| Acronym | rojkohgyt |
| Name | rojak honeycomb gyrotrigonism |
|
Lace city in approx. ASCII-art |
where:
B A = o3x3o3o3o *c3o3o (rojkoh)
A B = o3o3o3x3o *c3o3o (gyro rojkoh)
C C = o3o3o3o3o *c3x3o (alt. gyro rojkoh)
|
This euclidean honeycomb uses rojkoh and rojkoha 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
ooo3xoo3ooo3oxo3ooo *c3oox3ooo&#x (N → ∞) → pw. heights = 1/sqrt(3) = 0.577350 o..3o..3o..3o..3o.. *c3o..3o.. & | 108N | 40 30 | 20 180 360 90 | 90 240 160 480 240 720 | 120 60 60 210 200 240 300 300 300 900 | 30 30 24 180 132 40 120 72 120 720 270 360 | 12 2 30 84 32 24 192 140 420 60 | 1 12 16 31 54 160 | 2 14 20 ------------------------------------+------+-------------+-------------------------+---------------------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+-------------------------------------------------+------------------------+---------- ... x.. ... ... ... ... ... & | 2 | 2160N * | 1 9 6 0 | 9 18 6 18 6 9 | 18 6 9 18 8 18 18 9 9 18 | 6 9 6 24 18 2 18 6 9 24 18 9 | 6 1 6 24 6 6 24 6 15 9 | 1 6 8 6 15 7 | 2 7 1 oo.3oo.3oo.3oo.3oo. *c3oo.3oo.&#x & | 2 | * 1620N | 0 0 16 6 | 0 0 8 24 16 60 | 0 0 0 12 16 24 16 28 28 96 | 0 0 0 24 8 4 16 4 13 88 40 48 | 0 0 6 16 2 4 36 20 64 16 | 0 4 4 8 20 28 | 1 8 4 ------------------------------------+------+-------------+-------------------------+---------------------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+-------------------------------------------------+------------------------+---------- o..3x.. ... ... ... ... ... & | 3 | 3 0 | 720N * * * | 9 0 6 0 0 0 | 18 0 0 18 6 0 0 9 0 0 | 6 9 0 18 18 2 0 0 9 18 0 0 | 6 0 6 18 6 0 18 6 9 0 | 1 6 6 6 9 6 | 2 6 1 ... x..3o.. ... ... ... ... & | 3 | 3 0 | * 6480N * * | 1 4 0 2 0 0 | 4 2 4 2 0 4 4 0 1 0 | 2 4 4 6 4 0 8 2 1 0 4 0 | 4 1 2 12 2 4 6 0 0 4 | 1 4 6 2 8 0 | 2 5 0 ... xo. ... ... ... ... ...&#x & | 3 | 1 2 | * * 12960N * | 0 0 1 3 2 3 | 0 0 0 3 3 6 3 3 3 9 | 0 0 0 9 3 1 6 1 3 13 9 6 | 0 0 3 9 1 2 13 4 11 6 | 0 3 3 4 11 6 | 1 6 1 ooo3ooo3ooo3ooo3ooo *c3ooo3ooo&#x | 3 | 0 3 | * * * 3240N | 0 0 0 0 0 12 | 0 0 0 0 0 0 0 6 6 24 | 0 0 0 0 0 0 0 0 3 24 12 16 | 0 0 0 0 0 0 12 6 24 8 | 0 0 0 3 12 12 | 0 6 2 ------------------------------------+------+-------------+-------------------------+---------------------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+-------------------------------------------------+------------------------+---------- o..3x..3o.. ... ... ... ... & ♦ 6 | 12 0 | 4 4 0 0 | 1620N * * * * * | 4 0 0 2 0 0 0 0 0 0 | 2 4 0 4 4 0 0 0 1 0 0 0 | 4 0 2 8 2 0 4 0 0 0 | 1 4 4 2 4 0 | 2 4 0 ... x..3o..3o.. ... ... ... & ♦ 4 | 6 0 | 0 4 0 0 | * 6480N * * * * | 1 1 2 0 0 1 1 0 0 0 | 1 2 3 2 1 0 4 1 0 0 1 0 | 3 1 1 7 1 3 2 0 0 2 | 1 3 5 1 5 0 | 2 4 0 oo.3xo. ... ... ... ... ...