In three-dimensional space , a PLATONIC SOLID is a regular , convex polyhedron . It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria:
Four faces Six faces Eight faces Twelve faces Twenty faces (Animation ) (Animation ) (Animation ) (Animation ) (Animation ) Geometers have studied the mathematical beauty and symmetry of the
Platonic solids for thousands of years. They are named for the
ancient Greek philosopher
CONTENTS * 1 History * 2 Cartesian coordinates * 3 Combinatorial properties * 4 Classification * 4.1 Geometric proof * 4.2 Topological proof * 5 Geometric properties * 5.1 Angles * 5.2 Radii, area, and volume * 6.1 Dual polyhedra
* 6.2
* 7 In nature and technology * 7.1 Liquid crystals with symmetries of Platonic solids * 8 Related polyhedra and polytopes * 8.1 Uniform polyhedra * 8.2 Regular tessellations * 8.3 Higher dimensions * 9 See also * 10 References * 11 Sources * 12 External links HISTORY _ Kepler\'s
The Platonic solids have been known since antiquity. Carved stone
balls created by the late Neolithic people of
The ancient Greeks studied the Platonic solids extensively. Some
sources (such as
The Platonic solids are prominent in the philosophy of
Of the fifth Platonic solid, the dodecahedron,
In the 16th century, the German astronomer
In the 20th century, attempts to link Platonic solids to the physical world were expanded to the electron shell model in chemistry by Robert Moon in a theory known as the "Moon model ". CARTESIAN COORDINATES For Platonic solids centered at the origin, simple Cartesian coordinates are given below. The Greek letter _φ_ is used to represent the golden ratio 1 + √5/2. Cartesian coordinates FIGURE TETRAHEDRON OCTAHEDRON CUBE ICOSAHEDRON DODECAHEDRON FACES 4 8 6 20 12 VERTICES 4 6 (2 × 3) 8 12 (4 × 3) 20 (8 + 4 × 3) Orientation set 1 2 1 2 1 2 COORDINATES (1, 1, 1) (1, −1, −1) (−1, 1, −1) (−1, −1, 1) (−1, −1, −1) (−1, 1, 1) (1, −1, 1) (1, 1, −1) (±1, 0, 0) (0, ±1, 0) (0, 0, ±1) (±1, ±1, ±1) (0, ±1, ±_φ_) (±1, ±_φ_, 0) (±_φ_, 0, ±1) (0, ±_φ_, ±1) (±_φ_, ±1, 0) (±1, 0, ±_φ_) (±1, ±1, ±1) (0, ±1/_φ_, ±_φ_) (±1/_φ_, ±_φ_, 0) (±_φ_, 0, ±1/_φ_) (±1, ±1, ±1) (0, ±_φ_, ±1/_φ_) (±_φ_, ±1/_φ_, 0) (±1/_φ_, 0, ±_φ_) IMAGE The coordinates for the tetrahedron, icosahedron, and dodecahedron are given in two orientation sets, each containing half of the sign and position permutation of coordinates. These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as {4,3} or , one of two sets of 4 vertices in dual positions, as h{4,3} or . Both tetrahedral positions make the compound stellated octahedron . The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron , t{3,4} or _, also called a snub octahedron _, as s{3,4} or , and seen in the compound of two icosahedra . Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the compound of five cubes . COMBINATORIAL PROPERTIES A convex polyhedron is a
* all its faces are congruent convex regular polygons , * none of its faces intersect except at their edges, and * the same number of faces meet at each of its vertices . Each
The symbol {_p_, _q_}, called the
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: p F = 2 E = q V . {displaystyle pF=2E=qV.,} The other relationship between these values is given by Euler\'s formula : V E + F = 2. {displaystyle V-E+F=2.,} This can be proved in many ways. Together these three relationships completely determine _V_, _E_, and _F_: V = 4 p 4 ( p 2 ) ( q 2 ) , E = 2 p q 4 ( p 2 ) ( q 2 ) , F = 4 q 4 ( p 2 ) ( q 2 ) . {displaystyle V={frac {4p}{4-(p-2)(q-2)}},quad E={frac {2pq}{4-(p-2)(q-2)}},quad F={frac {4q}{4-(p-2)(q-2)}}.} Swapping _p_ and _q_ interchanges _F_ and _V_ while leaving _E_ unchanged. For a geometric interpretation of this property, see § Dual polyhedra below. CLASSIFICATION 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. GEOMETRIC PROOF Polygon nets around a vertex {3,3} Defect 180° {3,4} Defect 120° {3,5} Defect 60° {3,6} Defect 0° {4,3} Defect 90° {4,4} Defect 0° {5,3} Defect 36° {6,3} Defect 0° A vertex needs at least 3 faces, and an angle defect . A 0° angle defect will fill the Euclidean plane with a regular tiling. By Descartes\' theorem , the number of vertices is 720°/_defect_. The following geometric argument is very similar to the one given by
* 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 less
than 360°. The amount less than 360° is called an angle defect .
