Solid of Plato

In Geometry, a solid of Plato is a regular Polyèdre convex. They are the three-dimensional analogues of the regular polygons convex. There are precisely five figures of this kind (shown below). They are single in the fact that the edge, the edges and the angles all are adequate.

The name of each solid is derived from the number of the faces the component: respectively 4,6,8,12 and 20.

Because of their esthetism and their Symmetry, the solids of Plato were a favorite subject of study of the Géomètre S since thousands of years. They were named in the honor of the Greek Philosophe Plato which made the theory stating that the traditional elements were built starting from the regular solids.

History

The solids of Plato were known since Antiquity. The five solids were certainly known Greek old and it is obvious that these figures were known quite front. The people Neolithic S of Scotland built stone models of the five solids at least: 1000 years before Plato (Atiyah and Sutcliffe 2003). These models are kept with the Ashmolean Museum with Oxford.

Taking into consideration Mathematical of ancient Greece, certain sources (such as Proclus) credit Pythagore with discovered with the five convex regular polyhedrons. Another obviousness suggests that it can have only been familiar with the tetrahedron, the cube and the dodecahedron and that the discovery of octahedral and the icosahedron belongs to Théétète, a contemporary of Plato. In all the cases, Théétète gave a mathematical description of the five solids and was that which transmitted the first known demonstration which there do not exist other convex regular polyhedrons.

The solids of Plato are very prominent in the Philosophie of Plato from which they were named. Plato wrote about them in the dialog '' Timée '' approx. .360 before J.C in which it associated each of the four traditional elements (the Ground, the Air, the Eau and the Feu) with a regular solid. The ground was assoiciée with the cube, the air with the octahedral one, water with the icosahedron and fire with the tetrahedron. There was a justification for these associations: the heat of fire seems pointed and like a dagger (like a little the tetrahedron). The air is consisted of the octahedral one; its tiny components are so soft that one can hardly feel them. Water, the icosahedron, escapes with the hand when it is seized as if it were made up small tiny balls. By contrast, a strongly spherical solid, the hexahedron (cubic) represents the ground. These small solids make dust when they are émiettés and break when one seizes oneself some, a great difference with the soft flow of water. For the fifth solid of Plato, the dodecahedron, Plato notice obscurely, " … the god used to arrange the constellations on all the sky ". Aristote added a fifth element, aithêr (aether in Latin, " éther" in French) and postulated that the skies were made of this element, but it did not have any interest to make it coincide with the fifth solid of Plato.

Euclide gave a complete mathematical description of the solids of Plato in the '' Élements ''; the last book (Book XIII) which is devoted to their properties. The 13&ndash proposals; 17 in Book XIII described the construction of the tetrahedron, of octahedral, the cube, the icosahedron and the dodecahedron in this order. For each solid, Euclide finds the report/ratio of the diameter to the sphere circumscribed with the length of the edges. In proposal 18, he argues that there do not exist more convex regular polyhedrons. Many information in Book XIII are probably derived from the work of Théétète.

With the 16th century, the German Astronome Johannes Kepler tried to find a relation between the five Planet S known at the time (by excluding the Earth) and the five solids of Plato. In the Mysterium Cosmographicum , published in 1596, Kepler presented a model of Solar system in which the five solids were fixed and one in another separated by a series from registered and circumscribed spheres. The six spheres corresponded each one to the planets (Mercure, Venus, the Ground, Mars, Jupiter and Saturn). The solids were ordered interior towards outside, the first being the octahedral one, followed icosahedron, dodecahedron, tetrahedron and finally the cube. In this manner, the structure of the solar system and the relations of distances between planets were dictated by the solids of Plato. Towards the end, the original idea of Kepler was given up, but from this research emerged the discovery of the solid of Kepler, the observation which the orbits of planets are not circles, and the laws of planetary movement of Kepler for which it is now famous.

Combinative properties

A convex polyhedron is a solid of Plato if and only if

1. All its faces are regular polygons convex adequate,
2. None of its faces are cut except on the edges and
3. the same number of face meet with each one of its top S.

Each solid of Plato can consequently be noted by a symbol { p , Q } where

p = the number of with dimensions of each face (or the number of top on each face) and

Q = the number of faces meeting at each top (or the number of edges meeting at each top).

