| Acronym | jakohgyt |
| Name | jak honeycomb gyrotrigonism |
|
Lace city in approx. ASCII-art |
where:
B A = x3o3o3o3o *c3o3o (jakoh)
A B = o3o3o3o3x *c3o3o (gyro jakoh)
C C = o3o3o3o3o *c3o3x (alt. gyro jakoh)
|
This euclidean honeycomb uses jakoh and jakoha in the sense of infinite horohedra. Note however, as a gyrotrigonism, within position space of the lace city display, it just happens to remain of finite reach. Thence it more looks like an infintely long triangular needle.
Incidence matrix according to Dynkin symbol
xoo3ooo3ooo3ooo3oxo *c3ooo3oox&#x (N → ∞) → pw. heights = 1/sqrt(3) = 0.577350 o..3o..3o..3o..3o.. *c3o..3o.. & | 3N | 72 54 | 720 1296 270 | 2160 5760 2160 4320 | 2160 10800 7200 10800 10800 | 432 270 2592 5184 6480 1440 12960 25920 6480 | 54 432 594 3024 1008 6048 15120 5040 15120 | 1 54 56 576 936 324 3456 4320 5760 | 2 55 504 720 ------------------------------------+----+----------+-----------------+-------------------------+-------------------------------+----------------------------------------------------+-------------------------------------------------+-----------------------------------------+--------------- x.. ... ... ... ... ... ... & | 2 | 108N * | 20 12 0 | 90 120 30 30 | 120 360 160 180 120 | 30 30 120 240 210 40 360 480 90 | 12 30 60 132 60 240 420 120 300 | 1 12 12 32 60 30 132 120 140 | 2 12 31 20 oo.3oo.3oo.3oo.3oo. *c3oo.3oo.&#x & | 2 | * 81N | 0 32 10 | 0 160 80 200 | 0 320 320 560 640 | 0 0 80 160 320 80 720 1760 480 | 0 40 20 160 32 352 1120 400 1280 | 0 10 2 32 112 21 272 400 560 | 1 12 72 80 ------------------------------------+----+----------+-----------------+-------------------------+-------------------------------+----------------------------------------------------+-------------------------------------------------+-----------------------------------------+--------------- x..3o.. ... ... ... ... ... & | 3 | 3 0 | 720N * * | 9 6 0 0 | 18 36 6 9 0 | 6 9 18 36 18 2 36 18 0 | 6 6 18 18 12 36 36 6 9 | 1 6 6 6 12 9 18 9 6 | 2 6 6 1 xo. ... ... ... ... ... ...&#x & | 3 | 1 2 | * 1296N * | 0 10 5 5 | 0 30 30 30 30 | 0 0 10 20 40 10 60 130 30 | 0 10 5 25 5 40 115 40 110 | 0 5 1 6 25 5 36 45 60 | 1 6 16 10 ooo3ooo3ooo3ooo3ooo *c3ooo3ooo#x | 3 | 0 3 | * * 270N | 0 0 0 24 | 0 0 0 72 96 | 0 0 0 0 0 0 96 288 96 | 0 0 0 0 0 48 192 72 288 | 0 0 0 0 24 3 48 96 144 | 0 3 24 24 ------------------------------------+----+----------+-----------------+-------------------------+-------------------------------+----------------------------------------------------+-------------------------------------------------+-----------------------------------------+--------------- x..3o..3o.. ... ... ... ... & ♦ 4 | 6 0 | 4 0 0 | 1620N * * * | 4 4 0 0 0 | 2 4 4 8 2 0 4 0 0 | 4 2 8 4 4 8 4 0 0 | 1 4 4 2 4 4 4 1 0 | 2 4 2 0 xo.3oo. ... ... ... ... ...