As a configuration

The elements of a polyhedron can be expressed in a configuration matrix. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole polyhedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Dual pairs of polyhedra have their configuration matrices rotated 180 degrees from each other.^[6]

Geometric proof

Polygon nets around a vertex

{3,3} Defect 180°	{3,4} Defect 120°	{3,5} Defect 60°	{3,6} Defect 0°
{4,3} Defect 90°	{4,4} Defect 0°	{5,3} Defect 36°	{6,3} Defect 0°
A vertex needs at least 3 faces, and an angle defect. A 0° angle defect will fill the Euclidean plane with a regular tiling. By Descartes' theorem, the number of vertices is 720°/defect.

The following geometric argument is very similar to the one given by Euclid in the Elements:

Topological proof

A purely topological proof can be made using only combinatorial information about the solids. The key is Euler's observation that V − E + F = 2, and the fact that pF = 2E = qV, where p stands for the number of edges of each face and q for the number of edges meeting at each vertex. Combining these equations one obtains the equation

Orthographic projections and Schlegel diagrams with Hamiltonian cycles of the vertices of the five platonic solids – only the octahedron has an Eulerian path or cycle, by extending its path with the dotted one

${\displaystyle {\frac {2E}{q}}-E+{\frac {2E}{p}}=2.}$

Simple algebraic manipulation then gives

${\displaystyle {1 \over q}+{1 \over p}={1 \over 2}+{1 \over E}.}$

Since E is strictly positive we must have

${\displaystyle {\frac {1}{q}}+{\frac {1}{p}}>{\frac {1}{2}}.}$

Using the fact that p and q must both be at least 3, one can easily see that there are only five possibilities for {p, q}:

{3, 3}, {4, 3}, {3, 4}, {5, 3}, {3, 5}.

Angles

There are a number of angles associated with each Platonic solid. The dihedral angle is the interior angle between any two face planes. The dihedral angle, θ, of the solid {p,q} is given by the formula

${\displaystyle \sin \left({\frac {\theta }{2}}\right)={\frac {\cos \left({\frac {\pi }{q}}\right)}{\sin \left({\frac {\pi }{p}}\right)}}.}$

This is sometimes more conveniently expressed in terms of the tangent by

${\displaystyle \tan \left({\frac {\theta }{2}}\right)={\frac {\cos \left({\frac {\pi }{q}}\right)}{\sin \left({\frac {\pi }{h}}\right)}}.}$

The quantity h (called the Coxeter number) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.

The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2π. The defect, δ, at any vertex of the Platonic solids {p,q} is

${\displaystyle \delta =2\pi -q\pi \left(1-{2 \over p}\right).}$

By a theorem of Descartes, this is equal to 4π divided by the number of vertices (i.e. the total defect at all vertices is 4π).

The three-dimensional analog of a plane angle is a solid angle. The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by

${\displaystyle \Omega =q\theta -(q-2)\pi .\,}$

This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron {p,q} is a regular q-gon.

The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4π steradians) divided by the number of faces. This is equal to the angular deficiency of its dual.

The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in steradians. The constant φ = 1 + √5/2 is the golden ratio.

Polyhedron	Dihedral angle (θ)	tan θ/2	Defect (δ)	Vertex solid angle (Ω)	Face solid angle
tetrahedron	70.53°	${\displaystyle 1 \over {\sqrt {2}}}$	${\displaystyle \pi }$	${\displaystyle \arccos \left({\frac {23}{27}}\right)\quad \approx 0.551286}$	${\displaystyle \pi }$
cube	90°	${\displaystyle 1}$	${\displaystyle \pi \over 2}$	${\displaystyle {\frac {\pi }{2}}\quad \approx 1.57080}$	${\displaystyle 2\pi \over 3}$
octahedron	109.47°	${\displaystyle {\sqrt {2}}}$	${\displaystyle {2\pi } \over 3}$	${\displaystyle 4\arcsin \left({1 \over 3}\right)\quad \approx 1.35935}$	${\displaystyle \pi \over 2}$
dodecahedron	116.57°	${\displaystyle \varphi }$	${\displaystyle \pi \over 5}$	${\displaystyle \pi -\arctan \left({\frac {2}{11}}\right)\quad \approx 2.96174}$	${\displaystyle \pi \over 3}$
icosahedron	138.19°	${\displaystyle \varphi ^{2}}$	${\displaystyle \pi \over 3}$	${\displaystyle 2\pi -5\arcsin \left({2 \over 3}\right)\quad \approx 2.63455}$	${\displaystyle \pi \over 5}$

Radii, area, and volume

Another virtue of regularity is that the Platonic solids all possess three concentric spheres:

• the circumscribed sphere that passes through all the vertices,
• the midsphere that is tangent to each edge at the midpoint of the edge, and
• the inscribed sphere that is tangent to each face at the center of the face.