&#x & ♦ 4 | 3 3 | 1 0 3 0 | * * 4320N * * * | 0 0 0 3 2 0 0 3 0 0 | 0 0 0 6 3 1 0 0 3 9 0 0 | 0 0 3 6 1 0 9 4 6 0 | 0 3 2 4 6 5 | 1 5 1 ... xo.3oo. ... ... ... ...&#x & ♦ 4 | 3 3 | 0 1 3 0 | * * * 12960N * * | 0 0 0 1 0 2 2 0 1 0 | 0 0 0 3 2 0 4 1 1 0 4 0 | 0 0 1 6 1 2 6 0 0 4 | 0 2 3 2 8 0 | 1 5 0 ... xo. ... ox. ... ... ...&#x & ♦ 4 | 2 4 | 0 0 4 0 | * * * * 6480N * | 0 0 0 0 2 3 0 0 0 3 | 0 0 0 6 0 1 3 0 0 6 3 3 | 0 0 3 6 0 1 6 3 7 3 | 0 3 2 3 7 5 | 1 5 1 ... xoo ... ... ... ... ...&#x & ♦ 4 | 1 5 | 0 0 2 2 | * * * * * 19440N | 0 0 0 0 0 0 0 1 1 4 | 0 0 0 0 0 0 0 0 1 6 4 4 | 0 0 0 0 0 0 6 2 8 4 | 0 0 0 2 8 5 | 0 5 1 ------------------------------------+------+-------------+-------------------------+---------------------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+-------------------------------------------------+------------------------+---------- o..3x..3o..3o.. ... ... ... & ♦ 10 | 30 0 | 10 20 0 0 | 5 5 0 0 0 0 | 1296N * * * * * * * * * | 1 2 0 1 1 0 0 0 0 0 0 0 | 3 0 1 4 1 0 1 0 0 0 | 1 3 3 1 2 0 | 2 3 0 ... x..3o..3o..3o.. ... ... & ♦ 5 | 10 0 | 0 10 0 0 | 0 5 0 0 0 0 | * 1296N * * * * * * * * | 1 0 2 1 0 0 0 1 0 0 0 0 | 2 1 1 2 1 2 1 0 0 0 | 1 2 4 1 2 0 | 2 3 0 ... x..3o..3o.. ... *c3o.. ... & ♦ 8 | 24 0 | 0 32 0 0 | 0 16 0 0 0 0 | * * 810N * * * * * * * | 0 1 2 0 0 0 2 0 0 0 0 0 | 2 1 0 4 0 2 0 0 0 1 | 1 2 4 0 3 0 | 2 3 0 oo.3xo.3oo. ... ... ... ...&#x & ♦ 7 | 12 6 | 4 4 12 0 | 1 0 4 4 0 0 | * * * 3240N * * * * * * | 0 0 0 2 2 0 0 0 1 0 0 0 | 0 0 1 4 1 0 4 0 0 0 | 0 2 2 2 4 0 | 1 4 0 oo.3xo. ... ox. ... ... ...&#x & ♦ 5 | 4 6 | 1 0 9 0 | 0 0 2 0 3 0 | * * * * 4320N * * * * * | 0 0 0 3 0 1 0 0 0 3 0 0 | 0 0 3 3 0 0 3 3 3 0 | 0 3 1 3 3 4 | 1 4 1 ... xo.3oo.3ox. ... ... ...&#x & ♦ 8 | 12 12 | 0 8 24 0 | 0 2 0 8 6 0 | * * * * * 3240N * * * * | 0 0 0 2 0 0 2 0 0 0 1 0 | 0 0 1 4 0 1 2 0 0 2 | 0 2 2 1 5 0 | 1 4 0 ... xo.3oo. ... ... *c3oo. ...&#x & ♦ 5 | 6 4 | 0 4 6 0 | 0 1 0 4 0 0 | * * * * * * 6480N * * * | 0 0 0 0 1 0 2 1 0 0 1 0 | 0 0 0 3 1 2 2 0 0 2 | 0 1 3 1 5 0 | 1 4 0 ooo3xoo ... ... ... ... ...