Regular polyhedra

Platonic solids

Convex polyhedra such that all faces are congruent regular polygons and such that all polyhedral angles at the vertices are regular and equal (Fig.1a–Fig.1e).

Figure: r080790a

Figure: r080790b

Figure: r080790c

Figure: r080790d

Figure: r080790e

In the Euclidean space there are five regular polyhedra, the data of which are given in Table 1, where the Schläfli symbol (cf. Polyhedron group) denotes the regular polyhedron with -gonal faces and -hedral angles'

<tbody> </tbody>
Figure Schläfli symbol # vertices # edges # faces

Tetrahedron 1a 4 6 4

Cube (hexahedron) 1b 8 12 6

Octahedron 1c 6 12 8

Dodecahedron 1d 20 30 12

Icosahedron 1e 12 30 20

Dual polyhedra, or reciprocal polyhedra, and are, by definition, those which transform into each other upon reciprocation with respect to any concentric sphere. The tetrahedron is dual to itself, the hexahedron to the octahedron and the dodecahedron to the icosahedron.
Polyhedra in spaces of more than three dimensions are called polytopes.

In there are six regular polytopes, the data of which are given in Table 2.'

<tbody> </tbody>
Schläfli symbol # vertices # edges # -dim. faces # -dim. faces

Simplex 5 10 10 5

4-Cube 16 32 24 8

16-cell 8 24 32 16

24-cell 24 96 96 24

120-cell 600 1200 720 120

600-cell 120 720 1200 600

In , , there are three regular polytopes: the analogues of the tetrahedron, the cube and the octahedron; their Schläfli symbols are: , and .
If one permits self-intersection, then there are more regular polyhedra, namely the Kepler–Poinsot solids or regular star polyhedra. In these polyhedra, either the faces intersect each other or the faces are self-intersecting polygons (Fig.2a–Fig.2d). The data of these solids are listed in Table 3.

Figure: r080790f

Figure: r080790g

Figure: r080790h

Figure: r080790i

'

<tbody> </tbody>
Figure # vertices # edges # faces

Great dodecahedron 2a 12 30 12

Small stellated dodecahedron 2b 12 30 12

Great stellated dodecahedron 2c 20 30 12

Great icosahedron 2d 12 30 20

References[edit]
[1] P.S. Alexandroff [P.S. Aleksandrov] (ed.) et al. (ed.) , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[2] L.A. Lyusternik, "Convex figures and polyhedra" , Moscow (1956) (In Russian)
[3] D.O. Shklyarskii, N.N. Chentsov, I.M. Yaglom, "The USSR Olympiad book: selected problems and theorems of elementary mathematics" , Freeman (1962) (Translated from Russian)
[4] H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1990)

Comments[edit]
At least 2 of the 4 Kepler–Poinsot solids were discovered long before Kepler and Poinsot's times: The small stellated dodecahedron by P. Uccello (around 1420) and the great dodecahedron in 1958 by W. Jamnitzer (cf. [a2], [a3]).
References[edit]
[a1] B. Grünbaum, "Regular polyhedra - old and new" Aequat. Math. , 16 (1970) pp. 1–20
[a2] G. Flede (ed.) , Shaping space , Birkhäuser (1988)
[a3] L. Saffaro, "Dai cinqui poliedri platonici all'infinito" Encicl. Sci. e Tecn. Mondadori , 76 (1976) pp. 474–484
[a4] L. Fejes Toth, "Regular figures" , Pergamon (1964) (Translated from German)

[1]	P.S. Alexandroff [P.S. Aleksandrov] (ed.) et al. (ed.) , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[2]	L.A. Lyusternik, "Convex figures and polyhedra" , Moscow (1956) (In Russian)
[3]	D.O. Shklyarskii, N.N. Chentsov, I.M. Yaglom, "The USSR Olympiad book: selected problems and theorems of elementary mathematics" , Freeman (1962) (Translated from Russian)
[4]	H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1990)

[a1]	B. Grünbaum, "Regular polyhedra - old and new" Aequat. Math. , 16 (1970) pp. 1–20
[a2]	G. Flede (ed.) , Shaping space , Birkhäuser (1988)
[a3]	L. Saffaro, "Dai cinqui poliedri platonici all'infinito" Encicl. Sci. e Tecn. Mondadori , 76 (1976) pp. 474–484
[a4]	L. Fejes Toth, "Regular figures" , Pergamon (1964) (Translated from German)