The radii of these spheres are called the circumradius, the midradius, and the inradius. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius R and the inradius r of the solid {p, q} with edge length a are given by

${\displaystyle {\begin{aligned}R&={\frac {a}{2}}\tan \left({\frac {\pi }{q}}\right)\tan \left({\frac {\theta }{2}}\right)\\[3pt]r&={\frac {a}{2}}\cot \left({\frac {\pi }{p}}\right)\tan \left({\frac {\theta }{2}}\right)\end{aligned}}}$

where θ is the dihedral angle. The midradius ρ is given by

${\displaystyle \rho ={\frac {a}{2}}\cos \left({\frac {\pi }{p}}\right)\,{\csc }{\biggl (}{\frac {\pi }{h}}{\biggr )}}$

where h is the quantity used above in the definition of the dihedral angle (h = 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric in p and q:

${\displaystyle {\frac {R}{r}}=\tan \left({\frac {\pi }{p}}\right)\tan \left({\frac {\pi }{q}}\right)={\frac {\sqrt {{\csc ^{2}}{\Bigl (}{\frac {\theta }{2}}{\Bigr )}-{\cos ^{2}}{\Bigl (}{\frac {\alpha }{2}}{\Bigr )}}}{\sin {\Bigl (}{\frac {\alpha }{2}}{\Bigr )}}}.}$

The surface area, A, of a Platonic solid {p, q} is easily computed as area of a regular p-gon times the number of faces F. This is:

${\displaystyle A={\biggl (}{\frac {a}{2}}{\biggr )}^{2}Fp\cot \left({\frac {\pi }{p}}\right).}$

The volume is computed as F times the volume of the pyramid whose base is a regular p-gon and whose height is the inradius r. That is,

${\displaystyle V={\frac {1}{3}}rA.}$

The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, a, to be equal to 2.

Polyhedron, a = 2	Radius			Surface area, A	Volume
	In-, r	Mid-, ρ	Circum-, R		V	Unit edges
tetrahedron	${\displaystyle 1 \over {\sqrt {6}}}$	${\displaystyle 1 \over {\sqrt {2}}}$	${\displaystyle {\sqrt {3 \over 2}}}$	${\displaystyle 4{\sqrt {3}}}$	${\displaystyle {\frac {\sqrt {8}}{3}}\approx 0.942809}$	${\displaystyle \approx 0.117851}$
cube	${\displaystyle 1\,}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle {\sqrt {3}}}$	${\displaystyle 24\,}$	${\displaystyle 8\,}$	${\displaystyle 1\,}$
octahedron	${\displaystyle {\sqrt {2 \over 3}}}$	${\displaystyle 1\,}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle 8{\sqrt {3}}}$	${\displaystyle {\frac {\sqrt {128}}{3}}\approx 3.771236}$	${\displaystyle \approx 0.471404}$
dodecahedron	${\displaystyle {\frac {\varphi ^{2}}{\xi }}}$	${\displaystyle \varphi ^{2}}$	${\displaystyle {\sqrt {3}}\,\varphi }$	${\displaystyle 12{\sqrt {25+10{\sqrt {5}}}}}$	${\displaystyle {\frac {20\varphi ^{3}}{\xi ^{2}}}\approx 61.304952}$	${\displaystyle \approx 7.663119}$
icosahedron	${\displaystyle {\frac {\varphi ^{2}}{\sqrt {3}}}}$	${\displaystyle \varphi }$	${\displaystyle \xi \varphi }$	${\displaystyle 20{\sqrt {3}}}$	${\displaystyle {\frac {20\varphi ^{2}}{3}}\approx 17.453560}$	${\displaystyle \approx 2.181695}$

The constants φ and ξ in the above are given by

${\displaystyle \varphi =2\cos {\pi \over 5}={\frac {1+{\sqrt {5}}}{2}},\qquad \xi =2\sin {\pi \over 5}={\sqrt {\frac {5-{\sqrt {5}}}{2}}}={\sqrt {3-\varphi }}.}$

Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume). The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.

