Geometers have studied the Platonic solids for thousands of years.[1] They are named for the ancient Greek philosopherPlato, who hypothesized in one of his dialogues, theTimaeus, that theclassical elements were made of these regular solids.[2]
The Platonic solids have been known since antiquity. It has been suggested that certaincarved stone balls created by thelate Neolithic people ofScotland 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 whose knobs match the 20 vertices of the dodecahedron, and the arrangement of the knobs was not always symmetrical.[3]
Theancient Greeks studied the Platonic solids extensively. Some sources (such asProclus) creditPythagoras 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 toTheaetetus, 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 ofPlato, their namesake. Plato wrote about them in the dialogueTimaeusc. 360 B.C. in which he associated each of the fourclassical elements (earth,air,water, andfire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven".Aristotle added a fifth element,aither (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.[5]
Euclid completely mathematically described the Platonic solids in theElements, 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 theElements.[6] Much of the information in Book XIII is probably derived from the work of Theaetetus.
In the 16th century, the German astronomerJohannes Kepler attempted to relate the five extraterrestrialplanets known at that time to the five Platonic solids. InMysterium Cosmographicum, published in 1596, Kepler proposed a model of theSolar 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 ofSaturn. 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 histhree laws of orbital dynamics, the first of which was thatthe orbits of planets are ellipses rather than circles, changing the course of physics and astronomy.[7] He also discovered theKepler solids, which are twononconvex regular polyhedra.
For Platonic solids centered at the origin, simpleCartesian coordinates of the vertices are given below. The Greek letter is used to represent thegolden ratio.
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 compoundstellated octahedron.
None of its faces intersect except at their edges.
The same number of faces meet at each of itsvertices.
Each Platonic solid can therefore be assigned a pair {p, q} of integers, wherep is the number of edges (or, equivalently, vertices) of each face, andq is the number of faces (or, equivalently, edges) that meet at each vertex. This pair {p, q}, called theSchläfli symbol, gives acombinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.
All other combinatorial information about these solids, such as total number of vertices (V), edges (E), and faces (F), can be determined fromp andq. Since any edge joins two vertices and has two adjacent faces we must have:
The other relationship between these values is given byEuler's formula:
This can be proved in many ways. Together these three relationships completely determineV,E, andF:
Swappingp andq interchangesF andV while leavingE unchanged. For a geometric interpretation of this property, see§ Dual polyhedra.
The elements of a polyhedron can be expressed in aconfiguration 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.[8]
The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate 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.
A vertex needs at least 3 faces, and anangle defect. A 0° angle defect will fill the Euclidean plane with a regular tiling. ByDescartes' theorem, the number of vertices is 720°/defect.
The following geometric argument is very similar to the one given byEuclid in theElements:
Each vertex of the solid must be a vertex for at least three faces.
At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be strictly less than 360°. The amount less than 360° is called anangle defect.
Regular polygons ofsix or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. For these different shapes of faces the following holds:
Each vertex of a regular triangle is 60°, so a shape may have three, four, or five triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
A purelytopological proof can be made using only combinatorial information about the solids. The key isEuler's observation thatV − E + F = 2, and the fact thatpF = 2E = qV, wherep stands for the number of edges of each face andq for the number of edges meeting at each vertex. Combining these equations one obtains the equation
There are a number ofangles associated with each Platonic solid. Thedihedral 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 thetangent by
The quantityh (called theCoxeter number) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.
Theangular 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
By a theorem of Descartes, this is equal to 4π divided by the number of vertices (i.e. the total defect at all vertices is 4π).
The three-dimensional analog of a plane angle is asolid angle. The solid angle,Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by
The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4π 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 insteradians. The constantφ =1 +√5/2 is thegolden ratio.
themidsphere that is tangent to each edge at the midpoint of the edge, and
theinscribed sphere that is tangent to each face at the center of the face.
