There are many relations among the uniform polyhedra.... Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron. Others share the same vertices and edges as other polyhedron. The grouping below exhibit some of these relations.
The vertex figure of a polyhedron The relations can be made apparent by examining the vertex figures obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example, the cube has vertex figure 4.4.4, which is to say, three adjacent square faces. The possible faces are 3 – equilateral triangle 4 – square 5 – regular pentagon 6 – regular hexagon 8 – regular octagon 10 – regular decagon 5/2 – pentagram 8/3 – octagram 10/3 – decagram Some faces will appear with reverse orientation which is written here as −3 – a triangle with reverse orientation (often written as 3/2) Others pass through the origin which we write as 6* – hexagon passing through the origin
The Wythoff symbol relates the polyhedron to spherical triangles. Wythoff symbols are written p|q r, p q|r, p q r| where the spherical triangle has angles π/p,π/q,π/r, the bar indicates the position of the vertices in relation to the triangle.
Johnson (2000) classified uniform polyhedra according to the following: Regular (regular polygonal vertex figures): pq, Wythoff symbol q|p 2 Quasi-regular (rectangular or ditrigonal vertex figures): p.q.p.q 2|p q, or p.q.p.q.p.q, Wythoff symbol 3|p q Versi-regular (orthodiagonal vertex figures), p.q*.−p.q*, Wythoff symbol q q|p Truncated regular (isosceles triangular vertex figures): p.p.q, Wythoff symbol q 2|p Versi-quasi-regular (dipteroidal vertex figures), p.q.p.r Wythoff symbol q r|p Quasi-quasi-regular (trapezoidal vertex figures): p*.q.p*.−r q.r|p or p.q*.−p.q* p q r| Truncated quasi-regular (scalene triangular vertex figures), p.q.r Wythoff symbol p q r| Snub quasi-regular (pentagonal, hexagonal, or octagonal vertex figures), Wythoff symbol p q r| Prisms (truncated hosohedra), Antiprisms and crossed antiprisms (snub dihedra)
The format of each figure follows the same basic pattern image of polyhedron name of polyhedron alternate names (in brackets) Wythoff symbol Numbering systems: W – number used by Wenninger in polyhedra models, U – uniform indexing, K – Kaleido indexing, C – numbering used in Coxeter et al. 'Uniform Polyhedra'. Number of vertices V, edges E, Faces F and number of faces by type. Euler characteristic χ = V − E + F The vertex figures are on the left, followed by the Point groups in three dimensions#The seven remaining point groups, either tetrahedral Td, octahedral Oh or icosahedral Ih.
Truncated forms Regular polyhedra and their truncated forms Column A lists all the regular polyhedra, column B list their truncated forms. Regular polyhedra all have vertex figures pr: p.p.p etc. and Wythoff symbol pq r. The truncated forms have vertex figure q.q.r (where q=2p and r) and Wythoff p qr. vertex figure group A: regular: p.p.p B: truncated regular: p.p.r3.3.3 3.6.6 TdTetrahedron 32 3 W1, U01, K06, C15 V 4,E 6,F 4=4{3} χ=2Truncated tetrahedron 2 33 W6, U02, K07, C16 V 12,E 18,F 8=4{3}+4{6} χ=2 3.3.3.3 4.6.6 OhOctahedron 42 3, 34 W2, U05, K10, C17 V 6,E 12,F 8=8{3} χ=2Truncated octahedron 2 43 W7, U08, K13, C20 V 24,E 36,F 14=6{4}+8{6} χ=2 4.4.4 3.8.8 OhHexahedron (Cube) 32 4 W3, U06, K11, C18 