Symmetrohedron

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The symmetrohedron I(*;2;3;e) has regular pentagons and hexagons, and trapezoidal gap faces.
Symmetrohedron with pyritohedral symmetry, order 24

In geometry, a symmetrohedron is a high-symmetry polyhedron containing convex regular polygons on symmetry axes with gaps on the convex hull filled by irregular polygons. The name was coined by Craig S. Kaplan and George W. Hart.[1]

The trivial cases are the Platonic solids, Archimedean solids with all regular polygons. A first class is called bowtie which contain pairs of trapezoidal faces. A second class has kite faces. Another class are called LCM symmetrohedra.

Symbolic notation[]

Each symmetrohedron is described by a symbolic expression G(l; m; n; α). G represents the symmetry group (T,O,I). The values l, m and n are the multipliers ; a multiplier of m will cause a regular km-gon to be placed at every k-fold axis of G. In the notation, the axis degrees are assumed to be sorted in descending order, 5,3,2 for I, 4,3,2 for O, and 3,3,2 for T . We also allow two special values for the multipliers: *, indicating that no polygons should be placed on the given axes, and 0, indicating that the final solid must have a vertex (a zero-sided polygon) on the axes. We require that one or two of l, m, and n be positive integers. The final parameter, α, controls the relative sizes of the non-degenerate axis-gons.

Conway polyhedron notation is another way to describe these polyhedra, starting with a regular form, and applying prefix operators. The notation doesn't imply which faces should be made regular beyond the uniform solutions of the Archimedean solids.

Duals
I(*;2;3;e) Pyritohedral
Dual Symmetrohedron I(*;2;3;e).svg Symmetrohedron Pyrito.svg

1-generator point[]

These symmetrohedra are produced by a single generator point within a fundamental domains, reflective symmetry across domain boundaries. Edges exist perpendicular to each triangle boundary, and regular faces exist centered on each of the 3 triangle corners.

The symmetrohedra can be extended to euclidean tilings, using the symmetry of the regular square tiling, and dual pairs of triangular and hexagonal tilings. Tilings, Q is square symmetry p4m, H is hexagonal symmetry p6m.

Coxeter-Dynkin diagrams exist for these uniform polyhedron solutions, representing the position of the generator point within the fundamental domain. Each node represents one of 3 mirrors on the edge of the triangle. A mirror node is ringed if the generator point is active, off the mirror, and creates new edges between the point and its mirror image.