Theradii of these spheres are called thecircumradius, themidradius, and theinradius. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradiusR and the inradiusr of the solid {p, q} with edge lengtha are given by
whereθ is the dihedral angle. The midradiusρ is given by
whereh is the quantity used above in the definition of the dihedral angle (h = 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric inp andq:
Thesurface area,A, of a Platonic solid {p, q} is easily computed as area of a regularp-gon times the number of facesF. This is:
Thevolume is computed asF times the volume of thepyramid whose base is a regularp-gon and whose height is the inradiusr. That is,
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.
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 and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume). 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.
For an arbitrary point in the space of a Platonic solid with circumradiusR, whose distances to the centroid of the Platonic solid and itsn vertices areL anddi respectively, and
A polyhedronP is said to have theRupert property if a polyhedron of the same or larger size and the same shape asP can pass through a hole inP.[10]All five Platonic solids have this property.[10][11][12]
Every polyhedron has adual (or "polar") polyhedronwith 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 isself-dual (i.e. its dual is another tetrahedron).
The cube and the octahedron form a dual pair.
The dodecahedron and the icosahedron form a dual pair.
If a polyhedron has Schläfli symbol {p, q}, then its dual has the symbol {q, p}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.
One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number of edges.
More generally, one can dualize a Platonic solid with respect to a sphere of radiusd concentric with the solid. The radii (R, ρ, r) of a solid and those of its dual (R*, ρ*, r*) are related by
Dualizing with respect to the midsphere (d = ρ) is often convenient because the midsphere has the same relationship to both polyhedra. Takingd2 = Rr yields a dual solid with the same circumradius and inradius (i.e.R* = R andr* = r).
In mathematics, the concept ofsymmetry is studied with the notion of amathematical group. Every polyhedron has an associatedsymmetry group, which is the set of all transformations (Euclidean isometries) which leave the polyhedron invariant. Theorder of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between thefull symmetry group, which includesreflections, and theproper symmetry group, which includes onlyrotations.
The symmetry groups of the Platonic solids are a special class ofthree-dimensional point groups known aspolyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under theaction of the symmetry group, as are the edges and faces. One says the action of the symmetry group istransitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron isregular if and only if it isvertex-uniform,edge-uniform, andface-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:
theoctahedral groupO (which is also the symmetry group of the cube), and
theicosahedral groupI (which is also the symmetry group of the dodecahedron).
The orders of the proper (rotation) groups are 12, 24, and 60 respectively – precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron arecentrally symmetric, meaning they are preserved underreflection through the origin.
The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parentheses (likewise for the number of symmetries).Wythoff's kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids.
The tetrahedron, cube, and octahedron all occur naturally incrystal 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 thepyritohedron (named for the group ofminerals 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.Allotropes of boron and manyboron compounds, such asboron carbide, include discrete B12 icosahedra within their crystal structures.Carborane acids also have molecular structures approximating regular icosahedra.
In the early 20th century,Ernst Haeckel described a number of species ofRadiolaria, some of whose skeletons are shaped like various regular polyhedra. Examples includeCircoporus octahedrus,Circogonia icosahedra,Lithocubus geometricus andCircorrhegma dodecahedra. The shapes of these creatures should be obvious from their names.[13]
Manyviruses, such as theherpes[14] virus, have the shape of a regular icosahedron. Viral structures are built of repeated identicalprotein 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 viralgenome.
Inmeteorology andclimatology, global numerical models of atmospheric flow are of increasing interest which employgeodesic grids that are based on an icosahedron (refined bytriangulation) instead of the more commonly usedlongitude/latitude grid. This has the advantage of evenly distributed spatial resolution withoutsingularities (i.e. the poles) at the expense of somewhat greater numerical difficulty.
Geometry ofspace frames is often based on platonic solids. In the MERO system, Platonic solids are used for naming convention of various space frame configurations. For example,1/2O+T refers to a configuration made of one half of octahedron and a tetrahedron.
For the intermediate material phase calledliquid crystals, the existence of such symmetries was first proposed in 1981 byH. Kleinert and K. Maki.[15][16]In aluminum the icosahedral structure was discovered three years after this byDan Shechtman, which earned him theNobel Prize in Chemistry in 2011.
