In three-dimensional space, a
Tetrahedron Cube Octahedron Dodecahedron Icosahedron Four faces Six faces Eight faces Twelve faces Twenty faces (Animation) (3D model) (Animation) (3D model) (Animation) (3D model) (Animation) (3D model) (Animation) (3D model) Geometers have studied the mathematical beauty and symmetry of the
Platonic solids for thousands of years.[1] They are named for the
ancient Greek philosopher
Contents 1 History 2 Cartesian coordinates 3 Combinatorial properties 3.1 As a configuration 4 Classification 4.1 Geometric proof 4.2 Topological proof 5 Geometric properties 5.1 Angles 5.2 Radii, area, and volume 6 Symmetry 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[edit] Kepler's
Assignment to the elements in Kepler's Mysterium Cosmographicum The Platonic solids have been known since antiquity. It has been
suggested that certain carved stone balls created by the late
Neolithic people of
Parameters 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 Vertex 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[edit]
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
p is the number of edges (or, equivalently, vertices) of each face, and q is the number of faces (or, equivalently, edges) that meet 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. 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 One possible
The above as a two-dimensional planar graph 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. As a configuration[edit] The elements of the platonic solids can be expressed in a configuration matrix. The diagonal elements represent the number of vertices, edges, and faces (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. For example the top row shows a vertex has q edges and q faces incident, and the bottom row shows a face has p vertices, and p edges. And the middle row show an edge has 2 vertices, and 2 faces. Dual pairs of platonic solids have their configurations matrixes rotated 180 degrees from each other.[7] p,q Platonic configurations group order: g = 8pq/(4-(p-2)(q-2)) g=24 g=48 g=120 v e f v g/2q q q e 2 g/4 2 f p p g/2p 3,3 4 3 3 2 6 2 3 3 4 3,4 6 4 4 2 12 2 3 3 8 4,3 8 3 3 2 12 2 4 4 6 3,5 12 5 5 2 30 2 3 3 20 5,3 20 3 3 2 30 2 5 5 12 Classification[edit] 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[edit]
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
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[edit] 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 = 2E = 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[edit] Angles[edit] 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 θ 2 = cos ( π q ) sin ( π 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 θ 2 = cos ( π q ) sin ( π h ) . displaystyle tan theta over 2 = frac cos left( frac pi q right) sin left( frac pi h right) . The quantity h (called the Coxeter number) 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 δ = 2 π − q π ( 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
Ω = q θ − ( q − 2 ) π . displaystyle Omega =qtheta -(q-2)pi ., 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 Dihedral angle (θ) tan θ/2 Defect (δ) Vertex solid angle (Ω) Face solid angle tetrahedron 70.53° 1 2 displaystyle 1 over sqrt 2 π displaystyle pi cos − 1 ( 23 27 ) ≈ 0.551286 displaystyle cos ^ -1 left( frac 23 27 right)quad approx 0.551286 π displaystyle pi cube 90° 1 displaystyle 1 π 2 displaystyle pi over 2 π 2 ≈ 1.57080 displaystyle frac pi 2 quad approx 1.57080 2 π 3 displaystyle 2pi over 3 octahedron 109.47° 2 displaystyle sqrt 2 2 π 3 displaystyle 2pi over 3 4 sin − 1 ( 1 3 ) ≈ 1.35935 displaystyle 4sin ^ -1 left( 1 over 3 right)quad approx 1.35935 π 2 displaystyle pi over 2 dodecahedron 116.57° φ displaystyle varphi π 5 displaystyle pi over 5 π − tan − 1 ( 2 11 ) ≈ 2.96174 displaystyle pi -tan ^ -1 left( frac 2 11 right)quad approx 2.96174 π 3 displaystyle pi over 3 icosahedron 138.19° φ 2 displaystyle varphi ^ 2 π 3 displaystyle pi over 3 2 π − 5 sin − 1 ( 2 3 ) ≈ 2.63455 displaystyle 2pi -5sin ^ -1 left( 2 over 3 right)quad approx 2.63455 π 5 displaystyle pi over 5 Radii, area, and volume[edit] 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 π q tan θ 2 displaystyle R=left( a over 2 right)tan frac pi q tan frac theta 2 r = ( a 2 ) cot π p tan θ 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 ρ = ( a 2 ) cos ( π p ) sin ( π 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 π p tan π q = sin − 2 ( θ 2 ) − cos 2 ( α 2 ) sin ( α 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
A = ( a 2 ) 2 F p cot π p . displaystyle A=left( a over 2 right)^ 2 Fpcot frac pi p . 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)
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 φ 2 ξ displaystyle frac varphi ^ 2 xi φ 2 displaystyle varphi ^ 2 3 φ displaystyle sqrt 3 ,varphi 12 25 + 10 5 displaystyle 12 sqrt 25+10 sqrt 5 20 φ 3 ξ 2 ≈ 61.304952 displaystyle frac 20varphi ^ 3 xi ^ 2 approx 61.304952 ≈ 7.663119 displaystyle approx 7.663119 icosahedron φ 2 3 displaystyle frac varphi ^ 2 sqrt 3 φ displaystyle varphi ξ φ displaystyle xi varphi 20 3 displaystyle 20 sqrt 3 20 φ 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 φ = 2 cos π 5 = 1 + 5 2 ξ = 2 sin π 5 = 5 − 5 2 = 5 1 4 φ − 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[edit] Dual polyhedra[edit] Dual compounds 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
d 2 = R ∗ r = r ∗ R = ρ ∗ ρ . displaystyle d^ 2 =R^ ast r=r^ ast R=rho ^ ast rho . Dualizing with respect to the midsphere (d = ρ) is often
convenient because the midsphere has the same relationship to both
polyhedra. Taking d2 = Rr yields a dual solid with the same
circumradius and inradius (i.e. R* = R and
r* = r).