Domain Edges Tetrahedral (3 3 2) Octahedral (4 3 2) Icosahedral (5 3 2) Triangular (6 3 2) Square (4 4 2)
Symbol Image Symbol Image Symbol Image Symbol Image Dual Symbol Image Dual
Symmetrohedron domain 1-0-0-e.png 1 T(1;*;*;e)
T, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-33-t0.png C, O(1;*;*;e)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t0.svg I(1;*;*;e)
D, CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t0.svg H(1;*;*;e)
H, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.svg Uniform tiling 63-t2.svg Q(1;*;*;e)
Q, CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg Uniform tiling 44-t2.svg
Symmetrohedron domain 0-1-0-e.png 1 T(*;1;*;e)
dT, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-33-t2.png O(*;1;*;e)
O, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t2.svg I(*;1;*;e)
I, CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-53-t2.svg H(*;1;*;e)
dH, CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t2.svg Uniform tiling 63-t0.svg Q(*;1;*;e)
dQ, CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t2.svg Uniform tiling 44-t0.svg
Symmetrohedron domain 1-1-0-e.png 2 T(1;1;*;e)
aT, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-33-t1.png O(1;1;*;e)
aC, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.svg I(1;1;*;e)
aD, CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t1.svg H(1;1;*;e)
aH, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.svg Symmetric Tiling Dual 2 Rhombille.svg Q(1;1;*;e)
aQ, CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t1.svg Symmetric Tiling Dual 3 Alt Square.svg
Symmetrohedron domain 2-1-0-e.png 3 T(2;1;*;e)
tT, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-33-t01.png O(2;1;*;e)
tC, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t01.svg I(2;1;*;e)
tD, CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t01.svg H(2;1;*;e)
tH, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t01.svg Symmetric Tiling Dual 4 Kisdeltille.svg Q(2;1;*;e)
tQ, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t01.svg Symmetric Tiling Dual 5 Kisquadrille I.svg
Symmetrohedron domain 1-2-0-e.png 3 T(1;2;*;e)
dtT, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-33-t12.png O(1;2;*;e)
tO, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t12.svg I(1;2;*;e)
tI, CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-53-t12.svg H(1;2;*;e)
dtH, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t12.svg Symmetric Tiling Dual 6 Hexakis Hexagonal.svg Q(1;2;*;e)
dtQ, CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t12.svg Symmetric Tiling Dual 7 Kisquadrille II.svg
Symmetrohedron domain 1-1-0-1.png 4 T(1;1;*;1)
eT, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-33-t02.png O(1;1;*;1)
eC, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t02.png I(1;1;*;1)
eD, CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-53-t02.png H(1;1;*;1)
eH, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t02.svg Symmetric Tiling Dual 8 Deltoidal.svg Q(1;1;*;1)
eQ, CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t02.svg Symmetric Tiling Dual 9 Ortho Square.svg
Symmetrohedron domain 2-2-0-e.png 6 T(2;2;*;e)
bT, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-33-t012.png O(2;2;*;e)
bC, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t012.png I(2;2;*;e)
bD, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-53-t012.png H(2;2;*;e)
bH, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t012.svg Symmetric Tiling Dual 10 Kisrhombille.svg Q(2;2;*;e)
bQ, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t012.svg Symmetric Tiling Dual 11 Kisquadrille III.svg

2-generator points[]

