler characteristic ($\chi$)	2					2
Coxeter number ($h$)	4	6		10		${\frac {2(p+q+2)}{10-p-q}}$
Vertex configuration	3.3.3	4.4.4	3.3.3.3	5.5.5	3.3.3.3.3	$p^{q}$
Wythoff symbol	3 | 3 2	3 | 4 2	4 | 3 2	3 | 5 2	5 | 3 2	q | p 2
Coxeter diagram
Dual polyhedron	tetrahedron	octahedron	hexahedron	icosahedron	dodecahedron	$\{q,p\}$
Symmetry group	tetrahedral group [3, 3]    *332    Td    A3	octahedral group [4, 3]    *432    Oh    BC3		icosahedral group [5, 3]    *532    Ih    H3		$[p,q]$
Symmetries	24	48		120		$4N_{1}$
Rotation symmetry group	[3, 3]+    332    T	[4, 3]+    432    O		[5, 3]+    532    I		$[p,q]^{+}$
Rotational symmetries	12	24		60		$2N_{1}$
Face solid angle	${\frac {\tau }{2}}$	${\frac {\tau }{3}}$	${\frac {\tau }{4}}$	${\frac {\tau }{6}}$	${\frac {\tau }{10}}$	${\frac {2\tau }{N_{2}}}$
Angular deficiency ($\delta$)	${\frac {\tau }{2}}$	${\frac {\tau }{4}}$	${\frac {\tau }{3}}$	${\frac {\tau }{10}}$	${\frac {\tau }{6}}$	${\frac {2\tau }{N_{0}}}=\tau -q\tau \left({1 \over 2}-{1 \over p}\right)$
Dihedral angle ($\theta$)	≈ 70.53°	${\frac {\tau }{4}}$	≈ 109.47°	≈ 116.56°	≈ 138.19°
$\tan {\theta \over 2}$	$1 \over {\sqrt {2}}$	$1\,$	${\sqrt {2}}$	$\varphi$	$\varphi ^{2}$	${\frac {\cos \left({\frac {\tau }{2q}}\right)}{\sin \left({\frac {\tau }{2h}}\right)}}$
$\cos \theta \,$	$1 \over 3$	$0\,$	$-{1 \over 3}$	$-{1 \over {\sqrt {5}}}$	$-{{\sqrt {5}} \over 3}$
Solid angle ($\Omega$)	≈ 0.551286	${\frac {\tau }{4}}$	≈ 1.35935	≈ 2.96174	≈ 2.63455	$q\theta -(q-2){\tau \over 2}$
Standard vertex coordinates	(1,1,−1) (1,−1,1) (−1,1,1) (−1,−1,−1)	(±1,±1,±1)	(±1,0,0) (0,±1,0) (0,0,±1)	(±1,±1,±1) (0,±1/φ,±φ) (±1/φ,±φ,0) (±φ,0,±1/φ)	(0,±1,±φ) (±1,±φ,0) (±φ,0,±1)
Edge length ($a$)	$2{\sqrt {2}}$	$2\,$	${\sqrt {2}}$	$2/\varphi$	$2\,$
Circumradius ($r_{0}$)	${\sqrt {3}}$	${\sqrt {3}}$	$1\,$	${\sqrt {3}}$	$\xi \varphi$
Midradius ($r_{1}$)	$1\,$	${\sqrt {2}}$	$1 \over {\sqrt {2}}$	$\varphi$	$\varphi$
Inradius ($r_{2}$)	$1 \over {\sqrt {3}}$	$1\,$	$1 \over {\sqrt {3}}$	$\varphi /\xi$	$\varphi ^{2} \over 3$
Surface area	$8{\sqrt {3}}$	$24\,$	$4{\sqrt {3}}$	$60 \over {\xi \varphi }$	$20{\sqrt {3}}$
Volume	$8 \over 3$	$8\,$	$4 \over 3$	$4(\varphi +2)$	${20 \over 3}\varphi ^{2}$