* The angles at all vertices of all faces of a
* 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. For these different shapes of faces the following holds: * 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. Altogether this makes 5 possible Platonic solids. TOPOLOGICAL PROOF A purely topological proof can be made using only combinatorial information about the solids. The key is Euler\'s observation that _V_ − _E_ + _F_ = 2, and the fact that _pF_ = 2_E_ = _qV_, where _p_ stands for the number of edges of each face and _q_ for the number of edges meeting at each vertex. Combining these equations one obtains the equation 2 E q E + 2 E p = 2. {displaystyle {frac {2E}{q}}-E+{frac {2E}{p}}=2.} Simple algebraic manipulation then gives 1 q + 1 p = 1 2 + 1 E . {displaystyle {1 over q}+{1 over p}={1 over 2}+{1 over E}.} Since _E_ is strictly positive we must have 1 q + 1 p > 1 2 . {displaystyle {frac {1}{q}}+{frac {1}{p}}>{frac {1}{2}}.} 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_}: {3, 3}, {4, 3}, {3, 4}, {5, 3}, {3, 5}. 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 sin 2 = cos ( q ) sin ( p ) . {displaystyle sin {theta over 2}={frac {cos left({frac {pi }{q}}right)}{sin left({frac {pi }{p}}right)}}.} This is sometimes more conveniently expressed in terms of the tangent by tan 2 = cos ( q ) sin ( h ) . {displaystyle tan {theta over 2}={frac {cos left({frac {pi }{q}}right)}{sin left({frac {pi }{h}}right)}}.} The quantity _h_ (called the
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 = 2 q ( 1 2 p ) . {displaystyle delta =2pi -qpi left(1-{2 over p}right).} 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 3-dimensional analog of a plane angle is a solid angle . The
solid angle, _Ω_, at the vertex of a
This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron {_p_,_q_} is a regular _q_-gon. The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4π steradians) divided by the number of faces. Note that this is equal to the angular deficiency of its dual. The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in steradians . The constant _φ_ = 1 + √5/2 is the golden ratio . POLYHEDRON
tetrahedron 70.53° 1 2 {displaystyle 1 over {sqrt {2}}} {displaystyle pi } cos 1 ( 23 27 ) 0.551286 {displaystyle cos ^{-1}left({frac {23}{27}}right)quad approx 0.551286} {displaystyle pi } cube 90° 1 {displaystyle 1} 2 {displaystyle pi over 2} 2 1.57080 {displaystyle {frac {pi }{2}}quad approx 1.57080} 2 3 {displaystyle 2pi over 3} octahedron 109.47° 2 {displaystyle {sqrt {2}}} 2 3 {displaystyle {2pi } over 3} 4 sin 1 ( 1 3 ) 1.35935 {displaystyle 4sin ^{-1}left({1 over 3}right)quad approx 1.35935} 2 {displaystyle pi over 2} dodecahedron 116.57° {displaystyle varphi } 5 {displaystyle pi over 5} tan 1 ( 2 11 ) 2.96174 {displaystyle pi -tan ^{-1}left({frac {2}{11}}right)quad approx 2.96174} 3 {displaystyle pi over 3} icosahedron 138.19° 2 {displaystyle varphi ^{2}} 3 {displaystyle pi over 3} 2 5 sin 1 ( 2 3 ) 2.63455 {displaystyle 2pi -5sin ^{-1}left({2 over 3}right)quad approx 2.63455} 5 {displaystyle pi over 5} RADII, AREA, AND VOLUME Another virtue of regularity is that the Platonic solids all possess three concentric spheres: * the circumscribed sphere that passes through all the vertices, * the midsphere that is tangent to each edge at the midpoint of the edge, and * the inscribed sphere that is tangent to each face at the center of the face. 