&#x & ♦ 4 | 3 3 | 1 3 0 | * 4320N * * | 0 6 2 3 0 | 0 0 3 6 6 1 12 9 0 | 0 3 3 6 2 12 18 4 6 | 0 3 1 2 8 3 9 6 5 | 1 4 5 1 xo. ... ... ... ox. ... ...&#x & ♦ 4 | 2 4 | 0 4 0 | * * 1620N * | 0 0 8 0 4 | 0 0 0 0 12 4 0 24 6 | 0 6 0 8 0 0 24 12 28 | 0 4 0 2 12 0 8 13 20 | 1 4 8 4 xoo ... ... ... ... ... ...&#x & ♦ 4 | 1 5 | 0 2 2 | * * * 3240N | 0 0 0 6 8 | 0 0 0 0 0 0 12 36 12 | 0 0 0 0 0 8 32 12 48 | 0 0 0 0 8 1 10 20 30 | 0 2 9 6 ------------------------------------+----+----------+-----------------+-------------------------+-------------------------------+----------------------------------------------------+-------------------------------------------------+-----------------------------------------+--------------- x..3o..3o..3o.. ... ... ... & ♦ 5 | 10 0 | 10 0 0 | 5 0 0 0 | 1296N * * * * | 1 2 1 2 0 0 0 0 0 | 3 1 4 1 2 2 0 0 0 | 1 3 3 1 2 2 1 0 0 | 2 3 1 0 xo.3oo.3oo. ... ... ... ...&#x & ♦ 5 | 6 4 | 4 6 0 | 1 4 0 0 | * 6480N * * * | 0 0 1 2 1 0 2 0 0 | 0 1 2 2 1 4 3 0 0 | 0 2 1 1 3 2 3 1 0 | 1 3 2 0 xo.3oo. ... ... ox. ... ...&#x & ♦ 5 | 4 6 | 1 9 0 | 0 2 3 0 | * * 4320N * * | 0 0 0 0 3 1 0 3 0 | 0 3 0 3 0 0 6 3 3 | 0 3 0 1 6 0 3 3 4 | 1 3 4 1 xoo3ooo ... ... ... ... ...&#x & ♦ 5 | 3 7 | 1 6 3 | 0 2 0 3 | * * * 6480N * | 0 0 0 0 0 0 4 4 0 | 0 0 0 0 0 4 8 2 4 | 0 0 0 0 4 1 4 4 4 | 0 2 4 1 xoo ... ... ... oxo ... ...&#x & ♦ 5 | 2 8 | 0 6 4 | 0 0 1 4 | * * * * 6480N | 0 0 0 0 0 0 0 6 3 | 0 0 0 0 0 0 6 3 15 | 0 0 0 0 3 0 2 7 12 | 0 1 5 3 ------------------------------------+----+----------+-----------------+-------------------------+-------------------------------+----------------------------------------------------+-------------------------------------------------+-----------------------------------------+--------------- x..3o..3o..3o..3o.. ... ... & ♦ 6 | 15 0 | 20 0 0 | 15 0 0 0 | 6 0 0 0 0 | 216N * * * * * * * * | 2 1 0 0 2 0 0 0 0 | 1 2 2 1 2 0 0 0 0 | 2 2 1 0 x..3o..3o..3o.. ... *c3o.. ... & ♦ 10 | 40 0 | 80 0 0 | 80 0 0 0 | 32 0 0 0 0 | * 81N * * * * * * * | 2 0 2 0 0 2 0 0 0 | 1 2 2 0 0 1 0 0 0 | 2 2 0 0 xo.3oo.3oo.3oo. ... ... ...&#x & ♦ 6 | 10 5 | 10 10 0 | 5 10 0 0 | 1 5 0 0 0 | * * 1296N * * * * * * | 0 1 2 0 0 2 0 0 0 | 0 2 1 0 2 2 1 0 0 | 1 3 1 0 xo.3oo.3oo. ... ... *c3oo. ...&#x & ♦ 6 | 10 5 | 10 10 0 | 5 10 0 0 | 1 5 0 0 0 | * * * 2592N * * * * * | 0 0 1 1 1 1 0 0 0 | 0 1 1 1 1 1 1 0 0 | 1 2 1 0 xo.3oo.3oo. ... ox. ... ...