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Symmetrohedron

{{Short description|High-symmetry polyhedron}}

File:Symmetrohedron i-0-2-3-e.png

File:Pyritohedral near-miss johnson.png 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.[https://cs.uwaterloo.ca/~csk/publications/Papers/kaplan_hart_2001.pdf Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons]

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.

class="wikitable"

|+Duals

!I(*;2;3;e)

!Pyritohedral

152x152px

|150x150px

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.

class=wikitable

!rowspan=2|Domain

!rowspan=2|Edges

!colspan=2|Tetrahedral (3 3 2)

!colspan=2|Octahedral (4 3 2)

!colspan=2|Icosahedral (5 3 2)

! colspan="3" |Triangular (6 3 2)

! colspan="3" |Square (4 4 2)

Symbol||Image||Symbol||Image||Symbol||Image||Symbol||Image

!Dual||Symbol||Image

!Dual

100px1

|T(1;*;*;e)
T, {{CDD|node_1|3|node|3|node}}

60px

|C, O(1;*;*;e)
{{CDD|node_1|4|node|3|node}}

60px

|I(1;*;*;e)
D, {{CDD|node_1|5|node|3|node}}

60px

|H(1;*;*;e)
H, {{CDD|node_1|6|node|3|node}}

60px

|60px

|Q(1;*;*;e)
Q, {{CDD|node_1|4|node|4|node}}

60px

|60px

100px1

|T(*;1;*;e)
dT, {{CDD|node|3|node|3|node_1}}

60px

|O(*;1;*;e)
O, {{CDD|node|4|node|3|node_1}}

60px

|I(*;1;*;e)
I, {{CDD|node|5|node|3|node_1}}

60px

|H(*;1;*;e)
dH, {{CDD|node|6|node|3|node_1}}

60px

|60px

|Q(*;1;*;e)
dQ, {{CDD|node|4|node|4|node_1}}

60px

|60px

100px2

|T(1;1;*;e)
aT, {{CDD|node|3|node_1|3|node}}

60px

|O(1;1;*;e)
aC, {{CDD|node|4|node_1|3|node}}

60px

|I(1;1;*;e)
aD, {{CDD|node|5|node_1|3|node}}

60px

|H(1;1;*;e)
aH, {{CDD|node|6|node_1|3|node}}

60px

|60x60px

|Q(1;1;*;e)
aQ, {{CDD|node|4|node_1|4|node}}

60px

|60x60px

100px3

|T(2;1;*;e)
tT, {{CDD|node_1|3|node_1|3|node}}

60px

|O(2;1;*;e)
tC, {{CDD|node_1|4|node_1|3|node}}

60px

|I(2;1;*;e)
tD, {{CDD|node_1|5|node_1|3|node}}

60px

|H(2;1;*;e)
tH, {{CDD|node_1|6|node_1|3|node}}

60px

|60x60px

|Q(2;1;*;e)
tQ, {{CDD|node_1|4|node_1|4|node}}

60px

|60x60px

100px3

|T(1;2;*;e)
dtT, {{CDD|node|3|node_1|3|node_1}}

60px

|O(1;2;*;e)
tO, {{CDD|node|4|node_1|3|node_1}}

60px

|I(1;2;*;e)
tI, {{CDD|node|5|node_1|3|node_1}}

60px

|H(1;2;*;e)
dtH, {{CDD|node|6|node_1|3|node_1}}

60px

|60x60px

|Q(1;2;*;e)
dtQ, {{CDD|node|4|node_1|4|node_1}}

60px

|60x60px

100px4

|T(1;1;*;1)
eT, {{CDD|node_1|3|node|3|node_1}}

60px

|O(1;1;*;1)
eC, {{CDD|node_1|4|node|3|node_1}}

60px

|I(1;1;*;1)
eD, {{CDD|node_1|5|node|3|node_1}}

60px

|H(1;1;*;1)
eH, {{CDD|node_1|6|node|3|node_1}}

60px

|60x60px

|Q(1;1;*;1)
eQ, {{CDD|node_1|4|node|4|node_1}}

60px

|60x60px

100px6

|T(2;2;*;e)
bT, {{CDD|node_1|3|node_1|3|node_1}}

60px

|O(2;2;*;e)
bC, {{CDD|node_1|4|node_1|3|node_1}}

60px

|I(2;2;*;e)
bD, {{CDD|node_1|4|node_1|3|node_1}}

60px

|H(2;2;*;e)
bH, {{CDD|node_1|6|node_1|3|node_1}}

60px

|60x60px

|Q(2;2;*;e)
bQ, {{CDD|node_1|4|node_1|4|node_1}}

60px

|60x60px

2-generator points

class="wikitable sortable"