Platonic solids are often used to makedice, because dice of these shapes can be madefair. 6-sided dice are very common, but the other numbers are commonly used inrole-playing games. Such dice are commonly referred to as dn wheren is the number of faces (d8, d20, etc.); seedice notation for more details.
These shapes frequently show up in other games or puzzles. Puzzles similar to aRubik's Cube come in all five shapes – seemagic polyhedra.
Architects liked the idea of Plato's timelessforms that can be seen by the soul in the objects of the material world, but turned these shapes into more suitable for constructionsphere,cylinder,cone, andsquare pyramid.[17] In particular, one of the leaders ofneoclassicism,Étienne-Louis Boullée, was preoccupied with the architects' version of "Platonic solids".[18]
The next most regular convex polyhedra after the Platonic solids are thecuboctahedron, which is arectification of the cube and the octahedron, and theicosidodecahedron, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are bothquasi-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 thirteenArchimedean solids, which are the convexuniform polyhedra with polyhedral symmetry. Their duals, therhombic dodecahedron andrhombic triacontahedron, are edge- and face-transitive, but their faces are not regular and their vertices come in two types each; they are two of the thirteenCatalan solids.
The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types ofregular orstar polygons for faces. These include all the polyhedra mentioned above together with an infinite set ofprisms, an infinite set ofantiprisms, and 53 other non-convex forms.
TheJohnson solids are convex polyhedra which have regular faces but are not uniform. Among them are five of the eight convexdeltahedra, which have identical, regular faces (all equilateral triangles) but are not uniform. (The other three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.)
The threeregular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of thesphere. This is done by projecting each solid onto a concentric sphere. The faces project onto regularspherical polygons which exactly cover the sphere. Spherical tilings provide two infinite additional sets of regular tilings, thehosohedra, {2,n} with 2 vertices at the poles, andlune faces, and the dualdihedra, {n,2} with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra.
Every regular tessellation of the sphere is characterized by a pair of integers {p, q} with1/p + 1/q > 1/2. Likewise, a regular tessellation of the plane is characterized by the condition1/p + 1/q = 1/2. There are three possibilities:
In a similar manner, one can consider regular tessellations of thehyperbolic plane. These are characterized by the condition1/p + 1/q < 1/2. There is an infinite family of such tessellations.
In more than three dimensions, polyhedra generalize topolytopes, with higher-dimensional convexregular polytopes being the equivalents of the three-dimensional Platonic solids.
In the mid-19th century the Swiss mathematicianLudwig Schläfli discovered the four-dimensional analogues of the Platonic solids, calledconvex regular 4-polytopes. There are exactly six of these figures; five are analogous to the Platonic solids :5-cell as {3,3,3},16-cell as {3,3,4},600-cell as {3,3,5},tesseract as {4,3,3}, and120-cell as {5,3,3}, and a sixth one, the self-dual24-cell, {3,4,3}.
In all dimensions higher than four, there are only three convex regular polytopes: thesimplex as {3,3,...,3}, thehypercube as {4,3,...,3}, and thecross-polytope as {3,3,...,4}.[19] In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.
^Wildberg (1988): Wildberg discusses the correspondence of the Platonic solids with elements inTimaeus but notes that this correspondence appears to have been forgotten inEpinomis, 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.
Gardner, Martin (1987).The 2nd Scientific American Book of Mathematical Puzzles & Diversions, University of Chicago Press, Chapter 1: The Five Platonic Solids,ISBN0226282538
Kepler. JohannesStrena seu de nive sexangula (On the Six-Cornered Snowflake), 1611 paper by Kepler which discussed the reason for the six-angled shape of the snow crystals and the forms and symmetries in nature. Talks about platonic solids.
Lloyd, David Robert (2012). "How old are the Platonic Solids?".BSHM Bulletin: Journal of the British Society for the History of Mathematics.27 (3):131–140.doi:10.1080/17498430.2012.670845.S2CID119544202.
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