&#x & ♦ 6 | 7 8 | 4 16 0 | 1 8 6 0 | 0 2 4 0 0 | * * * * 3240N * * * * | 0 1 0 2 0 0 2 0 0 | 0 2 0 1 2 0 2 1 0 | 1 2 2 0 xo.3oo. ... oo.3ox. ... ...&#x ♦ 6 | 6 9 | 2 18 0 | 0 6 9 0 | 0 0 6 0 0 | * * * * * 720N * * * | 0 3 0 0 0 0 0 3 0 | 0 3 0 0 6 0 0 0 3 | 1 3 3 1 xoo3ooo3ooo ... ... ... ...&#x & ♦ 6 | 6 9 | 4 12 4 | 1 8 0 6 | 0 2 0 4 0 | * * * * * * 6480N * * | 0 0 0 0 0 2 2 0 0 | 0 0 0 0 2 1 2 1 0 | 0 2 2 0 xoo3ooo ... ... oxo ... ...&#x & ♦ 6 | 4 11 | 1 13 6 | 0 3 3 9 | 0 0 1 2 3 | * * * * * * * 12960N * | 0 0 0 0 0 0 2 1 2 | 0 0 0 0 2 0 1 2 3 | 0 1 3 1 xoo ... ... ... oxo ... oox&#x ♦ 6 | 3 12 | 0 12 8 | 0 0 3 12 | 0 0 0 0 6 | * * * * * * * * 3240N | 0 0 0 0 0 0 0 0 6 | 0 0 0 0 0 0 0 3 6 | 0 0 3 2 ------------------------------------+----+----------+-----------------+-------------------------+-------------------------------+----------------------------------------------------+-------------------------------------------------+-----------------------------------------+--------------- x..3o..3o..3o..3o.. *c3o.. ... & ♦ 27 | 216 0 | 720 0 0 | 1080 0 0 0 | 648 0 0 0 0 | 72 27 0 0 0 0 0 0 0 | 6N * * * * * * * * | 1 1 1 0 0 0 0 0 0 | 2 1 0 0 xo.3oo.3oo.3oo.3ox. ... ...&#x & ♦ 12 | 30 30 | 40 120 0 | 30 120 90 0 | 12 60 120 0 0 | 2 0 12 0 30 20 0 0 0 | * 108N * * * * * * * | 0 2 0 0 2 0 0 0 0 | 1 2 1 0 xo.3oo.3oo.3oo. ... *c3oo. ...&#x & ♦ 11 | 40 10 | 80 40 0 | 80 80 0 0 | 32 80 0 0 0 | 0 1 16 16 0 0 0 0 0 | * * 162N * * * * * * | 0 1 1 0 0 1 0 0 0 | 1 2 0 0 xo.3oo.3oo. ... ox. *c3oo. ...&#x & ♦ 7 | 11 10 | 10 25 0 | 5 20 10 0 | 1 10 10 0 0 | 0 0 0 2 5 0 0 0 0 | * * * 1296N * * * * * | 0 1 0 1 0 0 1 0 0 | 1 1 1 0 xo.3oo.3oo. ... ... *c3oo.3oo.&#x & ♦ 7 | 15 6 | 20 15 0 | 15 20 0 0 | 6 15 0 0 0 | 1 0 0 6 0 0 0 0 0 | * * * * 432N * * * * | 0 0 1 1 1 0 0 0 0 | 1 1 1 0 xoo3ooo3ooo3ooo ... ... ...&#x & ♦ 7 | 10 11 | 10 20 5 | 5 20 0 10 | 1 10 0 10 0 | 0 0 1 1 0 0 5 0 0 | * * * * * 2592N * * * | 0 0 0 0 1 1 1 0 0 | 0 2 1 0 xoo3ooo3ooo ... oxo ... ...&#x & ♦ 7 | 7 14 | 4 23 8 | 1 12 6 16 | 0 3 4 8 6 | 0 0 0 0 1 0 2 4 0 | * * * * * * 6480N * * | 0 0 0 0 1 0 1 1 0 | 0 1 2 0 xoo3ooo ... ooo3oxo ... ...&#x & ♦ 7 | 6 15 | 2 24 9 | 0 8 9 18 | 0 0 6 6 9 | 0 0 0 0 0 1 0 6 0 | * * * * * * * 2160N * | 0 0 0 0 2 0 0 0 2 | 0 1 2 1 xoo3ooo ... ... oxo ... oox&#x & ♦ 7 | 5 16 | 1 22 12 | 0 4 7 24 | 0 0 2 4 15 | 0 0 0 0 0 0 0 4 3 | * * * * * * * * 6480N | 0 0 0 0 0 0 0 1 2 | 0 0 2 1 ------------------------------------+----+----------+-----------------+-------------------------+-------------------------------+----------------------------------------------------+-------------------------------------------------+-----------------------------------------+--------------- x..3o..3o..3o..3o.. *c3o..3o.. & ♦ N | 36N 0 | 240N 0 0 | 540N 0 0 0 | 432N 0 0 0 0 | 72N 27N 0 0 0 0 0 0 0 | 2N 0 0 0 0 0 0 0 0 | 3 * * * * * * * * | 2 0 0 0 xo.3oo.3oo.3oo.3ox. *c3oo. ...