The symbol { p , Q }, called the Symbol of Schläfli, gives a description Combinatoire polyhedron. The symbols of Schläfli of the five solids of Plato are given in the table below.

All other combinative information in connection with these solids, such as the full number of tops ( S ), of the edges ( has ) and of the faces ( F ) can be given starting from p and Q . Since any edge joint two tops and has two adjacent faces, we must have:

$pF\; =\; 2A\; =\; qS.\; \backslash ,$

The other relation between these values is given by the formula of Euler:

$S\; -\; HAS\; +\; F\; =\; 2.\; \backslash ,$

This not-commonplace fact can be shown of a large variety in manners (in algebraic Topologie it rises from this fact that the characteristic of Euler of the Sphère is 2). Settings together, these three relations determine S completely, has and F :

$S\; =\; \backslash \; frac\; \{4p\}\; \{4\; -\; (p-2)\; (q-2)\},\; \backslash \; quad\; has\; =\; \backslash \; frac\; \{2pq\}\; \{4\; -\; (p-2)\; (q-2)\},\; \backslash \; quad\; F\; =\; \backslash \; frac\; \{4q\}\; \{4\; -\; (p-2)\; (q-2)\}.$

Note: to exchange p and Q inverts F and S leaving has unchanged (for a geometrical interpretation so to see the section on the duaux polyhedrons below).

Classification

It is a traditional result which there exist only five convex regular polyhedrons. Two common arguments are given below. Both show only that there cannot be more than five solids of Plato. That each of the five really exists is a séparée&mdash question; who can be answered by an explicit construction.

Geometrical demonstration

The following geometrical argument is very similar to that given by Euclide in the Élements :

1. Each top of the solid must coincide with a top each one on at least three faces.
2. At each top of the solid, the total, among the adjacent faces, of the angles between their with dimensions adjacent respective must be less 360°.
3. the angles of all the tops of all the faces of a solid of Plato are identical, therefore each top of each face must contribute less than 360°/3=120°.
4. the dimensioned regular polygons of six or more have only angles of 120° or more, therefore the common face must be triangle, the square or the pentagon. And for:
5. *les triangular faces : each top of a regular triangle has an angle of 60°, whose form must have 3,4 or 5 triangles meeting at a top; those are the tetrahedron, the octahedral one and the icosahedron respectively.
6. *les faces Carré be: each top of a square has an angle of 90°, therefore there exists only one possible arrangement with three faces at a top, the cube.
7. *les pentagonal faces : each top has an angle of 108°; again, only one arrangement, of three faces at a top is possible, the dodecahedron.

Topological demonstration

A topological demonstration purely can be given by using only combinative information on the solids. The key is the observation of Euler that $S\; -\; has\; +\; F\; =\; 2$, and the fact that $pF\; =\; 2A\; =\; qS$. By combining these equations, one obtains the equation

$\backslash \; frac\; \{2A\}\; \{Q\}\; -\; has\; +\; \backslash \; frac\; \{2A\}\; \{p\}\; =\; 2.$

A simple algebraic handling gives then

$\{1\; \backslash \; over\; Q\}\; +\; \{1\; \backslash \; over\; p\}\; =\; \{1\; \backslash \; over\; 2\}\; +\; \{1\; \backslash \; over\; has\}.$

Since $A$ is strictly positive, we must have

$\backslash \; frac\; \{1\}\; \{Q\}\; +\; \backslash \; frac\; \{1\}\; \{p\}\; >\; \backslash \; frac\; \{1\}\; \{2\}.$

By using the fact that p and Q must, both, being at least equal to 3, one can easily see that there exist only five possibilities for { p , Q }:

$\backslash \; \{3,\; 3\; \backslash \},\; \backslash \; quad\; \backslash \; \{4,\; 3\; \backslash \},\; \backslash \; quad\; \backslash \; \{3,\; 4\; \backslash \},\; \backslash \; quad\; \backslash \; \{5,\; 3\; \backslash \},\; \backslash \; quad\; \backslash \; \{3,5\; \backslash \}.$

Geometrical properties

Angles

There exists a number of Angle S associated with each solid with Plato. The Plane angle is the interior angle between two unspecified plane faces. The plane angle, θ, of the solid { p , Q } is given by the formula

