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Encyclopedia > Platonic solid

In geometry, a Platonic solid is a convex regular polyhedron. These are the three-dimensional analogs of the convex regular polygons. There are precisely five such figures (shown below). They are unique in that the faces, edges and angles are all congruent. For other uses, see Geometry (disambiguation). ... Look up Convex set in Wiktionary, the free dictionary. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... An example of congruence. ...

The Five Convex Regular Polyhedra (Platonic solids)
Tetrahedron Hexahedron
or Cube
Octahedron Dodecahedron Icosahedron

( Animation) A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A hexahedron is a polyhedron with six faces. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Tetrahedron. ... Image File history File links Tetrahedron. ...


( Animation) Image File history File links Hexahedron. ... Spinning hexahedron, made by me using POV-Ray, see image:poly. ...


( Animation) Image File history File links No higher resolution available. ... Spinning octahedron, made by me using POV-Ray, see image:poly. ...


( Animation) Image File history File links This is a lossless scalable vector image. ... Spinning dodecahedron, first 17 frames accidentally rendered with flashiness=1, while last 43 are rendered with flashiness=0. ...


( Animation) Image File history File links Icosahedron. ... Spinning icosahedron, made by me using POV-Ray, see image:poly. ...

The name of each figure is derived from the number of its faces: respectively 4, 6, 8, 12, and 20.[1]


The aesthetic beauty and symmetry of the Platonic solids have made them a favorite subject of geometers for thousands of years. They are named for the ancient Greek philosopher Plato who theorized the classical elements were constructed from the regular solids. An example of beauty in method - a simple and elegant proof of the Pythagorean theorem. ... Sphere symmetry group o. ... A geometer is a mathematician whose area of study is geometry. ... Greek philosophy focused on the role of reason and inquiry. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... . Bön . Hinduism (Tattva) and Buddhism (Mahābhūta) Prithvi / Bhumi — Earth Ap / Jala — Water Vayu / Pavan — Air / Wind Agni/Tejas — Fire Akasha — Aether Japanese (Godai) Earth (地) Water (水) Air / Wind (風) Fire (火) Void / Sky / Heaven (空) Chinese (Wu Xing) . Modern Many ancient philosophies used a set of archetypal classical elements to explain...

Contents

History

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

The Platonic solids have been known since antiquity. Ornamented models of them can be found among the carved stone balls created by the late neolithic people of Scotland at least 1000 years before Plato (Atiyah and Sutcliffe 2003). Download high resolution version (704x774, 142 KB)Keplers Platonic solid model of the Solar system from Mysterium Cosmographicum (1596). ... Download high resolution version (704x774, 142 KB)Keplers Platonic solid model of the Solar system from Mysterium Cosmographicum (1596). ... Mysterium Cosmographicum, (The Sacred Mystery of the Cosmos [Explained]) (alternately translated Cosmic Mystery, The Secret of the World or some variation) is an astronomy book by the German astronomer Johannes Kepler, published at Tübingen in 1596. ... A Carved Stone Ball from Towie, Aberdeenshire with exceptional decoration. ... An array of Neolithic artifacts, including bracelets, axe heads, chisels, and polishing tools. ... This article is about the country. ...


The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests 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 there are no other convex regular polyhedra. Ancient Greece is the term used to describe the Greek_speaking world in ancient times. ... This article is about Proclus Diadochus, the Neoplatonist philosopher. ... Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... Theaetetus (ca. ...


