User:Tomruen/Convex uniform tetracomb
Regular and uniform honeycombs
[edit]There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 4-space.[1]
# | Coxeter group | Coxeter-Dynkin diagram | ||
---|---|---|---|---|
1 | A~4 | [(3,3,3,3,3)] | [3[5]] | ![]() ![]() ![]() |
2 | B~4 | [4,3,3,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | C~4 | [4,3,31,1] | h[4,3,3,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | D~4 | [31,1,1,1] | q[4,3,3,4] | ![]() ![]() ![]() ![]() ![]() |
5 | F~4 | [3,4,3,3] | h[4,3,3,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
There are three regular honeycomb of Euclidean 4-space:
- tesseractic honeycomb, with symbols {4,3,3,4},
=
. There are 19 uniform honeycombs in this family.
- 24-cell honeycomb, with symbols {3,4,3,3},
. There are 31 uniform honeycombs in this family.
- 16-cell honeycomb, with symbols {3,3,4,3},
Other families that generate uniform honeycombs:
- There are 23 uniform honeycombs, 4 unique in the 16-cell honeycomb family. With symbols h{4,32,4} it is geometrically identical to the 16-cell honeycomb,
=
- There are 7 uniform honeycombs from the A~4,
family, all unique.
- There are 9 uniform honeycombs in the D~4: [31,1,1,1]
family, all repeated in other families, including the 16-cell honeycomb.
Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.
The single-ringed tessellations are given below, indexed by Olshevsky's listing.
Duoprismatic forms
- B~2xB~2: [4,4]x[4,4] = [4,3,3,4]
=
(Same as tesseractic honeycomb family)
- B~2xH~2: [4,4]x[6,3]
- H~2xH~2: [6,3]x[6,3]
- A~2xB~2: [Δ]x[4,4]
(Same forms as [6,3]x[4,4])
- A~2xH~2: [Δ]x[6,3]
(Same forms as [6,3]x[6,3])
- A~2xA~2: [Δ]x[Δ]
(Same forms as [6,3]x[6,3])
Prismatic forms
Noncompact prismatic forms
- A3xI~1: [3,3]x[∞] -
- B3xI~1: [4,3]x[∞] -
- H3xI~1: [5,3]x[∞] -
- I~1xI~1xI2r: [∞] x [∞] x [r] = [4,4]x[r] -
=
Non-Wythoffian forms
[edit]The non-Wythoffian forms are built as stacked composites of these prismatic noncompact groups:
- I2pxI~1xA1: [p]x[∞]x[ ] -
(Prism column)
- D~3xA1: [4,31,1]x[ ]
(Prism slab)
- A~3xA1:
(Prism slab)
- A~2xI2p: [Δ]x[p]
(Prism slab)
- B~2xI2p: [4,4]x[p]
(Prism slab)
- H~2xI2p: [6,3]x[p]
(Prism slab)
B~4 [4,3,3,4] family
[edit]# | Coxeter-Dynkin andSchläfli symbols |
Name | Facets by location: [4,3,3,4] | ||||
---|---|---|---|---|---|---|---|
4 | 3 | 2 | 1 | 0 | |||
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4,3,3] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4,3]×[ ] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4]×[4] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [ ]×[3,4] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3,4] | |||
1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0{4,3,3,4} |
Tesseractic honeycomb | {4,3,3}![]() |
- | - | - | - |
87 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1{4,3,3,4} |
Rectified tesseractic honeycomb | t1{4,3,3}![]() |
- | - | - | {3,3,4}![]() |
88 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2{4,3,3,4} |
Birectified tesseractic honeycomb (Same as 24-cell honeycomb {3,4,3,3}) |
t1{3,3,4}![]() |
- | - | - | t1{3,3,4}![]() |
89 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1{4,3,3,4} |
Truncated tesseractic honeycomb | t0,1{4,3,3} | - | - | - | {3,3,4} |
90 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,2{4,3,3,4} |
Cantellated tesseractic honeycomb (Small prismatotesseractic honeycomb) |
t0,2{4,3,3} | - | - | {}x{3,4} | t1{3,3,4} |
91 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,3{4,3,3,4} |
Runcinated tesseractic honeycomb (Small diprismatotesseractic honeycomb) |
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92 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1,2{4,3,3,4} |
Bitruncated tesseractic honeycomb | |||||
93 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1,3{4,3,3,4} |
Bicantellated tesseractic honeycomb (Same as Rectified 24-cell honeycomb t1{3,4,3,3}) |
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[1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,4{4,3,3,4} |
Stericated tesseractic honeycomb (same as tesseractic honeycomb) |
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94 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,2{4,3,3,4} |
Cantitruncated tesseractic honeycomb (Great prismatotesseractic honeycomb) |
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95 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,3{4,3,3,4} |
Runcitruncated tesseractic honeycomb (Small tomocubic-diprismatotesseractic honeycomb) |
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96 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,4{4,3,3,4} |
Steritruncated tesseractic honeycomb (Tomotesseractic-diprismatotesseractic honeycomb) |
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97 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,2,3{4,3,3,4} |
Runcicantellated tesseractic honeycomb (Rhombitesseractic-diprismatotesseractic honeycomb) |
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98 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,2,4{4,3,3,4} |
Stericantellated tesseractic honeycomb (Small rhombitesseractic-prismatotesseractic honeycomb) |
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99 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1,2,3{4,3,3,4} |
Bicantitruncated tesseractic honeycomb (Same as Truncated 24-cell honeycomb, t0,1{3,4,3,3} ) |
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100 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,2,3{4,3,3,4} |
Runcicantitruncated tesseractic honeycomb (Great diprismatotesseractic honeycomb) |
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101 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,2,4{4,3,3,4} |
Stericantitruncated tesseractic honeycomb (Great rhombitesseractic-prismatotesseractic honeycomb) |
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102 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,3,4{4,3,3,4} |
Steriruncitruncated tesseractic honeycomb (Great tomocubic-diprismatotesseractic honeycomb) |
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103 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,2,3,4{4,3,3,4} |
Omnitruncated tesseractic honeycomb |
C~4 [31,1,3,4] family
[edit]There are 23 honeycombs in this family,[2] all listed below.