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 R = ( a 2 ) tan q tan 2 {displaystyle R=left({a over 2}right)tan {frac {pi }{q}}tan {frac {theta }{2}}} r = ( a 2 ) cot p tan 2 {displaystyle r=left({a over 2}right)cot {frac {pi }{p}}tan {frac {theta }{2}}} where _θ_ is the dihedral angle. The midradius _ρ_ is given by = ( a 2 ) cos ( p ) sin ( h ) {displaystyle rho =left({a over 2}right){frac {cos left({frac {pi }{p}}right)}{sin left({frac {pi }{h}}right)}}} 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_: R r = tan p tan q = sin 2 ( 2 ) cos 2 ( 2 ) sin ( 2 ) . {displaystyle {R over r}=tan {frac {pi }{p}}tan {frac {pi }{q}}={frac {sqrt {{sin ^{-2}{left({frac {theta }{2}}right)}}-{cos ^{2}{left({frac {alpha }{2}}right)}}}}{sin {left({frac {alpha }{2}}right)}}}.} The surface area , _A_, of a
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, V = 1 3 r A . {displaystyle V={tfrac {1}{3}}rA.} 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) INRADIUS (_R_) MIDRADIUS (_ρ_) CIRCUMRADIUS (_R_) SURFACE AREA (_A_) VOLUME (_V_) Volume (unit edges) tetrahedron 1 6 {displaystyle 1 over {sqrt {6}}} 1 2 {displaystyle 1 over {sqrt {2}}} 3 2 {displaystyle {sqrt {3 over 2}}} 4 3 {displaystyle 4{sqrt {3}}} 8 3 0.942809 {displaystyle {frac {sqrt {8}}{3}}approx 0.942809} 0.117851 {displaystyle approx 0.117851} cube 1 {displaystyle 1,} 2 {displaystyle {sqrt {2}}} 3 {displaystyle {sqrt {3}}} 24 {displaystyle 24,} 8 {displaystyle 8,} 1 {displaystyle 1,} octahedron 2 3 {displaystyle {sqrt {2 over 3}}} 1 {displaystyle 1,} 2 {displaystyle {sqrt {2}}} 8 3 {displaystyle 8{sqrt {3}}} 128 3 3.771236 {displaystyle {frac {sqrt {128}}{3}}approx 3.771236} 0.471404 {displaystyle approx 0.471404} dodecahedron 2 {displaystyle {frac {varphi ^{2}}{xi }}} 2 {displaystyle varphi ^{2}} 3 {displaystyle {sqrt {3}},varphi } 12 25 + 10 5 {displaystyle 12{sqrt {25+10{sqrt {5}}}}} 20 3 2 61.304952 {displaystyle {frac {20varphi ^{3}}{xi ^{2}}}approx 61.304952} 7.663119 {displaystyle approx 7.663119} icosahedron 2 3 {displaystyle {frac {varphi ^{2}}{sqrt {3}}}} {displaystyle varphi } {displaystyle xi varphi } 20 3 {displaystyle 20{sqrt {3}}} 20 2 3 17.453560 {displaystyle {frac {20varphi ^{2}}{3}}approx 17.453560} 2.181695 {displaystyle approx 2.181695} The constants _φ_ and _ξ_ in the above are given by = 2 cos 5 = 1 + 5 2 = 2 sin 5 = 5 5 2 = 5 1 4 1 2 . {displaystyle varphi =2cos {pi over 5}={frac {1+{sqrt {5}}}{2}}qquad xi =2sin {pi over 5}={sqrt {frac {5-{sqrt {5}}}{2}}}=5^{frac {1}{4}}varphi ^{-{frac {1}{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. SYMMETRY DUAL POLYHEDRA A dual pair: cube and octahedron. Every polyhedron has a dual (or "polar") polyhedron WITH FACES AND
VERTICES INTERCHANGED. The dual of every
* 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
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
Dualizing with respect to the midsphere (_d_ = _ρ_) is often convenient because the midsphere has the same relationship to both polyhedra. Taking _d_2 = _Rr_ yields 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 . The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral 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 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 . 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 tetrahedral group _T_, * the octahedral group _O_ (which is also the symmetry group of the cube), and * the icosahedral group _I_ (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 are _centrally symmetric,_ meaning they are preserved under reflection through the origin . The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parenthesis (likewise for the number of symmetries). Wythoff\'s kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids. POLYHEDRON Schläfli symbol Wythoff symbol Dual polyhedron SYMMETRY GROUP (REFLECTION, ROTATION) POLYHEDRAL SCHöN. COX. ORB. ORDER tetrahedron {3, 3} 3 2 3 tetrahedron Tetrahedral _ T_d _T_ + *332 332 24 12 cube {4, 3} 3 2 4 octahedron Octahedral _ O_h _O_ + *432 432 48 24 octahedron {3, 4} 4 2 3 cube dodecahedron {5, 3} 3 2 5 icosahedron Icosahedral _ I_h _I_ + *532 532 120 60 icosahedron {3, 5} 5 2 3 dodecahedron 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. Circogonia icosahedra, a species of radiolaria , shaped like a regular icosahedron . In the early 20th century,