V 8,E 12,F 6=6{4} χ=2Truncated hexahedron 2 34 W8, U09, K14, C21 V 24,E 36,F 14=8{3}+6{8} χ=23.3.3.3.3 5.6.6 IhIcosahedron 52 3 W4, U22, K27, C25 V 12,E 30,F 20=20{3} χ=2Truncated icosahedron 2 53 W9, U25, K30, C27 E 60,V 90,F 32=12{5}+20{6} χ=2 5.5.5 3.10.10 IhDodecahedron 32 5 W5, U23, K28, C26 V 20,E 30,F 12=12{5} χ=2Truncated dodecahedron 2 35 W10, U26, K31, C29 V 60,E 90,F 32=20{3}+12{10} χ=25.5.5.5.5 5/2.10.10 IhGreat dodecahedron 5/22 5 W21, U35, K40, C44 V 12,E 30,F 12=12{5} χ=−6Truncated great dodecahedron 2 5/25 W75, U37, K42, C47 V 60,E 90,F 24=12{5/2}+12{10} χ=−6 3.3.3.3.3 5/2.6.6. IhGreat icosahedron (16th stellation of icosahedron) 5/22 3 W41, U53, K58, C69 V 12,E 30,F 20=20{3} χ=2Great truncated icosahedron 2 5/23 W95, U55, K60, C71 V 60,E 90,F 32=12{5/2}+20{6} χ=25/2.5/2.5/2.5/2.5/2 IhSmall stellated dodecahedron 52 5/2 W20, U34, K39, C43 V 12,E 30,F 12=12{5/2} χ=−65/2.5/2.5/2 IhGreat stellated dodecahedron 32 5/2 W22, U52, K57, C68 V 20,E 30,F 12=12{5/2} χ=2
In addition there are three quasi-truncated forms. These also class as truncated-regular polyhedra. vertex figuresGroup OhGroup IhGroup Ih3.8/3.8/3 5.10/3.10/3 3.10/3.10/3Stellated truncated hexahedron (Quasitruncated hexahedron) (stellatruncated cube) 2 34/3 W92, U19, K24, C66 V 24,E 36,F 14=8{3}+6{8/3} χ=2Small stellated truncated dodecahedron (Quasitruncated small stellated dodecahedron) (Small stellatruncated dodecahedron) 2 55/3 W97, U58, K63 V 60,E 90,F 24=12{5}+12{10/3} χ=−6Great stellated truncated dodecahedron (Quasitruncated great stellated dodecahedron) (Great stellatruncated dodecahedron) 2 35/3 W104, U66, K71, C83 V 60,E 90,F 32=20{3}+12{10/3} χ=2
Truncated forms of quasi-regular polyhedra Column A lists some quasi-regular polyhedra, column B lists normal truncated forms, column C shows quasi-truncated forms, column D shows a different method of truncation. These truncated forms all have a vertex figure p.q.r and a Wythoff symbol p q r. vertex figure group A: quasi-regular: p.q.p.q B: truncated quasi-regular: p.q.r C: truncated quasi-regular: p.q.r D: truncated quasi-regular: p.q.r 3.4.3.4 4.6.8 4.6.8/3 8.6.8/3 OhCuboctahedron 23 4 W11, U07, K12, C19 V 12,E 24,F 14=8{3}+6{4} χ=2Truncated cuboctahedron (Great rhombicuboctahedron) 2 3 4 W15, U11, K16, C23 V 48,E 72,F 26=12{4}+8{6}+6{8} χ=2Great truncated cuboctahedron (Quasitruncated cuboctahedron) 2 3 4/3 W93, U20, K25, C67 V 48,E 72,F 26=12{4}+8{6}+6{8/3} χ=2Cubitruncated cuboctahedron (Cuboctatruncated cuboctahedron) 3 4 4/3 W79, U16, K21, C52 V 48,E 72,F 20=8{6}+6{8}+6{8/3} χ=−4 3.5.3.5 4.6.10 4.6.10/3 10.6.10/3 IhIcosidodecahedron 23 5 W12, U24, K29, C28 V 30,E 60,F 32=20{3}+12{5} χ=2Truncated icosidodecahedron (Great rhombicosidodecahedron) 2 3 5 W16, U28, K33, C31 V 120,E 180,F 62=30{4}+20{6}+12{10} χ=2Great truncated icosidodecahedron (Great quasitruncated icosidodecahedron) 2 3 5/3 W108, U68, K73, C87 V 120,E 180,F 62=30{4}+20{6}+12{10/3} χ=2Icositruncated dodecadodecahedron (Icosidodecatruncated icosidodecahedron) 3 5 5/3 W84, U45, K50, C57 V 120,E 180,F 44=20{6}+12{10}+12{10/3} χ=−165/2.5.5/2.5
4.10.10/3IhDodecadodecahedron 2 55/2 W73, U36, K41, C45 V 30,E 60, F 24=12{5}+12{5/2} χ=−6Truncated dodecadodecahedron (Quasitruncated dodecahedron) 2 5 5/3 W98, U59, K64, C75 V 120,E 180,F 54=30{4}+12{10}+12{10/3} χ=−6 3.5/2.3.5/2 IhGreat icosidodecahedron 2 35/2 W94, U54, K59, C70 V 30,E 60, F 32=20{3}+12{5/2} χ=2
