Platonic solid

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: a regular tetrahedron (four triangular faces), a cube (six square faces), a regular octahedron (eight triangular faces), a regular dodecahedron (twelve pentagonal faces), and a regular icosahedron (twenty triangular faces).

Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the Timaeus, that the classical elements were made of these regular solids.

The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes. However, these balls have rounded knobs rather than being polyhedral. The number of knobs frequently differed from the numbers of vertices of the Platonic solids. No ball's knobs matched the 20 vertices of the dodecahedron, and the arrangement of the knobs was not always symmetrical.

The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.

The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus c. 360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "... the god used [it] for arranging the constellations on the whole heaven". Aristotle added a fifth element, aither (Latin: aether, 'aether' or 'ether' in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.

Euclid completely mathematically described the Platonic solids in the Elements, the last book (Book XIII) of which is devoted to their properties. Propositions 13-17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid, Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the five regular solids is the chief goal of the deductive system canonized in the Elements. Much of the information in Book XIII is probably derived from the work of Theaetetus.

Condensed: the full section is in Wikipedia.

For Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below. The Greek letter \(\varphi\) is used to represent the golden ratio \(\frac{1+\sqrt{5}}{2}\approx 1.6180\).

The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two positions such that each can be deduced from the other: in the case of the tetrahedron, by changing all coordinates of sign (central symmetry), or, in the other cases, by exchanging two coordinates (reflection with respect to any of the three diagonal planes).

These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as {4,3} or , one of two sets of 4 vertices in dual positions, as h{4,3} or . Both tetrahedral positions make the compound stellated octahedron.

The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t{3,4} or , also called a snub octahedron, as s{3,4} or , and seen in the compound of two icosahedra.

Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the compound of five cubes.

A convex polyhedron is a Platonic solid if and only if all three of the following requirements are met.

• All of its faces are congruent convex regular polygons.
• None of its faces intersect except at their edges.
• The same number of faces meet at each of its vertices.

Each Platonic solid can therefore be assigned a pair {p, q} of integers, where p is the number of edges (or, equivalently, vertices) of each face, and q is the number of faces (or, equivalently, edges) that meet at each vertex. This pair {p, q}, called the Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.

All other combinatorial information about these solids, such as total number of vertices (V), edges (E), and faces (F), can be determined from p and q. Since any edge joins two vertices and has two adjacent faces we must have:

\[pF = 2E = qV.\,\]

The other relationship between these values is given by Euler's formula:

\[V - E + F = 2.\,\]

This can be proved in many ways. Together these three relationships completely determine V, E, and F:

Condensed: the full section is in Wikipedia.

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.

The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate that no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question: one that requires an explicit construction.

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

1. Each vertex of the solid must be a vertex for at least three faces.
2. At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be strictly less than 360°. The amount less than 360° is called an angle defect.
3. Regular polygons of six or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. For these different shapes of faces the following holds:

   Triangular faces

   Each vertex of a regular triangle is 60°, so a shape may have three, four, or five triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.

   Square faces

   Each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.

   Pentagonal faces

   Each vertex is 108°; again, only one arrangement of three faces at a vertex is possible, the dodecahedron.

   Altogether this makes five possible Platonic solids.

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

\[\frac{2E}{q} - E + \frac{2E}{p} = 2.\]

Simple algebraic manipulation then gives

\[{1 \over q} + {1 \over p}= {1 \over 2} + {1 \over E}.\]

Since E is strictly positive we must have

\[\frac{1}{q} + \frac{1}{p} > \frac{1}{2}.\]

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}:

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

\[\sin(\theta/2) = \frac{\cos(\pi/q)}{\sin(\pi/p)}.\]

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

\[\tan(\theta/2) = \frac{\cos(\pi/q)}{\sin(\pi/h)}.\]

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

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

Condensed: the full section is in Wikipedia.

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

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

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

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

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:

\[\frac{R}{r} = \tan\left(\frac{\pi}{p}\right) \tan\left(\frac{\pi}{q}\right) = \frac{{\sqrt{{\csc^{2}}\Bigl(\frac\theta2\Bigr) - {\cos^{2}}\Bigl(\frac\alpha2\Bigr)}}}{\sin\Bigl(\frac{\alpha}{2}\Bigr)}.\]

Condensed: the full section is in Wikipedia.

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

\[S^{(2m)}_{[n]}= \frac 1n\sum_{i=1}^n d_i^{2m}\],

we have

\[\begin{aligned} S^{(2)}_{[4]} = S^{(2)}_{[6]} = S^{(2)}_{[8]}= S^{(2)}_{[12]}= S^{(2)}_{[20]} &= R^2+L^2, \\[4px] S^{(4)}_{[4]} = S^{(4)}_{[6]} = S^{(4)}_{[8]}= S^{(4)}_{[12]}= S^{(4)}_{[20]} &= \left(R^2+L^2\right)^2 + \frac 43 R^2L^2, \\[4px] S^{(6)}_{[6]} = S^{(6)}_{[8]} = S^{(6)}_{[12]}= S^{(6)}_{[20]}&= \left(R^2+L^2\right)^3 + 4R^2L^2 \left(R^2+L^2\right), \\[4px] S^{(8)}_{[12]} = S^{(8)}_{[20]} &= \left(R^2+L^2\right)^4 + 8R^2L^2 \left(R^2+L^2\right)^2+\frac {16}{5} R^4L^4, \\[4px] S^{(10)}_{[12]} = S^{(10)}_{[20]} &= \left(R^2+L^2\right)^5 +\frac {40}{3}R^2L^2\left(R^2+L^2\right)^3+16R^4L^4\left(R^2+L^2\right). \end{aligned}\] For all five Platonic solids, we have

\[S^{(4)}_{[n]}+\frac {16}{9}R^4= \left(S^{(2)}_{[n]}+ \frac 23R^2\right)^2.\]

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

\[4\left(\sum_{i=1}^n d_i^2\right)^2=3n \sum_{i=1}^n d_i^4.\]

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. All five Platonic solids have this property.

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

\[d^2 = R^\ast r = r^\ast R = \rho^\ast\rho.\]

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

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.