The Platonic solids feature prominently in the philosophy of Plato for whom they are named. 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. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithêr (aether in Latin, "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. PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... Timaeus (Greek: Τίμαιος, Timaios) is a theoretical treatise of Plato in the form of a Socratic dialogue, written circa 360 BC. The work puts forward speculation on the nature of the physical world. ... . Bön . Hinduism (Tattva) and Buddhism (MahābhÅ«ta) Prithvi / Bhumi — Earth Ap / Jala — Water Vayu / Pavan — Air / Wind Agni/Tejas — Fire Akasha — Aether Japanese (Godai) Earth (地) Water (æ°´) Air / Wind (風) Fire (火) Void / Sky / Heaven (空) Chinese (Wu Xing) . Modern Many ancient philosophies used a set of archetypal classical elements to explain... Chinese (Wu Xing) Japanese (Godai) Earth (地) | Water (æ°´) | Fire (火) | Air / Wind (風) | Void / Sky / Heaven (空) Hinduism (Tattva) and Buddhism (MahābhÅ«ta) Vayu / Pavan — Air / Wind Agni/Tejas — Fire Akasha — Aether Prithvi / Bhumi — Earth Ap / Jala — Water Bön Māori Earth, home and origin of humanity, has often been worshipped in... Chinese (Wu Xing) Japanese (Godai) Earth (地) | Water (æ°´) | Fire (火) | Air / Wind (風) | Void / Sky / Heaven (空) Hinduism (Tattva) and Buddhism (MahābhÅ«ta) Vayu / Pavan — Air / Wind Agni / Tejas — Fire Akasha — Aether Prithvi / Bhumi — Earth Ap / Jala — Water Bön New Zealand According to modern science, Earth’s atmosphere is a mixture of... Chinese Wood (木) | Fire (火) Earth (土) | Metal (金) | Water (æ°´) Japanese Earth (地) | Water (æ°´) | Fire (火) | Air / Wind (風) | Void / Sky / Heaven (空) Hinduism and Buddhism Vayu / Pavan — Air / Wind Agni / Tejas — Fire Akasha — Aether Prithvi / Bhumi — Earth Ap / Jala — Water Water has been important to all peoples of the earth, and it is rich in spiritual tradition. ... Chinese (Wu Xing) Japanese (Godai) Earth (地) | Water (æ°´) | Fire (火) | Air / Wind (風) | Void / Sky / Heaven (空) Hinduism (Tattva) and Buddhism (MahābhÅ«ta) Vayu / Pavan — Air / Wind Agni/Tejas — Fire Akasha — Aether Prithvi / Bhumi — Earth Ap / Jala — Water Bön Māori Fire has been important to all people of the earth, and... For other uses, see Aristotle (disambiguation). ... Hinduism (Tattva) and Buddhism (MahābhÅ«ta) Vayu / Pavan — Air / Wind Agni/Tejas — Fire Akasha — Aether Prithvi / Bhumi — Earth Ap / Jala — Water Chinese (Wu Xing) Japanese (Godai) Earth (地) | Water (æ°´) | Fire (火) | Air / Wind (風) | Void / Sky / Heaven (空) Bön Māori According to ancient and medieval science, Aether (Greek αἰθήρ, aithÄ“r[1...


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. Much of the information in Book XIII is probably derived from the work of Theaetetus. For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...


In the 16th century, the German astronomer Johannes Kepler attempted to find a relation between the five known planets at that time (excluding the Earth) 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. 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 the discovery of the Kepler solids, the realization that the orbits of planets are not circles, and Kepler's laws of planetary motion for which he is now famous. (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... An astronomer or astrophysicist is a person whose area of interest is astronomy or astrophysics. ... Kepler redirects here. ... This article is about the astronomical term. ... Mysterium Cosmographicum, (The Sacred Mystery of the Cosmos [Explained]) (alternately translated Cosmic Mystery, The Secret of the World or some variation) is an astronomy book by the German astronomer Johannes Kepler, published at Tübingen in 1596. ... Events February 5 - 26 catholics crucified in Nagasaki, Japan. ... This article is about the Solar System. ... This article is about the planet. ... (*min temperature refers to cloud tops only) Atmospheric characteristics Atmospheric pressure 9. ... This article is about Earth as a planet. ... Mars is the fourth planet from the Sun in the solar system, named after the Roman god of war (the counterpart of the Greek Ares), on account of its blood red color as viewed in the night sky. ... Atmospheric characteristics Atmospheric pressure 70 kPa Hydrogen ~86% Helium ~14% Methane 0. ... Atmospheric characteristics Atmospheric pressure 140 kPa Hydrogen >93% Helium >5% Methane 0. ... A single face is colored yellow and outlined in red to help identify the faces. ... Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ...