Polyhedra sharing edges and vertices Regular These are all mentioned elsewhere, but this table shows some relations. They are all regular apart from the tetrahemihexahedron which is versi-regular. vertex figure V E group regular regular/versi-regular 3.3.3.3 3.4*.−3.4* 6 12 OhOctahedron 42 3 W2, U05, K10, C17 F 8=8{3} χ=2Tetrahemihexahedron 3/232 W67, U04, K09, C36 F 7=4{3}+3{4} χ=13.3.3.3.3 5.5.5.5.5 12 30 IhIcosahedron 52 3 W4, U22, K27 F 20=20{3} χ=2Great dodecahedron 5/22 5 W21, U35, K40, C44 F 12=12{5} χ=−65/2.5/2.5/2.5/2.5/2 3.3.3.3.3 12 30 IhSmall stellated dodecahedron 52 5/2 W20, U34, K39, C43 F 12=12{5/2} χ=−6Great icosahedron (16th stellation of icosahedron) 5/22 3 W41, U53, K58, C69 F 20=20{3} χ=2
Quasi-regular and versi-regular Rectangular vertex figures, or crossed rectangles first column are quasi-regular second and third columns are hemihedra with faces passing through the origin, called versi-regular by some authors. vertex figure V E group quasi-regular: p.q.p.q versi-regular: p.s*.−p.s* versi-regular: q.s*.−q.s* 3.4.3.4 3.6*.−3.6* 4.6*.−4.6* 12 24 OhCuboctahedron 23 4 W11, U07, K12, C19 F 14=8{3}+6{4} χ=2Octahemioctahedron 3/2 33 W68, U03, K08, C37 F 12=8{3}+4{6} χ=0Cubohemioctahedron 4/3 43 W78, U15, K20, C51 F 10=6{4}+4{6} χ=−2 3.5.3.5 3.10*.−3.10* 5.10*.−5.10* 30 60 IhIcosidodecahedron 23 5 W12, U24, K29, C28 F 32=20{3}+12{5} χ=2Small icosihemidodecahedron 3/2 35 W89, U49, K54, C63 F 26=20{3}+6{10} χ=−4Small dodecahemidodecahedron 5/4 55 W91, U51, K56, 65 F 18=12{5}+6{10} χ=−12 3.5/2.3.5/2 3.10*.−3.10* 5/2.10*.−5/2.10* 30 60 IhGreat icosidodecahedron 25/2 3 W94, U54, K59, C70 F 32=20{3}+12{5/2} χ=2Great icosihemidodecahedron 3 35/3 W106, U71, K76, C85 F 26=20{3}+6{10/3} χ=−4Great dodecahemidodecahedron 5/3 5/25/3 W107, U70, K75, C86 F 18=12{5/2}+6{10/3} χ=−12 5.5/2.5.5/2 5.6*.−5.6* 5/2.6*.−5/2.6* 30 60 IhDodecadodecahedron 25/2 5 W73, U36, K41, C45 F 24=12{5}+12{5/2} χ=−6Great dodecahemicosahedron 5/4 53 W102, U65, K70, C81 F 22=12{5}+10{6} χ=−8Small dodecahemicosahedron 5/3 5/23 W100, U62, K67, C78 F 22=12{5/2}+10{6} χ=−8
Ditrigonal regular and versi-regular
Ditrigonal (that is di(2) -tri(3)-ogonal) vertex figures are the 3-fold analog of a rectangle. These are all quasi-regular as all edges are isomorphic. The compound of 5-cubes shares the same set of edges and vertices. The cross forms have a non-orientable vertex figure so the "−" notation has not been used and the "*" faces pass near rather than through the origin. vertex figure V E group ditrigonal crossed-ditrigonal crossed-ditrigonal 5/2.3.5/2.3.5/2.3 5/2.5*.5/2.5*.5/2.5* 3.5*.3.5*.3.5* 20 60 IhSmall ditrigonal icosidodecahedron 35/2 3 W70, U30, K35, C39 F 32=20{3}+12{5/2} χ=−8Ditrigonal dodecadodecahedron 35/3 5 W80, U41, K46, C53 F 24=12{5}+12{5/2} χ=−16Great ditrigonal icosidodecahedron 3/23 5 W87, U47, K52, C61 F 32=20{3}+12{5} χ=−8
versi-quasi-regular and quasi-quasi-regular Group III: trapezoid or crossed trapezoid vertex figures. The first column include the convex rhombic polyhedra, created by inserting two squares into the vertex figures of the Cuboctahedron and Icosidodecahedron.