!rowspan=2|Domain

!rowspan=2|Edges

!colspan=2|Tetrahedral (3 3 2)

!colspan=2|Octahedral (4 3 2)

!colspan=2|Icosahedral (5 3 2)

! colspan="3" |Triangular (6 3 2)

! colspan="3" |Square (4 4 2)

Symbol||Image||Symbol||Image||Symbol||Image||Symbol||Image

!Dual||Symbol||Image

!Dual

100px6

|rowspan=2|T(1;2;*;[2])
atT

rowspan=2|60px

|O(1;2;*;[2])
atO

60px

|I(1;2;*;[2])
atI

60px

|H(1;2;*;[2])
atΔ

80px

|File:Symmetric Tiling Dual 12 Rhombille II.svg

|rowspan=2|Q(1;2;*;[2])
Q(2;1;*;[2])
atQ

rowspan=2|120px

| rowspan="2" |120x120px

100px6

|O(2;1;*;[2])
atC

60px

|I(2;1;*;[2])
atD

60px

|H(2;1;*;[2])
atH

80px

|80x80px

100px7

|rowspan=2|T(3;*;*;[2])
T(*;3;*;[2])
dKdT

rowspan=2|60px

|O(3;*;*;[2])
dKdC

60px

|I(3;*;*;[2])
dKdD

60px

|H(3;*;*;[2])
dKdH

80px

|80x80px

|rowspan=2|Q(3;*;*;[2])
Q(*;3;*;[2])
dKQ

rowspan=2|120px

| rowspan="2" |120x120px

100px7

|O(*;3;*;[2])
dKdO

60px

|I(*;3;*;[2])
dKdI

60px

|H(*;3;*;[2])
dKdΔ

80px

|80x80px

100px8

|rowspan=2|T(2;3;*;α)
T(3;2;*;α)
dM0T

rowspan=2|60px

|O(2;3;*;α)
dM0dO

60px

|I(2;3;*;α)
dM0dI

60px

|H(2;3;*;α)
dM0dΔ

80px

|80x80px

|rowspan=2|Q(2;3;*;α)
Q(3;2;*;α)
dM0Q

rowspan=2|120px

| rowspan="2" |120x120px

100px8

|O(3;2;*;α)
dM0dC

60px

|I(3;2;*;α)
dM0dD

60px

|H(3;2;*;α)
dM0dH

80px

|80x80px

100px9

|rowspan=2|T(2;4;*;e)
T(4;2;*;e)
[https://levskaya.github.io/polyhedronisme/?recipe=C100ttT ttT]

rowspan=2|60px

|O(2;4;*;e)
[https://levskaya.github.io/polyhedronisme/?recipe=C100ttO ttO]

60px

|I(2;4;*;e)
[https://levskaya.github.io/polyhedronisme/?recipe=C100ttI ttI]

60px

|H(2;4;*;e)
ttΔ

80px

|80x80px

|rowspan=2|Q(4;2;*;e)
Q(2;4;*;e)
ttQ

rowspan=2|120px

| rowspan="2" |120x120px

100px9

|O(4;2;*;e)
[https://levskaya.github.io/polyhedronisme/?recipe=C100ttC ttC]

60px

|I(4;2;*;e)
[https://levskaya.github.io/polyhedronisme/?recipe=C100ttD ttD]