Combinatorial properties

A convex polyhedron is a Platonic solid if and only if

  1. all its faces are congruent convex 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 See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ...

p = the number of sides 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 Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below. In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ...

Polyhedron Vertices Edges Faces Schläfli symbol Vertex
configuration
tetrahedron 4 6 4 {3, 3} 3.3.3
cube 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

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: In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In polyhedral geometry a vertex configuration is a short-hand notation for representing a vertex as the sequence of faces around a vertex. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Tetrahedron. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Hexahedron. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links No higher resolution available. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... Image File history File links This is a lossless scalable vector image. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Icosahedron. ...

The other relationship between these values is given by Euler's formula: It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

This nontrivial fact can be proved in a great variety of ways (in algebraic topology it follows from the fact that the Euler characteristic of the sphere is 2). Together these three relationships completely determine V, E, and F: Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... For other uses, see Sphere (disambiguation). ...

Note that swapping p and q interchanges F and V while leaving E unchanged (For a geometric interpretation of this fact see the section on dual polyhedra below).


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 Euclid in the 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 six or 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.

A triangle. ... For other uses, see Square. ... Look up pentagon in Wiktionary, the free dictionary. ...

Topological proof

A purely topological proof can be made using only combinatorial information about the solids. The key is Euler's observation that VE + F = 2, and the fact that pF = 2E = qV. Combining these equations one obtains the equation A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

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 article is about angles in geometry. ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ...

This is sometimes more conveniently expressed in terms of the tangent by For other uses, see tangent (disambiguation). ...

The quantity h 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 In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. ...

By Descartes' theorem, this is equal to 4π divided by the number of vertices (i.e. the total defect at all vertices is 4π). In geometry, Descartes theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. ...


The 3-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 A solid angle is the three dimensional analog of the ordinary angle. ...

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. Right spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...


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. The steradian (ste from Greek stereos, solid) is the SI derived unit of solid angle, and the 3-dimensional equivalent of the radian. ... Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ...

Polyhedron Dihedral angle
Defect Solid angle
tetrahedron 70.53°
cube 90°
octahedron 109.47°
dodecahedron 116.57°
icosahedron 138.19°

In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. ... A solid angle is the three dimensional analog of the ordinary angle. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

Radii, area, and volume

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

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 A circumscribed sphere is a sphere that encloses another solid object, such as a polyhedron. ... A sphere containing on its surface one point from each edge of a semiregular or regular polyhedron. ... An inscribed sphere is a sphere that is enclosed by another solid object, such as a polyhedron. ... This article is about an authentication, authorization, and accounting protocol. ...

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

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

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: Area is the measure of how much exposed area any two dimensional object has. ...

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, For other uses, see Volume (disambiguation). ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ...

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)
r ρ R A V
tetrahedron
cube
octahedron
dodecahedron
icosahedron

The constants φ and ξ in the above are given by A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

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, the largest dihedral angle, and it hugs its inscribed sphere the tightest. 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.


Symmetry

Dual polyhedra

A dual cube-octahedron.

Every polyhedron has a dual 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. Image File history File links Dual_Cube-Octahedron. ... Image File history File links Dual_Cube-Octahedron. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...

  • 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. Polyhedra for which the dual polyhedron is a congruent figure. ...


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

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 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. Sphere symmetry group o. ... This picture illustrates how the hours on a clock form a group under modular addition. ... The symmetry group of an object (e. ... In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In erik, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. ... In the three-dimensional space, the possible moves of a rigid body are rotations and translations. ...


The symmetry groups of the Platonic solids are known as polyhedral groups (which are a special class of the point groups in three dimensions). 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. A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... In mathematics, a symmetry group describes all symmetries of objects. ... In grammar, a verb is transitive if it takes an object. ... In geometry, a polyhedron (or tiling) is vertex-uniform if all its vertices are the same, that is, if each vertex is surrounded by the same faces, in the same order. ... In geometry, a form is edge-uniform if its symmetries act transitively on its edges. ... In geometry, a polyhedron is face-uniform when all its faces have the same shape and size (technically, when all faces are congruent). ...


