Platonic Solid

A solid with equivalent faces composed of congruent regular convex Polygons. There are exactly five such solids: the Cube, Dodecahedron, Icosahedron, Octahedron, and Tetrahedron, as was proved by Euclid in the last proposition of the Elements.

The Platonic solids were known to the ancient Greeks, and were described by Plato in his Timaeus ca. 350 BC . In this work, Plato equated the Tetrahedron with the ``element'' fire, the Cube with earth, the Icosahedron with water, the Octahedron with air, and the Dodecahedron with the stuff of which the constellations and heavens were made (Cromwell 1997).

The Platonic solids are sometimes also known as the Regular Polyhedra of Cosmic Figures (Cromwell 1997), although the former term is sometimes used to refer collectively to both the Platonic solids and Kepler-Poinsot Solids (Coxeter 1973).

If $P$ is a Polyhedron with congruent (convex) regular polygonal faces, then Cromwell (1997, pp. 77-78) shows that the following statements are equivalent.

1. The vertices of $P$ all lie on a Sphere.

2. All the Dihedral Angles are equal.

3. All the Vertex Figures are Regular Polygons.

4. All the Solid Angles are equivalent.

5. All the vertices are surrounded by the same number of Faces.

Let $v$ (sometimes denoted $N_0$) be the number of Vertices, $e$ (or $N_1$) the number of Edges, and $f$ (or $N_2$) the number of Faces. The following table gives the Schläfli Symbol, Wythoff Symbol, and C&R symbol, the number of vertices $v$, edges $e$, and faces $f$, and the Point Groups for the Platonic solids (Wenninger 1989).

Solid	Schläfli Symbol	Wythoff Symbol	C&R Symbol	$v$	$e$	$f$	Group
Cube	$\{4,3\}$	3 $\vert$ 2 4	$4^3$	8	12	6	$O_h$
Dodecahedron	$\{5,3\}$	3 $\vert$ 2 5	$5^3$	20	30	12	$I_h$
Icosahedron	$\{3,5\}$	5 $\vert$ 2 3	$3^5$	12	30	20	$I_h$
Octahedron	$\{3,4\}$	4 $\vert$ 2 3	$3^4$	6	12	8	$O_h$
Tetrahedron	$\{3,3\}$	3 $\vert$ 2 3	$3^3$	4	6	4	$T_d$

Let $r$ be the Inradius, $\rho$ the Midradius, and $R$ the Circumradius. The following two tables give the analytic and numerical values of these distances for Platonic solids with unit side length.

Solid	$r$	$\rho$	$R$
Cube	${\textstyle{1\over 2}}$	${\textstyle{1\over 2}}\sqrt{2}$	${\textstyle{1\over 2}}\sqrt{3}$
Dodecahedron	${\textstyle{1\over 20}}\sqrt{250+110\sqrt{5}}$	${\textstyle{1\over 4}}(3+\sqrt{5}\,)$	${\textstyle{1\over 4}}(\sqrt{15}+\sqrt{3}\,)$
Icosahedron	${\textstyle{1\over 12}}(3\sqrt{3}+\sqrt{15}\,)$	${\textstyle{1\over 4}}(1+\sqrt{5}\,)$	${\textstyle{1\over 4}}\sqrt{10+2\sqrt{5}}$
Octahedron	${\textstyle{1\over 6}}\sqrt{6}$	${\textstyle{1\over 2}}$	${\textstyle{1\over 2}}\sqrt{2}$
Tetrahedron	${\textstyle{1\over 12}}\sqrt{6}$	${\textstyle{1\over 4}}\sqrt{2}$	${\textstyle{1\over 4}}\sqrt{6}$

Solid	$r$	$\rho$	$R$
Cube	0.5	0.70711	0.86603
Dodecahedron	1.11352	1.30902	1.40126
Icosahedron	0.75576	0.80902	0.95106
Octahedron	0.40825	0.5	0.70711
Tetrahedron	0.20412	0.35355	0.61237

Finally, let $A$ be the Area of a single Face, $V$ be the Volume of the solid, the Edges be of unit length on a side, and $\alpha$ be the Dihedral Angle. The following table summarizes these quantities for the Platonic solids.

Solid	$A$	$V$	$\alpha$
Cube	1	1	${\textstyle{1\over 2}}\pi$
Dodecahedron	${\textstyle{1\over 4}}\sqrt{25+10\sqrt{5}}$	${\textstyle{1\over 4}}(15+7\sqrt{5}\,)$	$\cos^{-1}(-{\textstyle{1\over 5}}\sqrt{5})$
Icosahedron	${\textstyle{1\over 4}}\sqrt{3}$	${\textstyle{5\over 12}}(3+\sqrt{5}\,)$	$\cos^{-1}(-{\textstyle{1\over 3}}\sqrt{5})$
Octahedron	${\textstyle{1\over 4}}\sqrt{3}$	${\textstyle{1\over 3}}a^3\sqrt{2}$	$\cos^{-1}(-{\textstyle{1\over 3}})$
Tetrahedron	${\textstyle{1\over 4}}\sqrt{3}$	${\textstyle{1\over 12}}\sqrt{2}$	$\cos^{-1}({\textstyle{1\over 3}})$

The number of Edges meeting at a Vertex is $2e/v$. The Schläfli Symbol can be used to specify a Platonic solid. For the solid whose faces are $p$-gons (denoted $\{p\}$), with $q$ touching at each Vertex, the symbol is $\{p,q\}$. Given $p$ and $q$, the number of Vertices, Edges, and faces are given by

$$N_0 = {4p\over 4-(p-2)(q-2)}$$

$$N_1 = {2pq\over 4-(p-2)(q-2)}$$

$$N_2 = {4q\over 4-(p-2)(q-2)}.$$

Minimal Surfaces for Platonic solid frames are illustrated in Isenberg (1992, pp. 82-83).

See also Archimedean Solid, Catalan Solid, Johnson Solid, Kepler-Poinsot Solid, Quasiregular Polyhedron, Uniform Polyhedron

References

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