&#x & ♦ 5 | 3 7 | 1 0 6 3 | 0 0 2 0 0 3 | * * * * * * * 6480N * * | 0 0 0 0 0 0 0 0 1 4 0 0 | 0 0 0 0 0 0 4 2 4 0 | 0 0 0 2 4 4 | 0 4 1 ... xoo3ooo ... ... ... ...&#x & ♦ 5 | 3 7 | 0 1 6 3 | 0 0 0 2 0 3 | * * * * * * * * 6480N * | 0 0 0 0 0 0 0 0 1 0 4 0 | 0 0 0 0 0 0 6 0 0 4 | 0 0 0 2 8 0 | 0 5 0 ... xoo ... oxo ... ... ...&#x & ♦ 5 | 2 8 | 0 0 6 4 | 0 0 0 0 1 4 | * * * * * * * * * 19440N | 0 0 0 0 0 0 0 0 0 2 1 2 | 0 0 0 0 0 0 2 1 5 2 | 0 0 0 1 5 4 | 0 4 1 ------------------------------------+------+-------------+-------------------------+---------------------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+-------------------------------------------------+------------------------+---------- o..3x..3o..3o..3o.. ... ... & ♦ 15 | 60 0 | 20 60 0 0 | 15 30 0 0 0 0 | 6 6 0 0 0 0 0 0 0 0 | 216N * * * * * * * * * * * | 2 0 1 0 1 0 0 0 0 0 | 1 2 2 1 0 0 | 2 2 0 o..3x..3o..3o.. ... *c3o.. ... & ♦ 40 | 240 0 | 80 320 0 0 | 80 160 0 0 0 0 | 32 0 10 0 0 0 0 0 0 0 | * 81N * * * * * * * * * * | 2 0 0 2 0 0 0 0 0 0 | 1 2 2 0 1 0 | 2 2 0 ... x..3o..3o..3o.. *c3o.. ... & ♦ 16 | 80 0 | 0 160 0 0 | 0 120 0 0 0 0 | 0 16 10 0 0 0 0 0 0 0 | * * 162N * * * * * * * * * | 1 1 0 1 0 1 0 0 0 0 | 1 1 3 0 1 0 | 2 2 0 oo.3xo.3oo.3ox. ... ... ...&#x & ♦ 15 | 40 30 | 10 30 90 0 | 5 10 20 30 30 0 | 1 1 0 5 10 5 0 0 0 0 | * * * 1296N * * * * * * * * | 0 0 1 2 0 0 1 0 0 0 | 0 2 1 1 2 0 | 1 3 0 oo.3xo.3oo. ... ... *c3oo. ...&#x & ♦ 11 | 30 10 | 10 20 30 0 | 5 5 10 20 0 0 | 1 0 0 5 0 0 5 0 0 0 | * * * * 1296N * * * * * * * | 0 0 0 2 1 0 1 0 0 0 | 0 1 2 1 2 0 | 1 3 0 oo.3xo. ... ox.3oo. ... ...&#x & ♦ 6 | 6 9 | 2 0 18 0 | 0 0 6 0 9 0 | 0 0 0 0 6 0 0 0 0 0 | * * * * * 720N * * * * * * | 0 0 3 0 0 0 0 3 0 0 | 0 3 0 3 0 3 | 1 3 1 ... xo.3oo.3ox. ... *c3oo. ...&#x & ♦ 16 | 48 32 | 0 64 96 0 | 0 32 0 64 24 0 | 0 0 2 0 0 8 16 0 0 0 | * * * * * * 810N * * * * * | 0 0 0 2 0 1 0 0 0 1 | 0 1 2 0 3 0 | 1 3 0 ... xo.3oo. ... ... *c3oo.3oo.&#x & ♦ 6 | 10 5 | 0 10 10 0 | 0 5 0 10 0 0 | 0 1 0 0 0 0 5 0 0 0 | * * * * * * * 1296N * * * * | 0 0 0 0 1 2 1 0 0 0 | 0 0 3 1 2 0 | 1 3 0 ooo3xoo3ooo ... ... ... ...&#x & ♦ 8 | 12 13 | 4 4 24 6 | 1 0 8 8 0 12 | 0 0 0 2 0 0 0 4 4 0 | * * * * * * * * 1620N * * * | 0 0 0 0 0 0 4 0 0 0 | 0 0 0 2 4 0 | 0 4 0 ooo3xoo ... oxo ... ... ...&#x & ♦ 6 | 4 11 | 1 0 13 6 | 0 0 3 0 3 9 | 0 0 0 0 1 0 0 2 0 3 | * * * * * * * * * 12960N * * | 0 0 0 0 0 0 1 1 2 0 | 0 0 0 1 2 3 | 0 3 1 ... xoo3ooo3oxo ... ... ...