Domain Edges Tetrahedral (3 3 2) Octahedral (4 3 2) Icosahedral (5 3 2) Triangular (6 3 2) Square (4 4 2)
Symbol Image Symbol Image Symbol Image Symbol Image Dual Symbol Image Dual
Symmetrohedron domain 1-2-0-b2.png 6 T(1;2;*;[2])
atT
Rectified truncated tetrahedron.png O(1;2;*;[2])
atO
Rectified truncated octahedron.png I(1;2;*;[2])
atI
Rectified truncated icosahedron.png H(1;2;*;[2])
atΔ
Conway tiling atdH.png Symmetric Tiling Dual 12 Rhombille II.svg Q(1;2;*;[2])
Q(2;1;*;[2])
atQ
Diagonal-square tiling truncation2.svg Symmetric Tiling Dual 14 Join Kisquadrille.svg
Symmetrohedron domain 2-1-0-b2.png 6 O(2;1;*;[2])
atC
Rectified truncated cube.png I(2;1;*;[2])
atD
Rectified truncated dodecahedron.png H(2;1;*;[2])
atH
Conway tiling atH.png Symmetric Tiling Dual 13 Join Kisdeltille.svg
Symmetrohedron domain 3-0-0-b2.png 7 T(3;*;*;[2])
T(*;3;*;[2])
dKdT
Conway polyhedron dKT.png O(3;*;*;[2])
dKdC
Conway polyhedron dKO.png I(3;*;*;[2])
dKdD
Conway polyhedron dKI.png H(3;*;*;[2])
dKdH
Conway tiling dKdH.png Symmetric Tiling 16 Join K(1).svg Q(3;*;*;[2])
Q(*;3;*;[2])
dKQ
Square lattice with dodecagons.svg Symmetric Tiling Dual 17 OT.svg
Symmetrohedron domain 0-3-0-b2.png 7 O(*;3;*;[2])
dKdO
Conway polyhedron dKC.png I(*;3;*;[2])
dKdI
Conway polyhedron dKD.png H(*;3;*;[2])
dKdΔ
Conway tiling dKH.png Symmetric Tiling Dual 15 Rhomb(1).svg
Symmetrohedron domain 2-3-s-a.png 8 T(2;3;*;α)
T(3;2;*;α)
dM0T
Conway polyhedra M0T.png O(2;3;*;α)
dM0dO
Conway polyhedra M0C.png I(2;3;*;α)
dM0dI
Conway polyhedra M0D.png H(2;3;*;α)
dM0
Conway tiling dM0H.png Symmetric Tiling Dual 18 Rhomb(2).svg Q(2;3;*;α)
Q(3;2;*;α)
dM0Q
Diagonal-square tiling truncation3.svg Symmetric Tiling Dual 19 Join KQ(2).svg
Symmetrohedron domain 3-2-s-a.png 8 O(3;2;*;α)
dM0dC
Conway polyhedra M0O.png I(3;2;*;α)
dM0dD
Conway polyhedra M0I.png H(3;2;*;α)
dM0dH
Conway tiling dM0dH.png Symmetric Tiling Dual 18 Join K(2).svg
Symmetrohedron domain 2-4-0-e.png 9 T(2;4;*;e)
T(4;2;*;e)
ttT
Conway polyhedron ttT.png O(2;4;*;e)
ttO
Conway polyhedron ttO.png I(2;4;*;e)
ttI
Conway polyhedron ttI.png H(2;4;*;e)
ttΔ
Conway tiling ttdH.png Symmetric Tiling Dual 20 Kisdeltille(2).svg Q(4;2;*;e)
Q(2;4;*;e)
ttQ
Square lattice with 16-gons.svg Symmetric Tiling Dual 22 Join KQ(3).svg
Symmetrohedron domain 4-2-0-e.png 9 O(4;2;*;e)
ttC
Conway polyhedron ttC.png I(4;2;*;e)
ttD
Conway polyhedron ttD.png H(4;2;*;e)
ttH
Conway tiling ttH.png Symmetric Tiling Dual 21 Join K(3).svg
Symmetrohedron domain 1-2-0-1.png 7 T(2;1;*;1)
T(1;2;*;1)
dM3T
Conway polyhedron dM3T.png O(1;2;*;1)
dM3O
Conway polyhedron dM3O.png I(1;2;*;1)
dM3I
Conway polyhedron dM3I.png H(1;2;*;1)
dM3Δ
Conway tiling dM3dH.png Symmetric Tiling Dual 23 Rhomb(4).svg Q(2;1;*;1)
Q(1;2;*;1)
dM3dQ
Square-octagon-bowtie tiling.svg Symmetric Tiling Dual 25 Join KQ(4).svg
Symmetrohedron domain 2-1-0-1.png 7 O(2;1;*;1)
dM3C
Conway polyhedron dM3C.png I(2;1;*;1)
dM3D
Conway polyhedron dM3D.png H(2;1;*;1)
dM3H
Conway tiling dM3H.png Symmetric Tiling Dual 24 Join K(4).svg
Symmetrohedron domain 2-3-0-e.png 9 T(2;3;*;e)
T(3;2;*;e)
dm3T
Conway polyhedron b3T.gif O(2;3;*;e)
dm3C
Conway polyhedron b3O.png I(2;3;*;e)
dm3D
Conway polyhedron b3I.png H(2;3;*;e)
dm3H
Conway tiling b3dH.png Symmetric Tiling Dual 27 Join K(5).svg Q(2;3;*;e)
Q(3;2;*;e)
dm3Q
12gon-octagon bowtie.svg Symmetric Tiling Dual 28 Join KQ(5).svg
Symmetrohedron domain 3-2-0-e.png 9 O(3;2;*;e)
dm3O
Conway polyhedron b3C.png I(3;2;*;e)
dm3I
Conway polyhedron b3D.png H(3;2;*;e)
dm3Δ
Conway tiling b3H.png Symmetric Tiling Dual 26 Rhomb(5).svg
Symmetrohedron domain 0-2-3-e.png 10 T(2;*;3;e)
T(*;2;3;e)
dXdT