60px

|H(4;2;*;e)
ttH

80px

|80x80px

100px7

|rowspan=2|T(2;1;*;1)
T(1;2;*;1)
dM3T

rowspan=2|60px

|O(1;2;*;1)
dM3O

60px

|I(1;2;*;1)
dM3I

60px

|H(1;2;*;1)
dM3Δ

80px

|80x80px

|rowspan=2|Q(2;1;*;1)
Q(1;2;*;1)
dM3dQ

rowspan=2|120px

| rowspan="2" |120x120px

100px7

|O(2;1;*;1)
dM3C

60px

|I(2;1;*;1)
dM3D

60px

|H(2;1;*;1)
dM3H

80px

|80x80px

100px9

|rowspan=2|T(2;3;*;e)
T(3;2;*;e)
dm3T

rowspan=2|60px

|O(2;3;*;e)
dm3C

60px

|I(2;3;*;e)
dm3D

60px

|H(2;3;*;e)
dm3H

80px

|80x80px

|rowspan=2|Q(2;3;*;e)
Q(3;2;*;e)
dm3Q

rowspan=2|120px

| rowspan="2" |120x120px

100px9

|O(3;2;*;e)
dm3O

60px

|I(3;2;*;e)
dm3I

60px

|H(3;2;*;e)
dm3Δ

80px

|80x80px

100px10

|rowspan=2|T(2;*;3;e)
T(*;2;3;e)
dXdT

3.4.6.6

| rowspan="2" |60px

|O(*;2;3;e)
dXdO

60px

|I(*;2;3;e)
dXdI

60px

|H(*;2;3;e)
dXdΔ

80px

|80x80px

|rowspan=2|Q(2;*;3;e)
Q(*;2;3;e)
dXdQ

rowspan=2|120px

| rowspan="2" |120x120px

100px10

|O(2;*;3;e)
dXdC

3.4.6.8

|60px

|I(2;*;3;e)
dXdD

3.4.6.10

|60px

|H(2;*;3;e)
dXdH

3.4.6.12

|80px

|80x80px

3-generator points

class="wikitable sortable"

!rowspan=2|Domain

!rowspan=2|Edges

!colspan=2|Tetrahedral (3 3 2)

!colspan=2|Octahedral (4 3 2)

!colspan=2|Icosahedral (5 3 2)

! colspan="3" |Triangular (6 3 2)

! colspan="3" |Square (4 4 2)

Symbol||Image||Symbol||Image||Symbol||Image||Symbol||Image

!Dual||Symbol||Image

!Dual

100px6

|rowspan=2|T(2;0;*;[1])

rowspan=2|80px

|O(0;2;*;[1])
dL0dO

80px

|I(0;2;*;[1])
dL0dI

80px

|H(0;2;*;[1])
dL0H

80px

|80x80px

|rowspan=2|Q(2;0;*;[1])
Q(0;2;*;[1])
dL0dQ

rowspan=2|120px

| rowspan="2" |120x120px

100px6

|O(2;0;*;[1])
dL0dC

80px

|I(2;0;*;[1])
dL0dD

80px

|H(2;0;*;[1])
dL0Δ

80px

|80x80px

100px7

|rowspan=2|T(3;0;*;[2])

rowspan=2|80px

|O(0;3;*;[2])
dLdO

80px

|I(0;3;*;[2])
dLdI

80px

|H(0;3;*;[2])
dLH

80px

|80x80px

|rowspan=2|Q(2;0;*;[1])
Q(0;2;*;[2])
dLQ

rowspan=2|120px

| rowspan="2" |120x120px

100px7

|O(3;0;*;[2])
dLdC

80px

|I(3;0;*;[2])
dLdD

80px

|H(3;0;*;[2])
dLΔ

80px

|80x80px

100px12

|T(2;2;*;a)
[https://levskaya.github.io/polyhedronisme/?recipe=C100amT amT]

80px

|O(2;2;*;a)
[https://levskaya.github.io/polyhedronisme/?recipe=C100amC amC]

80px

|I(2;2;*;a)
[https://levskaya.github.io/polyhedronisme/?recipe=C100amD amD]

80px

|H(2;2;*;a)
amH

80px

|80x80px

|Q(2;2;*;a)
amQ

80px

|80x80px

See also

References

{{reflist}}

External links

  • [http://www.antiprism.com/download/index.html Antiprism] Free software that includes Symmetro for generating and viewing these polyhedra with Kaplan-Hart notation.

Category:Polyhedra