&#x & ♦ 9 | 12 20 | 0 8 36 12 | 0 2 0 16 6 24 | 0 0 0 0 0 1 2 0 8 6 | * * * * * * * * * * 3240N * | 0 0 0 0 0 0 2 0 0 2 | 0 0 0 1 5 0 | 0 4 0 ... xoo ... oxo ... oox ...&#x ♦ 6 | 3 12 | 0 0 12 8 | 0 0 0 0 3 12 | 0 0 0 0 0 0 0 0 0 6 | * * * * * * * * * * * 6480N | 0 0 0 0 0 0 0 0 3 1 | 0 0 0 0 3 3 | 0 3 1 ------------------------------------+------+-------------+-------------------------+---------------------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+-------------------------------------------------+------------------------+---------- o..3x..3o..3o..3o.. *c3o.. ... & ♦ 216 | 2160 0 | 720 4320 0 0 | 1080 3240 0 0 0 0 | 648 432 270 0 0 0 0 0 0 0 | 72 27 27 0 0 0 0 0 0 0 0 0 | 6N * * * * * * * * * | 1 1 1 0 0 0 | 2 1 0 ... x..3o..3o..3o.. *c3o..3o.. & ♦ 72 | 720 0 | 0 2160 0 0 | 0 2160 0 0 0 0 | 0 432 270 0 0 0 0 0 0 0 | 0 0 54 0 0 0 0 0 0 0 0 0 | * 3N * * * * * * * * | 1 0 2 0 0 0 | 2 1 0 oo.3xo.3oo.3ox.3oo. ... ...&#x & ♦ 30 | 120 90 | 40 120 360 0 | 30 60 120 120 180 0 | 12 12 0 30 120 30 0 0 0 0 | 2 0 0 12 0 20 0 0 0 0 0 0 | * * 108N * * * * * * * | 0 2 0 1 0 0 | 1 2 0 oo.3xo.3oo.3ox. ... *c3oo. ...&#x & ♦ 56 | 320 160 | 80 480 720 0 | 80 280 160 480 240 0 | 32 16 20 80 80 80 120 0 0 0 | 0 1 1 16 16 0 10 0 0 0 0 0 | * * * 162N * * * * * * | 0 1 1 0 1 0 | 1 2 0 oo.3xo.3oo. ... ... *c3oo.3oo.&#x & ♦ 16 | 60 15 | 20 60 60 0 | 15 30 20 60 0 0 | 6 6 0 15 0 0 30 0 0 0 | 1 0 0 0 6 0 0 6 0 0 0 0 | * * * * 216N * * * * * | 0 0 2 1 0 0 | 1 2 0 ... xo.3oo.3ox. ... *c3oo.3oo.&#x & ♦ 32 | 160 80 | 0 320 320 0 | 0 240 0 320 80 0 | 0 32 20 0 0 40 160 0 0 0 | 0 0 2 0 0 0 10 32 0 0 0 0 | * * * * * 81N * * * * | 0 0 2 0 1 0 | 1 2 0 ooo3xoo3ooo3oxo ... ... ...&#x & ♦ 16 | 40 45 | 10 30 130 30 | 5 10 30 60 30 90 | 1 1 0 10 10 5 10 20 30 30 | 0 0 0 1 1 0 0 1 5 10 5 0 | * * * * * * 1296N * * * | 0 0 0 1 2 0 | 0 3 0 ooo3xoo ... oxo3ooo ... ...&#x & ♦ 7 | 6 15 | 2 0 24 9 | 0 0 8 0 9 18 | 0 0 0 0 6 0 0 6 0 9 | 0 0 0 0 0 1 0 0 0 6 0 0 | * * * * * * * 2160N * * | 0 0 0 1 0 2 | 0 2 1 ooo3xoo ... oxo ... oox ...&#x & ♦ 7 | 5 16 | 1 0 22 12 | 0 0 4 0 7 24 | 0 0 0 0 2 0 0 4 0 15 | 0 0 0 0 0 0 0 0 0 4 0 3 | * * * * * * * * 6480N * | 0 0 0 0 1 2 | 0 2 1 ... xoo3ooo3oxo ... *c3oox ...