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 . In meteorology and climatology , global numerical models of atmospheric flow are of increasing interest which employ geodesic 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. Geometry of space 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. Several
* A set of polyhedral dice. 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 d_n_ where _n_ is the number of faces (d8, d20, etc.); see dice notation for more details. These shapes frequently show up in other games or puzzles. Puzzles
similar to a Rubik\'s
LIQUID CRYSTALS WITH SYMMETRIES OF PLATONIC SOLIDS For the intermediate material phase called liquid crystals , the
existence of such symmetries was first proposed in 1981 by H. Kleinert
and K. Maki. In aluminum the icosahedral structure was discovered
three years after this by
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. cuboctahedron icosidodecahedron The next most regular convex polyhedra after the Platonic solids are the cuboctahedron , which is a rectification of the cube and the octahedron, and the icosidodecahedron , which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both _quasi-regular_, meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen Archimedean solids , which are the convex uniform polyhedra with polyhedral symmetry. The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of 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. The Johnson solids are convex polyhedra which have regular faces but are not uniform. REGULAR TESSELLATIONS Regular spherical tilings PLATONIC TILINGS {3,3} {4,3} {3,4} {5,3} {3,5} REGULAR DIHEDRAL TILINGS {2,2} {3,2} {4,2} {5,2} {6,2}... REGULAR HOSOHEDRAL TILINGS {2,2} {2,3} {2,4} {2,5} {2,6}... The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as 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. Spherical tilings provide two infinite additional sets of regular tilings, the hosohedra , {2,_n_} with 2 vertices at the poles, and lune faces, and the dual dihedra , {_n_,2} with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra. One can show that every regular tessellation of the sphere is characterized by a pair of integers {_p_, _q_} with 1/_p_ + 1/_q_ > 1/2. Likewise, a regular tessellation of the plane is characterized by the condition 1/_p_ + 1/_q_ = 1/2. There are three possibilities: The three regular tilings of the Euclidean plane {4, 4} {3, 6} {6, 3} In a similar manner one can consider regular tessellations of the hyperbolic plane . These are characterized by the condition 1/_p_ + 1/_q_ < 1/2. There is an infinite family of such tessellations. Example regular tilings of the hyperbolic plane {5, 4} {4, 5} {7, 3} {3, 7} HIGHER DIMENSIONS Further information:
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 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
In all dimensions higher than four, there are only three convex regular polytopes: the simplex as {3,3,...,3}, the hypercube as {4,3,...,3}, and the cross-polytope as {3,3,...,4}. In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}. SEE ALSO *
REFERENCES * ^ Gardner (1987):
* Boyer , Carl; 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 . *
* Gardner, |