vertex figure V E group trapezoid: p.q.r.q crossed-trapezoid: p.s*.−r.s* crossed-trapezoid: q.s*.−q.s* 3.4.4.4 3.8*.−4.8* 4.8*.−4.8* 24 48 OhSmall rhombicuboctahedron (rhombicuboctahedron) 3 42 W13, U10, K15, C22 F 26=8{3}+(6+12){4} χ=2Small cubicuboctahedron 3/2 44 W69, U13, K18, C38 F 20=8{3}+6{4}+6{8} χ=−4Small rhombihexahedron 2 3/2 4 W86, U18, K23, C60 F 18=12{4}+6{8} χ=−6 3.8/3.4.8/3 3.4*.−4.4* 8/3.4*.−8/3.4* 24 48 OhGreat cubicuboctahedron 3 44/3 W77, U14, K19, C50 F 20=8{3}+6{4}+6{8/3} χ=−4Nonconvex great rhombicuboctahedron (Quasirhombicuboctahedron) 3/2 42 W85, U17, K22, C59 F 26=8{3}+(6+12){4} χ=2Great rhombihexahedron 2 4/3 3/2 W103, U21, K26, C82 F 18=12{4}+6{8/3} χ=−6 3.4.5.4 3.10*.−5.10* 4.10*.−4.10* 60 120 IhSmall rhombicosidodecahedron (rhombicosidodecahedron) 3 52 W14, U27, K32, C30 F 62=20{3}+30{4}+12{5} χ=2Small dodecicosidodecahedron 3/2 55 W72, U33, K38, C42 F 44=20{3}+12{5}+12{10} χ=−16Small rhombidodecahedron 2 5/2 5 W74, U39, K44, C46 F 42=30{4}+12{10} χ=−18 5/2.4.5.4 5/2.6*.−5.6* 4.6*.−4.6* 60 120 IhRhombidodecadodecahedron 5/2 52 W76, U38, K43, C48 F 54=30{4}+12{5}+12{5/2} χ=−6Icosidodecadodecahedron 5/3 53 W83, U44, K49, C56 F 44=12{5}+12{5/2}+20{6} χ=−16Rhombicosahedron 2 3 5/2 W96, U56, K61, C72 F 50=30{4}+20{6} χ=−10 3.10/3.5/2.10/3 3.4*.−5/2.4* 10/3.4*.−10/3.4* 60 120 IhGreat dodecicosidodecahedron 5/2 35/3 W99, U61, K66, C77 F 44=20{3}+12{5/2}+12{10/3 } χ=−16Nonconvex great rhombicosidodecahedron (Quasirhombicosidodecahedron) 5/3 32 W105, U67, K72, C84 F 62=20{3}+30{4}+12{5/2} χ=2Great rhombidodecahedron 2 3/2 5/3 W109, U73, K78, C89 F 42=30{4}+12{10/3} χ=−18 3.6.5/2.6 3.10*.−5/2.10* 6.10*.−6.10* 60 120 IhSmall icosicosidodecahedron 5/2 33 W71, U31, K36, C40 F 52=20{3}+12{5/2}+20{6} χ=−8Small ditrigonal dodecicosidodecahedron 5/3 35 W82, U43, K48, C55 F 44=20{3}+12{5/2}+12{10} χ=−16Small dodecicosahedron 3 3/2 5 W90, U50, K55, C64 F 32=20{6}+12{10} χ=−28 3.10/3.5.10/3 3.6*.−5.6*