&#x ♦ 24 | 72 96 | 0 96 288 96 | 0 48 0 192 72 288 | 0 0 3 0 0 24 48 0 96 144 | 0 0 0 0 0 0 3 0 0 0 24 24 | * * * * * * * * * 270N | 0 0 0 0 3 0 | 0 3 0 ------------------------------------+------+-------------+-------------------------+---------------------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+-------------------------------------------------+------------------------+---------- o..3x..3o..3o..3o.. *c3o..3o.. & ♦ 36N | 720N 0 | 240N 2160N 0 0 | 540N 2160N 0 0 0 0 | 432N 432N 270N 0 0 0 0 0 0 0 | 72N 27N 54N 0 0 0 0 0 0 0 0 0 | 2N N 0 0 0 0 0 0 0 0 | 3 * * * * * | 2 0 0 oo.3xo.3oo.3ox.3oo. *c3oo. ...&#x & ♦ 432 | 4320 2160 | 1440 8640 12960 0 | 2160 6480 4320 8640 6480 0 | 1296 864 540 2160 4320 2160 2160 0 0 0 | 144 54 54 864 432 720 270 0 0 0 0 0 | 2 0 72 54 0 0 0 0 0 0 | * 3N * * * * | 1 1 0 oo.3xo.3oo.3ox. ... *c3oo.3oo.&#x & ♦ 288 | 2880 1080 | 720 6480 6480 0 | 1080 5400 1440 6480 2160 0 | 648 864 540 1080 720 1080 3240 0 0 0 | 72 27 81 216 432 0 270 648 0 0 0 0 | 1 1 0 27 72 27 0 0 0 0 | * * 6N * * * | 1 1 0 ooo3xoo3ooo3oxo3ooo ... ...&#x & ♦ 31 | 120 120 | 40 120 480 90 | 30 60 160 240 180 360 | 12 12 0 60 120 30 60 120 120 180 | 2 0 0 12 12 20 0 12 30 120 30 0 | 0 0 1 0 2 0 12 20 0 0 | * * * 108N * * | 0 2 0 ooo3xoo3ooo3oxo ... *c3oox ...&#x & ♦ 72 | 400 400 | 80 640 1760 480 | 80 400 320 1280 560 1920 | 32 32 30 160 160 200 400 320 640 1200 | 0 1 2 32 32 0 30 32 80 320 200 240 | 0 0 0 2 0 1 32 0 80 10 | * * * * 81N * | 0 2 0 ooo3xoo ... oxo3ooo oox ...&#x & ♦ 8 | 7 21 | 2 0 36 18 | 0 0 10 0 15 45 | 0 0 0 0 8 0 0 12 0 36 | 0 0 0 0 0 1 0 0 0 18 0 9 | 0 0 0 0 0 0 0 2 6 0 | * * * * * 2160N | 0 1 1 ------------------------------------+------+-------------+-------------------------+---------------------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+-------------------------------------------------+------------------------+---------- oo.3xo.3oo.3ox.3oo. *c3oo.3oo.&#x & ♦ 72N | 1440N 540N | 480N 4320N 4320N 0 | 1080N 4320N 1440N 4320N 2160N 0 | 864N 864N 540N 1080N 1440N 1080N 2160N 0 0 0 | 144N 54N 108N 432N 432N 240N 270N 432N 0 0 0 0 | 4N 2N 36N 54N 72N 27N 0 0 0 0 | 2 N 2N 0 0 0 | 3 * * ooo3xoo3ooo3oxo3ooo *c3oox ...&#x & ♦ 504 | 5040 4320 | 1440 10800 25920 6480 | 2160 8640 7200 21600 10800 32400 | 1296 1296 810 4320 5760 4320 8640 8640 10800 25920 | 144 54 108 1296 1296 720 810 1296 2160 12960 4320 6480 | 2 1 72 108 144 54 1296 1440 4320 270 | 0 1 2 72 54 720 | * 3N * ooo3xoo ... oxo3ooo oox3ooo&#x ♦ 9 | 9 27 | 3 0 54 27 | 0 0 18 0 27 81 | 0 0 0 0 18 0 0 27 0 81 | 0 0 0 0 0 3 0 0 0 54 0 27 | 0 0 0 0 0 0 0 9 27 0 | 0 0 0 0 0 9 | * * 240N
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