# | Coxeter-Dynkin andSchläfli symbols File:CDel B5 nodes.png |
Name | Facets by location: [31,1,3,4] | ||||
---|---|---|---|---|---|---|---|
4 | 3 | 2 | 1 | 0 | |||
![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3,4] |
![]() ![]() ![]() ![]() ![]() [31,1,1] |
![]() ![]() ![]() ![]() ![]() [3,3]×[ ] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() [ ]×[3]×[ ] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3,4] | |||
104 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() {31,1,3,4} |
16-cell honeycomb | {3,3,4}![]() |
{31,1,1}![]() |
- | - | - |
105 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1{31,1,3,4} |
Truncated 16-cell honeycomb (Same as truncated 16-cell honeycomb, t0,1{3,3,4,3}) |
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106 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,2{31,1,3,4} |
Cantellated 16-cell honeycomb (Same as birectified 16-cell honeycomb) |
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107 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,2{31,1,3,4} |
Cantitruncated 16-cell honeycomb (Same as bitruncated 16-cell honeycomb) |
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108 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,3{31,1,3,4} |
Runcinated 16-cell honeycomb (Small diprismatodemitesseractive honeycomb) |
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109 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,3{31,1,3,4} |
Runcitruncated 16-cell honeycomb (Small prismato16-cell honeycomb) |
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110 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,2,3{31,1,3,4} |
Runcicantellated 16-cell honeycomb (Great prismato16-cell honeycomb) |
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111 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,2,3{31,1,3,4} |
Runcicantitruncated 16-cell honeycomb (Great diprismato16-cell honeycomb) |
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[88] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1{31,1,3,4} |
Rectified 16-cell honeycomb (Same as 24-cell honeycomb, {3,4,3,3}) (Also birectified tesseractic honeycomb) |
t1{3,3,4}![]() |
t1{31,1,1} | - | - | t1{3,3,4}![]() |
[87] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2{31,1,3,4} |
Same as rectified tesseractic honeycomb | t1{4,3,3}![]() |
{31,1,1}![]() |
- | - | t1{4,3,3}![]() |
[1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t3{31,1,3,4} |
Same as tesseractic honeycomb | {4,3,3}![]() |
- | - | - | {4,3,3}![]() |
[87] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,4{31,1,3,4} |
(Same as Rectified tesseractive honeycomb) | |||||
[92] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1,2{31,1,3,4} |
(Same as bitruncated tesseractic honeycomb) | |||||
[90] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1,3{31,1,3,4} |
(Same as cantellated tesseractic honeycomb) | |||||
[89] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2,3{31,1,3,4} |
(Same as truncated tesseractic honeycomb) | |||||
[92] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,4{31,1,3,4} |
(Same as bitruncated tesseractic honeycomb) | |||||
[93] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,2,4{31,1,3,4} |
(Same as rectified 24-cell honeycomb) (Also bicantellated tesseractic honeycomb) |
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[91] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,3,4{31,1,3,4} |
(Same as runcinated tesseractic honeycomb) | |||||
[94] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1,2,3{31,1,3,4} |
(Same as cantitruncated tesseractic honeycomb) | |||||
[99] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,2,4{31,1,3,4} |
(Same as bicantitruncated tesseractic honeycomb) | |||||
[97] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,3,4{31,1,3,4} |
(Same as runcicantellated tesseractic honeycomb) | |||||
[95] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,2,3,4{31,1,3,4} |
(Same as runcitruncated tesseractic honeycomb) | |||||
[100] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,1,2,3,4{31,1,3,4} |
(Same as runcicantitruncated tesseractic honeycomb) |
F~4 [3,4,3,3] family
[edit]There are 32 honeycombs in this family, 31 reflective forms and one snub.[3] They are named as truncated forms from the regular 16-cell honeycomb and 24-cell honeycomb. These 31 forms are listed by the regular generators in two groups of 19, with 7 shared between.