Point in space

For an arbitrary point in the space of a Platonic solid with circumradius R, whose distances to the centroid of the Platonic solid and its n vertices are L and d_i respectively, and

${\displaystyle S_{[n]}^{(2m)}={\frac {1}{n}}\sum _{i=1}^{n}d_{i}^{2m}}$,

we have^[7]

${\displaystyle {\begin{aligned}S_{[4]}^{(2)}=S_{[6]}^{(2)}=S_{[8]}^{(2)}=S_{[12]}^{(2)}=S_{[20]}^{(2)}&=R^{2}+L^{2},\\[4px]S_{[4]}^{(4)}=S_{[6]}^{(4)}=S_{[8]}^{(4)}=S_{[12]}^{(4)}=S_{[20]}^{(4)}&=\left(R^{2}+L^{2}\right)^{2}+{\frac {4}{3}}R^{2}L^{2},\\[4px]S_{[6]}^{(6)}=S_{[8]}^{(6)}=S_{[12]}^{(6)}=S_{[20]}^{(6)}&=\left(R^{2}+L^{2}\right)^{3}+4R^{2}L^{2}\left(R^{2}+L^{2}\right),\\[4px]S_{[12]}^{(8)}=S_{[20]}^{(8)}&=\left(R^{2}+L^{2}\right)^{4}+8R^{2}L^{2}\left(R^{2}+L^{2}\right)^{2}+{\frac {16}{5}}R^{4}L^{4},\\[4px]S_{[12]}^{(10)}=S_{[20]}^{(10)}&=\left(R^{2}+L^{2}\right)^{5}+{\frac {40}{3}}R^{2}L^{2}\left(R^{2}+L^{2}\right)^{3}+16R^{4}L^{4}\left(R^{2}+L^{2}\right).\end{aligned}}}$

For all five Platonic solids, we have^[7]

${\displaystyle S_{[n]}^{(4)}+{\frac {16}{9}}R^{4}=\left(S_{[n]}^{(2)}+{\frac {2}{3}}R^{2}\right)^{2}.}$

If d_i are the distances from the n vertices of the Platonic solid to any point on its circumscribed sphere, then^[7]

${\displaystyle 4\left(\sum _{i=1}^{n}d_{i}^{2}\right)^{2}=3n\sum _{i=1}^{n}d_{i}^{4}.}$

Rupert property

A polyhedron P is said to have the Rupert property if a polyhedron of the same or larger size and the same shape as P can pass through a hole in P.^[8] All five Platonic solids have this property.^[8][9][10]

Dual polyhedra

Dual compounds

Every polyhedron has a dual (or "polar") polyhedron with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs.

• The tetrahedron is self-dual (i.e. its dual is another tetrahedron).
• The cube and the octahedron form a dual pair.
• The dodecahedron and the icosahedron form a dual pair.

If a polyhedron has Schläfli symbol {p, q}, then its dual has the symbol {q, p}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.

One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number of edges.

More generally, one can dualize a Platonic solid with respect to a sphere of radius d concentric with the solid. The radii (R, ρ, r) of a solid and those of its dual (R*, ρ*, r*) are related by

${\displaystyle d^{2}=R^{\ast }r=r^{\ast }R=\rho ^{\ast }\rho .}$

Dualizing with respect to the midsphere (d = ρ) is often convenient because the midsphere has the same relationship to both polyhedra. Taking d² = Rr yields a dual solid with the same circumradius and inradius (i.e. R* = R and r* = r).

Symmetry groups

In mathematics, the concept of symmetry is studied with the notion of a mathematical group. Every polyhedron has an associated symmetry group, which is the set of all transformations (Euclidean isometries) which leave the polyhedron invariant. The order of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the full symmetry group, which includes reflections, and the proper symmetry group, which includes only rotations.

The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is regular if and only if it is vertex-uniform, edge-uniform, and face-uniform.

There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are:

• the tetrahedral group T,
• the octahedral group O (which is also the symmetry group of the cube), and
• the icosahedral group I (which is also the symmetry group of the dodecahedron).

The orders of the proper (rotation) groups are 12, 24, and 60 respectively – precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron are centrally symmetric, meaning they are preserved under reflection through the origin.