$\backslash \; sin\; \{\backslash \; theta\; \backslash \; over\; 2\}\; =\; \backslash \; frac\; \{\backslash \; cos\; (\backslash \; pi/q)\}\{\backslash \; sin\; (\backslash \; pi/p)\}.$

This is sometimes expressed in a more practical way in terms of tangent by

$\backslash \; tan\; \{\backslash \; theta\; \backslash \; over\; 2\}\; =\; \backslash \; frac\; \{\backslash \; cos\; (\backslash \; pi/q)\}\{\backslash \; sin\; (\backslash \; pi/h)\}.$

The quantity H is 4,6,6,10 and 10 for the tetrahedron, the cube, the octahedral one, the dodecahedron and the icosahedron respectively.

The angular Déficience at the top of a polyhedron is the difference between the sum of the angles of a face and 2π. Deficiency, δ, at an unspecified top of the tops of Platons { p , Q } is

$\backslash \; delta\; =\; 2\; \backslash \; pi\; -\; Q\; \backslash \; pi\; \backslash \; left\; (1\; \{2\; \backslash \; over\; p\}\; \backslash \; right).$

By the Theorem of Descartes, this equal to 4π is divided by the number of tops (i.e the total deficiency of all the tops is 4π).

The three-dimensional analog of a plane angle is a solid Angle. The solid angle, Ω, to the top of a solid of Plato is given in term of angle diedric by

$\backslash \; Omega\; =\; Q\; \backslash \; theta\; -\; (q-2)\; \backslash \; pi.\; \backslash ,$

This comes from the formula from the spherical Excès for a spherical Polygone and the fact that the Figure of top of the polyhedron { p , Q } is a Q - gone regular.

The various angles associated with the solids with Plato are given below. The numerical values of the solid angles are given in Stéradian S. the constant $\backslash \; varphi\; =\; \backslash \; frac\; \{(1+\; \backslash \; sqrt\; \{5\})\}\{2\}\; \backslash ,$ is the Golden section.

Rays, surfaces and volumes

Another virtue of the regularity is that the solids of Plato have all three concentric spheres:

• the circumscribed Sphere which passes through all the tops,
• the average Sphère which is tangent with each edge in the middle of this one and
• the registered Sphère which is tangent with each vis-a-vis the center of this one.

The ray S of these spheres are called the circumscribed rays , the average radii and the internal rays . Those are the distances starting from the center of the polyhedron at the tops, the mediums of the edges and the centers of faces respectively. The ray circumscribed R and the ray intern R solid { p , Q } with a length of edge has are given by

$R\; =\; \backslash \; left\; (\{has\; \backslash \; over\; 2\}\; \backslash \; right)\; \backslash \; tan\; \backslash \; frac\; \{\backslash \; pi\}\; \{Q\}\; \backslash \; tan\; \backslash \; frac\; \{\backslash \; theta\}\; \{2\}$

$r\; =\; \backslash \; left\; (\{has\; \backslash \; over\; 2\}\; \backslash \; right)\; \backslash \; cot\; \backslash \; frac\; \{\backslash \; pi\}\; \{p\}\; \backslash \; tan\; \backslash \; frac\; \{\backslash \; theta\}\; \{2\}$

where θ is the plane angle. The average radius ρ is given by

$\backslash \; rho\; =\; \backslash \; left\; (\{has\; \backslash \; over\; 2\}\; \backslash \; right)\; \backslash \; frac\; \{\backslash \; cos\; (\backslash \; pi/p)\}\{\backslash \; sin\; (\backslash \; pi/h)\}$

where H is the quantity used above in the definition of the plane angle ( H = 4,6,6,10 or 10). To note that the report/ratio of the ray circumscribed with the rays internal is symmetrical in p and Q :

$\{R\; \backslash \; over\; R\}\; =\; \backslash \; tan\; \backslash \; frac\; \{\backslash \; pi\}\; \{p\}\; \backslash \; tan\; \backslash \; frac\; \{\backslash \; pi\}\; \{Q\}.$

The Superficie has of a solid of Plato { p , Q } is easily calculated, it is the surface of a p - gone regular time the number of faces F . I.e.:

$A\; =\; \backslash \; left\; (\{has\; \backslash \; over\; 2\}\; \backslash \; right)\; ^2\; FP\; \backslash \; cot\; \backslash \; frac\; \{\backslash \; pi\}\; \{p\}.$

The Volume is calculated as being F time the volume of the pyramid whose base is a p - gone regular and the height is the internal ray R . I.e.:

$V\; =\; \{1\; \backslash \; over\; 3\}\; rA.$

The following table lists the various rays of the solids of Plato like their surfaces and their volumes. The total size is fixed by taking the length of edge, has , equalizes to 2.