From the regular 24-cell honeycomb, 19 forms are:
From the regular 16-cell honeycomb, 19 forms are:
A~4 [3[5]] family
[edit]There are 7 honeycombs in this family,[4] all unique to this family, all given below.
# | Coxeter-Dynkin andSchläfli symbols |
Name | Facets by location | ||||
---|---|---|---|---|---|---|---|
4 | 3 | 2 | 1 | 0 | |||
![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3,3] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() [ ]x[3,3] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() [3]x[3] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3]x[ ] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3,3] | |||
134 | ![]() ![]() ![]() ![]() ![]() {3[5]} |
5-cell honeycomb | |||||
135 | ![]() ![]() ![]() ![]() ![]() t0,1{3[5]} |
Truncated 5-cell honeycomb | |||||
136 | ![]() ![]() ![]() ![]() ![]() t0,2{3[5]} |
Cantellated 5-cell honeycomb | |||||
137 | ![]() ![]() ![]() ![]() ![]() t0,1,2{3[5]} |
Cantitruncated 5-cell honeycomb | |||||
138 | ![]() ![]() ![]() ![]() ![]() t0,1,3{3[5]} |
Runcitruncated 5-cell honeycomb | |||||
139 | ![]() ![]() ![]() ![]() ![]() t0,1,2,3{3[5]} |
Runcicantitruncated 5-cell honeycomb | |||||
140 | ![]() ![]() ![]() ![]() ![]() t0,1,2,3,4{3[5]} |
Omnitruncated 5-cell honeycomb |
D~4 [31,1,1,1] family
[edit]There are 9 honeycombs in this family,[5] all repeated, with all 9 forms given below.
# | Coxeter-Dynkin andSchläfli symbols |
Name | Facets by location: [3,4,3,3] | ||||
---|---|---|---|---|---|---|---|
4 | 3 | 2 | 1 | 0 | |||
![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() | |||
[?] | ![]() ![]() ![]() ![]() ![]() {31,1,1,1} |
- | |||||
[104] | ![]() ![]() ![]() ![]() ![]() t4{31,1,1,1} |
Same as 24-cell honeycomb | - | t1{31,1,1} | t1{31,1,1} | t1{31,1,1} | t1{31,1,1} |
[?] | ![]() ![]() ![]() ![]() ![]() t0,4{31,1,1,1} |
- | |||||
[?] | ![]() ![]() ![]() ![]() ![]() t0,1{31,1,1,1} |
- | |||||
[?] | ![]() ![]() ![]() ![]() ![]() t0,1,4{31,1,1,1} |
- | |||||
[?] | ![]() ![]() ![]() ![]() ![]() t0,1,2{31,1,1,1} |
- | |||||
[?] | ![]() ![]() ![]() ![]() ![]() t0,1,4{31,1,1,1} |
- | |||||
[?] | ![]() ![]() ![]() ![]() ![]() t0,1,2,3{31,1,1,1} |
{}x{}x{}x{} ![]() |
t0,2,3{31,1,1} | t0,2,3{31,1,1} | t0,2,3{31,1,1} | t0,2,3{31,1,1} | |
[?] | ![]() ![]() ![]() ![]() ![]() t0,1,2,3,4{31,1,1,1} |
{}x{}x{}x{}![]() |
t0,1,2,3{31,1,1} | t0,1,2,3{31,1,1} | t0,1,2,3{31,1,1} | t0,1,2,3{31,1,1} |
Duoprismatic forms
[edit]Coxeter groups:
- B~2xB~2: [4,4]x[4,4]
- B~2xH~2: [4,4]x[6,3]
- H~2xH~2: [6,3]x[6,3]
[4,4]×[4,4]
[edit]There are 15 reflective combinatoric forms, but only 3 unique ones.
# | Coxeter-Dynkin andSchläfli symbols |
Name |
---|---|---|
1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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63 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[63] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[6] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[63] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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10 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[10] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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67 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[10] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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[67] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[4,4]x[6,3]
[edit]There are 35 reflective combinatoric forms.
# | Coxeter-Dynkin | Name |
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[6,3]x[6,3]
[edit]There are 28 reflective combinatoric forms.
# | Coxeter-Dynkin diagram | Name |
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References
[edit]- ^ George Olshevsky (2006), Uniform Panoploid Tetracombs, manuscript. Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs.
- ^ Olshevsky section V
- ^ Olshevsky section VI
- ^ Olshevsky section VII
- ^ Olshevsky section VII