The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parentheses (likewise for the number of symmetries). Wythoff's kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids.

Polyhedron	Schläfli symbol	Wythoff symbol	Dual polyhedron	Symmetry group (reflection, rotation)
				Polyhedral	Schön.	Cox.	Orb.	Order
tetrahedron	{3, 3}	3 | 2 3	tetrahedron	Tetrahedral	Td T	[3,3] [3,3]+	*332 332	24 12
cube	{4, 3}	3 | 2 4	octahedron	Octahedral	Oh O	[4,3] [4,3]+	*432 432	48 24
octahedron	{3, 4}	4 | 2 3	cube
dodecahedron	{5, 3}	3 | 2 5	icosahedron	Icosahedral	Ih I	[5,3] [5,3]+	*532 532	120 60
icosahedron	{3, 5}	5 | 2 3	dodecahedron

Liquid crystals with symmetries of Platonic solids

For the intermediate material phase called liquid crystals, the existence of such symmetries was first proposed in 1981 by H. Kleinert and K. Maki.^[12][13] In aluminum the icosahedral structure was discovered three years after this by Dan Shechtman, which earned him the Nobel Prize in Chemistry in 2011.

Uniform polyhedra

There exist four regular polyhedra that are not convex, called Kepler–Poinsot polyhedra. These all have icosahedral symmetry and may be obtained as stellations of the dodecahedron and the icosahedron.

cuboctahedron

icosidodecahedron

The next most regular convex polyhedra after the Platonic solids are the cuboctahedron, which is a rectification of the cube and the octahedron, and the icosidodecahedron, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both quasi-regular, meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen Archimedean solids, which are the convex uniform polyhedra with polyhedral symmetry. Their duals, the rhombic dodecahedron and rhombic triacontahedron, are edge- and face-transitive, but their faces are not regular and their vertices come in two types each; they are two of the thirteen Catalan solids.

The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of regular or star polygons for faces. These include all the polyhedra mentioned above together with an infinite set of prisms, an infinite set of antiprisms, and 53 other non-convex forms.

The Johnson solids are convex polyhedra which have regular faces but are not uniform. Among them are five of the eight convex deltahedra, which have identical, regular faces (all equilateral triangles) but are not uniform. (The other three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.)

Regular tessellations

Regular spherical tilings

Platonic
{3,3}	{4,3}	{3,4}	{5,3}	{3,5}
Regular dihedral
{2,2}	{3,2}	{4,2}	{5,2}	{6,2}...
Regular hosohedral
{2,2}	{2,3}	{2,4}	{2,5}	{2,6}...

The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the sphere. This is done by projecting each solid onto a concentric sphere. The faces project onto regular spherical polygons which exactly cover the sphere. Spherical tilings provide two infinite additional sets of regular tilings, the hosohedra, {2,n} with 2 vertices at the poles, and lune faces, and the dual dihedra, {n,2} with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra.

One can show that every regular tessellation of the sphere is characterized by a pair of integers {p, q} with 1/p + 1/q > 1/2. Likewise, a regular tessellation of the plane is characterized by the condition 1/p + 1/q = 1/2. There are three possibilities:

The three regular tilings of the Euclidean plane

{4, 4}	{3, 6}	{6, 3}

In a similar manner, one can consider regular tessellations of the hyperbolic plane. These are characterized by the condition 1/p + 1/q < 1/2. There is an infinite family of such tessellations.

Example regular tilings of the hyperbolic plane

{5, 4}	{4, 5}	{7, 3}	{3, 7}

Higher dimensions

In more than three dimensions, polyhedra generalize to polytopes, with higher-dimensional convex regular polytopes being the equivalents of the three-dimensional Platonic solids.

In the mid-19th century the Swiss mathematician Ludwig Schläfli discovered the four-dimensional analogues of the Platonic solids, called convex regular 4-polytopes. There are exactly six of these figures; five are analogous to the Platonic solids : 5-cell as {3,3,3}, 16-cell as {3,3,4}, 600-cell as {3,3,5}, tesseract as {4,3,3}, and 120-cell as {5,3,3}, and a sixth one, the self-dual 24-cell, {3,4,3}.

In all dimensions higher than four, there are only three convex regular polytopes: the simplex as {3,3,...,3}, the hypercube as {4,3,...,3}, and the cross-polytope as {3,3,...,4}.^[14] In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.