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: the regular tetrahedron (four triangular faces), the cube (six square faces), the regular octahedron (eight triangular faces), the regular dodecahedron (twelve pentagonal faces), and the regular icosahedron (twenty triangular faces).
Geometers have studied the Platonic solids for thousands of years.Gardner (1987): Martin Gardner wrote a popular account of the five solids in his December 1958 Mathematical Games column in Scientific American. 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.
History 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 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, (, or 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.Wildberg (1988): Wildberg discusses the correspondence of the Platonic solids with elements in Timaeus but notes that this correspondence appears to have been forgotten in Epinomis, which he calls "a long step towards Aristotle's theory", and he points out that Aristotle's ether is above the other four elements rather than on an equal footing with them, making the correspondence less apposite.
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.
In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn. The six spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. He also discovered the Kepler solids, which are two nonconvex regular polyhedra.
Cartesian coordinates For Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below. The Greek letter is used to represent the golden ratio .
+ Parameters Figure Tetrahedron Octahedron Cube Icosahedron Dodecahedron Faces4862012 Vertices46 (2 × 3)812 (4 × 3)20 (8 + 4 × 3) Position 12 12 12 Vertex coordinates 0, ±φ)
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.
Combinatorial properties 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.
+Properties of Platonic solids PolyhedronVerticesEdgesFacesSchläfli symbolVertex configuration Regular tetrahedron 4 6 4 {3, 3} 3.3.3 Cube 8 12 6 {4, 3} 4.4.4 Regular octahedron 6 12 8 {3, 4} 3.3.3.3 Regular dodecahedron 20 30 12 {5, 3} 5.5.5 Regular icosahedron 12 30 20 {3, 5} 3.3.3.3.3
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:
The other relationship between these values is given by Euler's formula:
This can be proved in many ways. Together these three relationships completely determine V, E, and F:
Swapping p and q interchanges F and V while leaving E unchanged. For a geometric interpretation of this property, see .
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.
{p,q} Platonic configurations Group order: g = 8pq/(4 − (p − 2)(q − 2)) g = 24 g = 48 g = 120 v e fv g/2q q q e 2 g/4 2 f p p g/2p+ {3,3} 4 3 3 2 6 2 3 3 4+ {3,4} 6 4 4 2 12 2 3 3 8+ {4,3} 8 3 3 2 12 2 4 4 6+ {3,5} 12 5 5 2 30 2 3 3 20+ {5,3} 20 3 3 2 30 2 5 5 12
Classification 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.
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
Simple algebraic manipulation then gives
Since E is strictly positive we must have
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}:
Geometric properties 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
This is sometimes more conveniently expressed in terms of the tangent by
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 2. The defect, δ, at any vertex of the Platonic solids {p,q} is
By a theorem of Descartes, this is equal to 4 divided by the number of vertices (i.e. the total defect at all vertices is 4).
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
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 (4 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 = is the golden ratio.
Polyhedron Dihedral angle (θ) tan Defect (δ) Vertex solid angle (Ω) Face solid angle tetrahedron 70.53° cube 90° octahedron 109.47° dodecahedron 116.57° icosahedron 138.19°
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.