Platonic Solids – Wikipedia info

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In Euclidean geometry, a Platonic solid is a regular, convex polyhedron. The faces are congruent, regular polygons, with the same number of faces meeting at each vertex. There are exactly five solids which meet those criteria; each is named according to its number of faces.

Tetrahedron
(four faces)
hexahedron
(six faces)
Octahedron
(eight faces)
Dodecahedron
(twelve faces)
Icosahedron
(twenty faces)
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The classical elements were constructed from the regular solids.

History

Kepler’s Platonic solid model of the solar system from Mysterium Cosmographicum (1596)

The Platonic solids have been known since antiquity. Ornamented models resembling them can be found among the carved stone balls created by the late neolithic people of Scotland, although there seems to be no special attention paid to the Platonic solids over less symmetrical objects, and some of the five solids do not appear.[1] Dice go back to the dawn of civilization with shapes that augured formal charting of Platonic solids.

The 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 there are no other convex regular polyhedra.

The Platonic solids feature prominently in the philosophy of Pythagorus.

Euclid gave a complete mathematical description of 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 5 regular solids is the chief goal of the deductive system canonized in the Elements.[2] 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 find a relation between the five extraterrestrial planets known at that time and the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler laid out 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. In this way the structure of the solar system and the distance relationships between the planets was dictated 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.

In the 20th century, attempts to link Platonic solids to the physical world were expanded to the [3]

Combinatorial properties

A convex polyhedron is a Platonic solid if and only if

  1. all its faces are regular polygons,
  2. none of its faces intersect except at their edges, and
  3. the same number of faces meet at each of its vertices.

Each Platonic solid can therefore be denoted by a symbol {p, q} where

p = the number of edges of each face (or the number of vertices of each face) and
q = the number of faces meeting at each vertex (or the number of edges meeting at each vertex).

The symbol {p, q}, called the combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.

Polyhedron Vertices Edges Faces Schläfli symbol Vertex
configuration
tetrahedron   4 6 4 {3, 3} 3.3.3
hexahedron   8 12 6 {4, 3} 4.4.4
octahedron   6 12 8 {3, 4} 3.3.3.3
dodecahedron   20 30 12 {5, 3} 5.5.5
icosahedron   12 30 20 {3, 5} 3.3.3.3.3

Classification

It is a classical result that there are only five convex regular polyhedra. Two common arguments are given below. Both of these arguments only show that there can be no more than five Platonic solids. That all five actually exist is a separate question—one that can be answered by an explicit construction.

Geometric proof

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

  1. Each vertex of the solid must coincide with one vertex each of 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 less than 360°.
  3. The angles at all vertices of all faces of a Platonic solid are identical, so each vertex of each face must contribute less than 360°/3 = 120°.
  4. Regular polygons of sixor more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. And for:
    • Triangular faces: each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 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.

 

Geometric properties

Symmetry

Dual polyhedra

A dual pair: cube and octahedron.

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. The edges of the dual are formed by connecting the centers of adjacent faces in the original. In this way, the number of faces and vertices is interchanged, while the number of edges stays the same.

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^astrho.

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

Symmetry groups

In mathematics, the concept of rotations.

The symmetry groups of the Platonic solids are known as 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 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 parenthesis (likewise for the number of symmetries). Wythoff’s kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. We list for reference Wythoff’s symbol for each of the Platonic solids.

Polyhedron Schläfli symbol Wythoff symbol Dual polyhedron Symmetries Symmetry group
tetrahedron {3, 3} 3 | 2 3 tetrahedron 24 (12) Td (T)
cube {4, 3} 3 | 2 4 octahedron 48 (24) Oh (O)
octahedron {3, 4} 4 | 2 3 cube
dodecahedron {5, 3} 3 | 2 5 icosahedron 120 (60) Ih (I)
icosahedron {3, 5} 5 | 2 3 dodecahedron

In nature and technology

The tetrahedron, cube, and octahedron all occur naturally in minerals of which it is typical) has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.

Circogonia icosahedra, a species of Radiolaria, shaped like a regular icosahedron.

In the early 20th century, Ernst Haeckel described (Haeckel, 1904) a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra. Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra. The shapes of these creatures should be obvious from their names.

Many genome.

In poles) at the expense of somewhat greater numerical difficulty.

Geometry of space frames is often based on platonic solids. In MERO system, Platonic solids are used for naming convention of various space frame configurations. For example ½O+T refers to a configuration made of one half of octahedron and a tetrahedron.

Several dodecahedrane.

Platonic solids are often used to make dice notation for more details.

These shapes frequently show up in other games or puzzles. Puzzles similar to a magic polyhedra.

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 and their structure was analyzed in.[4] See the review article here. In aluminum the icosahedral structure was discovered three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011.

Related polyhedra and polytopes

Uniform polyhedra

There exist four regular polyhedra which are not convex, called 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.

The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of antiprisms, and 53 other non-convex forms.

The Johnson solids are convex polyhedra which have regular faces but are not uniform.

Tessellations

The three spherical polygons which exactly cover the sphere. 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:

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.

Higher dimensions

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

In the mid-19th century the Swiss mathematician Hexagon.

In all dimensions higher than four, there are only three convex regular polytopes: the cross-polytope. In three dimensions, these coincide with the tetrahedron, the cube, and the octahedron.

See also

Notes

  1. ^ Hart, George. “Neolithic Carved Stone Polyhedra”. http://www.georgehart.com/virtual-polyhedra/neolithic.html.
  2. ^ Weyl H. (1952). Symmetry. Princeton. p. 74.
  3. Hecht & Stevens 2004
  4. ^ Kleinert, H. and Maki, K. (1981), “Lattice Textures in Cholesteric Liquid Crystals”, Fortschritte der Physik 29 (5): 219–259, doi:10.1002/prop.19810290503, http://www.physik.fu-berlin.de/~kleinert/75/75.pdf

References

External links

This article uses material from the Wikipedia article Platonic Solids, which is released under the Creative Commons Attribution-Share-Alike License 3.0.