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
Polyhedral Schön. Cox. Orb. Order tetrahedron 3, 3 3 2 3 tetrahedron Tetrahedral Td T [3,3] [3,3]+ *332 332 24 12 cube 4, 3 3 2 4 octahedron Octahedral Oh O [4,3] [4,3]+ *432 432 48 24 octahedron 3, 4 4 2 3 cube dodecahedron 5, 3 3 2 5 icosahedron Icosahedral Ih I [5,3] [5,3]+ *532 532 120 60 icosahedron 3, 5 5 2 3 dodecahedron In nature and technology[edit] 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,
Tetrahedrane Cubane Dodecahedrane 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 dn 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
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[edit] 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[edit]
Further information: List of regular polytopes
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
Archimedean solid Catalan solid Deltahedron Johnson solid Goldberg solid Kepler solids List of regular polytopes Regular polytopes Regular skew polyhedron Toroidal polyhedron References[edit] ^ Gardner (1987):
Sources[edit] Atiyah, Michael; Sutcliffe, Paul (2003). "Polyhedra in Physics, Chemistry and Geometry". Milan J. Math. 71: 33–58. arXiv:math-ph/0303071 . doi:10.1007/s00032-003-0014-1. 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, Martin(1987). The 2nd Scientific American Book of Mathematical Puzzles & Diversions, University of Chicago Press, Chapter 1: The Five Platonic Solids, ISBN 0226282538 Haeckel, Ernst, E. (1904). Kunstformen der Natur. Available as Haeckel, E. (1998); Art forms in nature, Prestel USA. ISBN 3-7913-1990-6. Hecht, Laurence; Stevens, Charles B. (Fall 2004). "New Explorations with The Moon Model" (PDF). 21st Century Science and Technology. p. 58. Kepler. Johannes Strena 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. Kleinert, Hagen and Maki, K. (1981). "Lattice Textures in Cholesteric Liquid Crystals" (PDF). Fortschritte der Physik. 29 (5): 219–259. doi:10.1002/prop.19810290503 CS1 maint: Multiple names: authors list (link) 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. Pugh, Anthony (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Weyl, Hermann (1952). Symmetry. Princeton, NJ: Princeton University Press. ISBN 0-691-02374-3. Wildberg, Christian (1988). John Philoponus' Criticism of Aristotle's Theory of Aether. Walter de Gruyter. pp. 11–12. ISBN 9783110104462 External links[edit] Wikimedia Commons has media related to Platonic solids. Platonic solids at Encyclopaedia of Mathematics Weisstein, Eric W. "Platonic solid". MathWorld. Weisstein, Eric W. "Isohedron". MathWorld. Book XIII of Euclid's Elements. Interactive 3D Polyhedra in Java Solid Body Viewer is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format. Interactive Folding/Unfolding Platonic Solids in Java Paper models of the Platonic solids created using nets generated by Stella software Platonic Solids Free paper models(nets) Grime, James; Steckles, Katie. "Platonic Solids". Numberphile. Brady Haran. Teaching Math with Art student-created models Teaching Math with Art teacher instructions for making models Frames of Platonic Solids images of algebraic surfaces Platonic Solids with some formula derivations How to make four platonic solids from a cube v t e Convex polyhedra Platonic solids (regular) tetrahedron cube octahedron dodecahedron icosahedron Archimedean solids (semiregular or uniform) truncated tetrahedron cuboctahedron truncated cube truncated octahedron rhombicuboctahedron truncated cuboctahedron snub cube icosidodecahedron truncated dodecahedron truncated icosahedron rhombicosidodecahedron truncated icosidodecahedron snub dodecahedron Catalan solids (duals of Archimedean) triakis tetrahedron rhombic dodecahedron triakis octahedron tetrakis hexahedron deltoidal icositetrahedron disdyakis dodecahedron pentagonal icositetrahedron rhombic triacontahedron triakis icosahedron pentakis dodecahedron deltoidal hexecontahedron disdyakis triacontahedron pentagonal hexecontahedron Dihedral regular dihedron hosohedron Dihedral uniform prisms antiprisms duals: bipyramids trapezohedra Dihedral others pyramids truncated trapezohedra gyroelongated bipyramid cupola bicupola pyramidal frusta bifrustum rotunda birotunda Degenerate polyhed |