The constants φ and ξ above are given by

$\backslash \; varphi\; =\; 2\; \backslash \; cos\; \{\backslash \; pi\; \backslash \; over\; 5\}\; =\; \backslash \; frac\; \{1+\; \backslash \; sqrt\; 5\}\; \{2\}\; \backslash \; qquad\; \backslash \; xi\; =\; 2\; \backslash \; sin\; \{\backslash \; pi\; \backslash \; over\; 5\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{5\; \backslash \; sqrt\; 5\}\; \{2\}\}\; =\; 5^\; \{1/4\}\; \backslash \; varphi^\; \{-\; 1/2\}.$

Among the solids of Plato, the dodecahedron or the icosahedron can be looked like the best approximation of the sphere. The icosahedron has the greatest number of face, the greatest plane angle, and its envelope is closest to its registered sphere. The dodecahedron, of another with dimensions, with the smallest angular deflection, the greatest solid angle at the top and it fills more its circumscribed sphere.

Symmetry

Dual polyhedron

Each polyhedron has a dual Polyèdre with the interchanged faces and tops. The dual one of each solid of Plato is another solid of Plato, i.e. we can arrange the five solids in dual pairs.

• the tetrahedron is car-dual (i.e its dual is another tetrahedron).
• the cube and the octahedral one form a dual pair.
• the dodecahedron and the icosahedron forms a dual pair.

If a polyhedron has a symbol of Schläfli { p , Q }, then its dual has the symbol { Q , p }. Indeed, each combinative property of a solid of Plato can be interpreted like another combinative property of the dual one.

Groups of symmetry

In mathematics, the concept of Symétrie is studied with the mathematical concept of group. Each polyhedron has a associated Groupe of symmetry, which is the whole of all the transformations (Euclidean isométries) which leave the polyhedron invariant. The order of the group of symmetry is the number of symmetries of the polyhedron. One often makes a distinction between the total group of symmetry , which includes the reflections, and the clean group of symmetry , which includes only the rotations.

The groups of symmetry of the solids of Plato are known under the name of polyhedric groups (which are a particular class of the groups of point in three dimensions). The high degree of symmetry of the solids of Plato can be interpreted various manners. For most important, the tops of each top are all equivalent under the action of the group of symmetry, as are the edges and the faces. It is said that the action of the group of symmetry is transitive on the tops, the edges and the faces. In fact, it is another manner of defining the regularity of a polyhedron: a polyhedron is regular if and only if it is of uniform top, uniform edge and uniform face.

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

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

The orders of the clean groups (rotation) are 12,24 and 60 respectively — precisely, twice the number of the edges in the respective polyhedron. The orders of the total groups of symmetry are twice again the preceding orders (24, 48 and 120). See (Coxeter 1973) for a deduction of these facts.

The following table lists the various properties of symmetry of the solids of Plato. The listed groups of symmetry are the total groups with the sub-groups of rotation given between brackets (as for the number of symmetries). The kaleidoscopic construction of Wythoff is a method for construction of the polyhedrons directly starting from the groups of symmetry. Us listels the reference of the symbol of Wythoff for each solid of Plato.

In kind and in technology

The tetrahedron, the cube and the octahedral one naturally appear all in the crystalline structures. Those by no means exhaust the numbers of the crystal shapes possible. Nevertheless, neither the regular icosahedron, nor the regular dodecahedron appear among them. One of these forms, called the Pyritoèdre (named in connection with the group of the mineral with which it is typical) has twelve pentagonal faces, arranged with the same reason as the faces of the regular dodecahedron. Nevertheless, the faces of the pyritoèdre are not regular, therefore, the pyritoèdre is not either regular.