&#x & ♦ 54 | 432 270 | 1440 2160 0 | 2160 4320 2160 0 | 1296 4320 4320 0 0 | 144 54 864 864 2160 720 0 0 0 | 2 72 54 432 0 0 0 0 0 | * 3N * * * * * * * | 1 1 0 0 xo.3oo.3oo.3oo. ... *c3oo.3oo.&#x & ♦ 28 | 216 27 | 720 216 0 | 1080 720 0 0 | 648 1080 0 0 0 | 72 27 216 432 0 0 0 0 0 | 1 0 27 0 72 0 0 0 0 | * * 6N * * * * * * | 1 1 0 0 xo.3oo.3oo. ... ox. *c3oo.3oo.&#x & ♦ 8 | 16 12 | 20 36 0 | 15 40 15 0 | 6 30 20 0 0 | 1 0 0 12 15 0 0 0 0 | 0 0 0 6 2 0 0 0 0 | * * * 216N * * * * * | 1 0 1 0 xoo3ooo3ooo3ooo3oxo ... ...&#x & ♦ 13 | 30 42 | 40 150 30 | 30 160 90 120 | 12 90 120 120 90 | 2 0 12 12 30 20 60 120 0 | 0 1 0 0 2 12 30 20 0 | * * * * 216N * * * * | 0 1 1 0 xoo3ooo3ooo3ooo ... *c3ooo ...&#x & ♦ 12 | 40 21 | 80 80 10 | 80 160 0 40 | 32 160 0 80 0 | 0 1 32 32 0 0 80 0 0 | 0 0 2 0 0 32 0 0 0 | * * * * * 81N * * * | 0 2 0 0 xoo3ooo3ooo3ooo ... ... oox&#x & ♦ 8 | 11 17 | 10 36 10 | 5 30 10 25 | 1 15 10 20 10 | 0 0 1 2 5 0 10 10 0 | 0 0 0 1 0 2 5 0 0 | * * * * * * 1296N * * | 0 1 1 0 xoo3ooo3ooo ... oxo ... oox&#x & ♦ 8 | 8 20 | 4 36 16 | 1 16 13 40 | 0 4 8 16 28 | 0 0 0 0 2 0 4 16 6 | 0 0 0 0 0 0 4 0 4 | * * * * * * * 1620N * | 0 0 2 0 xoo3ooo ... ooo3oxo ... oox&#x & ♦ 8 | 7 21 | 2 36 18 | 0 10 15 45 | 0 0 8 12 36 | 0 0 0 0 0 1 0 18 9 | 0 0 0 0 0 0 0 2 6 | * * * * * * * * 2160N | 0 0 1 1 ------------------------------------+----+----------+-----------------+-------------------------+-------------------------------+----------------------------------------------------+-------------------------------------------------+-----------------------------------------+--------------- xo.3oo.3oo.3oo.3ox. *c3oo.3oo.&#x & ♦ 2N | 72N 27N | 480N 432N 0 | 1080N 1440N 540N 0 | 864N 2160N 1440N 0 0 | 144N 54N 432N 864N 1080N 240N 0 0 0 | 4N 36N 54N 432N 144N 0 0 0 0 | 2 N 2N 72N 0 0 0 0 0 | 3 * * * xoo3ooo3ooo3ooo3oxo *c3ooo ...&#x & ♦ 55 | 432 324 | 1440 2592 270 | 2160 5760 2160 2160 | 1296 6480 4320 4320 2160 | 144 54 1296 1728 2160 720 4320 4320 0 | 2 72 108 432 144 1728 2160 720 0 | 0 1 2 0 72 54 432 0 0 | * 3N * * xoo3ooo3ooo3ooo3oxo ... oox&#x & ♦ 14 | 31 54 | 40 192 60 | 30 200 120 270 | 12 120 160 240 300 | 2 0 12 24 60 20 120 360 90 | 0 1 0 12 4 24 120 40 120 | 0 0 0 2 2 0 12 30 20 | * * 108N * xoo3ooo ... ooo3oxo ooo3oox&#x ♦ 9 | 9 27 | 3 54 27 | 0 18 27 81 | 0 0 18 27 81 | 0 0 0 0 0 3 0 54 27 | 0 0 0 0 0 0 0 9 27 | 0 0 0 0 0 0 0 0 9 | * * * 240N
© 2004-2026 | top of page |