3.4.6.6

Conway dual cross tetrahedron.png O(*;2;3;e)
dXdO
Conway dual cross cube.png I(*;2;3;e)
dXdI
Symmetrohedron i-0-2-3-e.png H(*;2;3;e)
dXdΔ
Conway tiling dXH.png Symmetric Tiling Dual 29 Rhomb(6).svg Q(2;*;3;e)
Q(*;2;3;e)
dXdQ
Octagon-hexagon-square-trap tiling.svg Symmetric Tiling Dual 31.svg
Symmetrohedron domain 2-0-3-e.png 10 O(2;*;3;e)
dXdC

3.4.6.8

Conway dual cross octahedron.png I(2;*;3;e)
dXdD

3.4.6.10

Conway dual crossed icosahedron.png H(2;*;3;e)
dXdH

3.4.6.12

Conway dXdH.png Symmetric Tiling Dual 30 3D.svg

3-generator points[]

Domain Edges Tetrahedral (3 3 2) Octahedral (4 3 2) Icosahedral (5 3 2) Triangular (6 3 2) Square (4 4 2)
Symbol Image Symbol Image Symbol Image Symbol Image Dual Symbol Image Dual
Symmetrohedron domain 0-2-s-b1.png 6 T(2;0;*;[1]) Conway polyhedron dL0T.png O(0;2;*;[1])
dL0dO
Conway polyhedron dL0C.png I(0;2;*;[1])
dL0dI
Conway polyhedron dL0D.png H(0;2;*;[1])
dL0H
Conway tiling dL0H.png Symmetric Tiling Dual 32 Rhomb(7).svg Q(2;0;*;[1])
Q(0;2;*;[1])
dL0dQ
Diagonalkite-square tiling truncation2.svg Symmetric Tiling Dual 34 Join KQ(7).svg
Symmetrohedron domain 2-0-s-b1.png 6 O(2;0;*;[1])
dL0dC
Conway polyhedron dL0O.png I(2;0;*;[1])
dL0dD
Conway polyhedron dL0I.png H(2;0;*;[1])
dL0Δ
Conway tiling dL0dH.png Symmetric Tiling Dual 33 Join K(6).svg
Symmetrohedron domain 0-3-s-b2.png 7 T(3;0;*;[2]) Conway polyhedron dLT.png O(0;3;*;[2])
dLdO
Conway polyhedron dLC.png I(0;3;*;[2])
dLdI
Conway polyhedron dLD.png H(0;3;*;[2])
dLH
Conway tiling dLH.png Symmetric Tiling Dual 35 Rhomb(8).svg Q(2;0;*;[1])
Q(0;2;*;[2])
dLQ
Octagon-rhomb tiling.svg Symmetric Tiling Dual 37 Join KQ(8).svg
Symmetrohedron domain 3-0-s-b2.png 7 O(3;0;*;[2])
dLdC
Conway polyhedron dLO.png I(3;0;*;[2])
dLdD
Conway polyhedron dLI.png H(3;0;*;[2])
dLΔ
Conway tiling dLdH.png Symmetric Tiling Dual 36 Join K(7).svg
Symmetrohedron domain 2-2-s-a.png 12 T(2;2;*;a)
amT
Conway polyhedron amT.png O(2;2;*;a)
amC
Conway polyhedron amC.png I(2;2;*;a)
amD
Conway polyhedron amD.png H(2;2;*;a)
amH
Conway tiling amH.png Symmetric Tiling Dual 38 Join Kisrhombille.svg Q(2;2;*;a)
amQ
Diagonal-square tiling truncation2.svg Symmetric Tiling Dual 14 Join Kisquadrille.svg

See also[]

References[]

External links[]

  • Symmetrohedra on RobertLovesPi.net.
  • Antiprism Free software that includes Symmetro for generating and viewing these polyhedra with Kaplan-Hart notation.
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