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. To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ...


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. In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ...

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 mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... The symmetry group of an object (e. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

In nature and technology

The tetrahedron, cube, and octahedron all occur naturally in crystal structures. These by no means exhaust the numbers of possible forms of crystals. However, neither the regular icosahedron nor the regular dodecahedron are amongst them. One of the forms, called the pyritohedron (named for the group of 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. Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... A pyritohedron is an irregular dodecahedron. ... The mineral pyrite, or iron pyrite, is iron sulfide, FeS2. ...

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. Circogonia Icosahedra from Haeckels 1904 Kunstformen der Natur. 157 by 175 pixels, 6172 bytes. ... Possible classes Polycystinea Acantharea Taxopodea Radiolaria are amoeboid protozoa that produce intricate mineral skeletons, typically with a central capsule dividing the cell into inner and outer portions, called endoplasm and ectoplasm. ... Ernst Haeckel. ... Possible classes Polycystinea Acantharea Taxopodea Radiolaria are amoeboid protozoa that produce intricate mineral skeletons, typically with a central capsule dividing the cell into inner and outer portions, called endoplasm and ectoplasm. ...


Many viruses, such as the herpes virus, have the shape of a regular icosahedron. Viral structures are built of repeated identical protein subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome. This article is about biological infectious particles. ... ... A representation of the 3D structure of myoglobin, showing coloured alpha helices. ... In biology the genome of an organism is the whole hereditary information of an organism that is encoded in the DNA (or, for some viruses, RNA). ...


In meteorology and climatology, global numerical models of atmospheric flow are of increasing interest which employ grids that are based on an icosahedron (refined by triangulation) instead of the more commonly used longitude/latitude grid. This has the advantage of evenly distributed spatial resolution without singularities (i.e. the poles) at the expense of somewhat greater numerical difficulty. // Meteorology (from Greek: μετέωρον, meteoron, high in the sky; and λόγος, logos, knowledge) is the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. ... Climatology is the study of climate, scientifically defined as weather conditions averaged over a period of time,[1] and is a branch of the atmospheric sciences. ... Triangulation can be used to find the distance from the shore to the ship. ... Longitude is the east-west geographic coordinate measurement most commonly utilized in cartography and global navigation. ... This article is about the geographical term. ... Singularity has several different meanings: mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways ill-behaved gravitational singularity - an infinity occurring in an astrophysical model, involving infinite curvature (a mathematical singularity) in the space/time continuum technological singularity - a predicted point in...


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. Simplified space frame roof with the half-octahedron highlighted in blue A space frame is a truss-like, lightweight rigid structure constructed from interlocking struts in a geometric pattern. ...


Platonic solids are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as dn where n is the number of faces (d8, d20, etc.); see dice notation for more details. Two standard six-sided pipped dice with rounded corners. ... This article is about games in which one plays the role of a character. ... A brass dice set. ...

These shapes frequently show up in other games or puzzles. Puzzles similar to a Rubik's Cube come in all five shapes — see magic polyhedra. Dice sold in sets are often identically colored, with matching die and marking colors. ... Dice sold in sets are often identically colored, with matching die and marking colors. ... Rolling dice Dice (the plural of the word die, probably from the Latin dare: to give) are, in general, small polyhedral objects with the faces marked with numbers or other symbols, thrown in order to choose one of the faces randomly. ... This article is about traditional role-playing games. ... Variations of Rubiks Cubes (from left to right: Rubiks Revenge, Rubiks Cube, Professors Cube, & Pocket Cube). ... Skewb Diamond Rubiks Cube Pocket Cube Pyraminx Magic polyhedra (also refered to as twisty puzzles) is a term for a specific type of puzzle, the most popular of which is the Rubiks Cube. ...