At the beginning of the 20th century, Ernst Haeckel described (Haeckel, 1904) the many ones of species of Radiolaire S, some comprising of the skeletons having the shape of various regular polyhedrons. Its examples include Circoporus octahedrus , Circogonia will icosahedra , Lithocubus geometricus and Circorrhegma will dodecahedra . The shapes of these crétures being obvious according to their names.

Many Virus, such as the virus of the Herpes, has the shape of a regular icosahedron. The viral structures are built on under-utités repeated identical Protéine S and the icosahedron is the easiest form to assemble by using these sub-units. A regular polyhedron is used because it can be built starting from a basic unit of protein used indefinitely, this generates a space in the viral Génome.

In Meteorology and Climatology, the total digital models of atmospheric flows are of an increasing interest. They use grids which are based on an icosahedron (refined by Triangulation) in the place of the grid Longitude/Latitude more commonly used. This with the advantage of having a space resolution also distributed without Singularity S (i.e the geographical poles) at the expense of a certain larger numerical difficulty.

The geometry of the reinforcements of space is often based on the solids of Plato. In system MERO, the solids of Plato are used for the convention of nomenclature of the various configurations of reinforcements of space. For example ½ O+T refers to a configuration made of half-octahedral and a tetrahedron.

The solids of Plato are often used to manufacture Dé S. the dice with 6 faces are very common, but the other numbers are commonly used in the roleplays. Such dice are often called D N where N is the number of faces (d8, d20, etc);

See also: Die

. These forms frequently appear in other plays or other puzzles. Puzzles similar to the Rubik' S Cube were born in all these forms — to see polyhedral magic.

Polyhedrons connected and polytopes

Uniform polyhedrons

There exist four regular polyhedrons which are not convex, called the solid of Kepler-Poinsot. Those have all the icosahedral Symétrie and can be obtained by Stellation S of the dodecahedron and the icosahedron.

The next convex polyhedrons most regular after the solids of Plato are the Cuboctaèdre, which are a correction cube and the octahedral one, and the Icosidodécaèdre, which is a correction of the dodecahedron and icosahedron (the correction of the car-dual polyhedron, the tetrahedron is octahedral regular). They are both quasi-regular what means that they are of top and edge uniforms and that they have regular faces, but the faces all are not adequate (coming from two different classes). They form two of the thirteen solid of Archimedes, which are polyhedral uniforms convex with a polyhedric symmetry.

The uniform polyhedrons form a class much larger polyhedrons. These solids are uniform tops and one one or more types of regular polygons or spangled for faces. Those include all the polyhedrons mentioned above with the infinite whole of the prisms, the infinite whole of the Antiprisme S like 53 other not-convex forms.

The solid of Johnson are convex polyhedrons which have regular faces but which are not uniform.

Pavings

The three regular pavings of the plan are strongly connected to the solids of Plato. Indeed, one can look at the solids of Plato like five regular pavings of the Sphère. This is carried out by projecting each solid on a concentric sphere. The faces project on spherical polygons regular which cover the sphere exactly. One can show that each regular paving of the sphere is characterized by a pair of entireties { p , Q } with 1 p + 1 Q > 1/2. In the same way, a regular paving of the plan is characterized by condition 1 p + 1 Q = 1/2. There exist three possibilities:

• {4, 4} which is the square Pavage,
• {3, 6} which is the triangular Pavage and
• {6, 3} which is the hexagonal Pavage (dual of triangular paving).

In a similar way, one can consider regular pavings on the hyperbolic Plan. They are characterized by condition 1 p + 1 Q < 1/2. There exists an infinite number of such pavings.

Higher dimensions

When there is more than three dimensions, the polyhedrons spread with the Polytope S. In the middle of the 19th century, the Mathématicien Suisse Ludwig Schläfli discovered the four-dimensional analogues of the solids of Plato, called the convex 4-polytopes regular. There exist exactly six of these figures; five are similar to the solids of Plato, while the sixth, the 24-cells, does not have an analog in lower dimension.

In dimensions higher than four, there exist only three polytopes regular convex: the Simplex, the Hypercube and the cross Polytope. In three dimensions, those coincide with the tetrahedron, the cube and the octahedral one.

See too

• regular Polytope
• List of the regular polytopes
• Cubic of Metatron - a symbol from which the solids of Plato can be derived
• Fleur de Vie - a historical and religious symbol from which the cube of Metatron can be derived

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