Related polyhedra and polytopes

Uniform polyhedra

There exist four regular polyhedra which are not convex, called Kepler-Poinsot polyhedra. These all have icosahedral symmetry and may be obtained as stellations of the dodecahedron and the icosahedron. A single face is colored yellow and outlined in red to help identify the faces. ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ...


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. Image File history File links This is a lossless scalable vector image. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Download high resolution version (841x861, 76 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...


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. A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...


The Johnson solids are convex polyhedra which have regular faces but are not uniform. The elongated square gyrobicupola (J37), a Johnson solid This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles. ...


Tessellations

The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as the five 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. 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 the condition 1/p + 1/q = 1/2. There are three possibilities: Plane tilings by regular polygons have been widely used since antiquity. ... For other uses, see Sphere (disambiguation). ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...

In a similar manner one can consider regular tessellations of the hyperbolic plane. These are characterized the condition 1/p + 1/q < 1/2. There is an infinite number of such tessellations. In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ...


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 geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... A dodecahedron, one of the five 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, while the sixth one, the 24-cell, has no lower-dimensional analogue. This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ...


In all dimensions higher than four, there are only three convex regular polytopes: the simplex, the hypercube, and the cross-polytope. In three dimensions, these coincide with the tetrahedron, the cube, and the octahedron. A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ...


See also

A dodecahedron, one of the five Platonic solids. ... This page lists the regular polytopes in Euclidean space. ... Metatron (Hebrew מטטרון or מיטטרון), is the name of an angel in Judaism and some branches of Christianity. ... The Flower of Life (click image for links to further images). ...

Notes

  1. ^ In the context of solid geometry the word regular is implied and usually omitted. The word irregular is also used to clarify that a polyhedron is not regular, although still assumed to have the same topology as the regular form. Other fully different topological forms, such as the rhombic dodecahedron which has 12 rhombic faces, or nonconvex star polyhedron, like the great dodecahedron, are never given with shortened names.

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. ... The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ... Two rhombi. ... In geometry, the term star polyhedron does not seem to have been propely defined, even though it is in common use. ... In geometry, the great dodecahedron is a Kepler-Poinsot solid. ...

References

  • Atiyah, Michael; and Sutcliffe, Paul (2003). "Polyhedra in Physics, Chemistry and Geometry". Milan J. Math 71: 33–58. 
  • Carl, Boyer; Merzbach, Uta (1989). A History of Mathematics, 2nd ed., Wiley. ISBN 0-471-54397-7. 
  • Coxeter, H. S. M. (1973). Regular Polytopes, 3rd ed., New York: Dover Publications. ISBN 0-486-61480-8. 
  • Euclid (1956). in Heath, Thomas L.: The Thirteen Books of Euclid's Elements, Books 10–13, 2nd unabr. ed., New York: Dover Publications. ISBN 0-486-60090-4. 
  • Haeckel, E. (1904). Kunstformen der Natur. Available as Haeckel, E. (1998); Art forms in nature, Prestel USA. ISBN 3-7913-1990-6, or online at [1].
  • Weyl, Hermann (1952). Symmetry. Princeton, NJ: Princeton University Press. ISBN 0-691-02374-3. 

Sir Michael Francis Atiyah, OM, FRS (b. ... Harold Scott MacDonald Donald Coxeter, CC , Ph. ... Stereographic projection of the 120-cell, a 4-dimensional regular polytope. ... For other uses, see Euclid (disambiguation). ... Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ...

External links


  Results from FactBites:
 
Platonic solid - Wikipedia, the free encyclopedia (1384 words)
Two-dimensional images of each of the Platonic solids are found within Metatron's Cube, a construct which originates from joining all the centres together from the Flower of Life.
The Platonic solids may be seen as increasingly better approximations to that sphere.
In terms of the "variation in altitude" (the ratio between the radius of the circumscribed sphere and the radius of the inscribed sphere), the Platonic solid that best fits the sphere is a tie between the icosahedron and the dodecahedron.
  More results at FactBites »

 
 

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