Acronym | trial triddip | |
Name |
{3} atop gyrated triddip, {3}-first cap of rix | |
Circumradius | sqrt(2/3) = 0.816497 | |
Lace city in approx. ASCII-art |
T t o l where: o = o o3o (point) l = x o3o (line) t = o x3o ({3}) T = x x3o (trip) | |
Lace hyper city in approx. ASCII-art |
o3x x3o x3o x3o(shallow triangular pyramid with {3}-tip and all dual {3}'s for base vertices) | |
where: o = o3o 3 = x3o | ||
Face vector | 12, 39, 52, 33, 10 | |
Confer |
|
Incidence matrix according to Dynkin symbol
xo3ox ox3oo&#x → height = 1/sqrt(3) = 0.577350
({3} || gyro triddip)
o.3o. o.3o. | 3 * ♦ 2 6 0 0 | 1 6 3 6 0 0 0 | 3 6 3 2 0 0 | 3 2 1 0
.o3.o .o3.o | * 9 | 0 2 2 2 | 0 1 2 4 1 4 1 | 1 2 4 2 2 2 | 2 1 2 1
---------------+-----+----------+----------------+-------------+--------
x. .. .. .. | 2 0 | 3 * * * | 1 3 0 0 0 0 0 | 3 3 0 0 0 0 | 3 1 0 0
oo3oo oo3oo&#x | 1 1 | * 18 * * | 0 1 1 2 0 0 0 | 1 2 2 1 0 0 | 2 1 1 0
.. .x .. .. | 0 2 | * * 9 * | 0 0 1 0 1 2 0 | 1 0 2 0 2 1 | 2 0 1 1
.. .. .x .. | 0 2 | * * * 9 | 0 0 0 2 0 2 1 | 0 1 2 2 1 2 | 1 1 2 1
---------------+-----+----------+----------------+-------------+--------
x.3o. .. .. | 3 0 | 3 0 0 0 | 1 * * * * * * | 3 0 0 0 0 0 | 3 0 0 0
xo .. .. ..&#x | 2 1 | 1 2 0 0 | * 9 * * * * * | 1 2 0 0 0 0 | 2 1 0 0
.. ox .. ..&#x | 1 2 | 0 2 1 0 | * * 9 * * * * | 1 0 2 0 0 0 | 2 0 1 0
.. .. ox ..&#x | 1 2 | 0 2 0 1 | * * * 18 * * * | 0 1 1 1 0 0 | 1 1 1 0
.o3.x .. .. | 0 3 | 0 0 3 0 | * * * * 3 * * | 1 0 0 0 2 0 | 2 0 0 1
.. .x .x .. | 0 4 | 0 0 2 2 | * * * * * 9 * | 0 0 1 0 1 1 | 1 0 1 1
.. .. .x3.o | 0 3 | 0 0 0 3 | * * * * * * 3 | 0 0 0 2 0 2 | 0 1 2 1
---------------+-----+----------+----------------+-------------+--------
xo3ox .. ..&#x ♦ 3 3 | 3 6 3 0 | 1 3 3 0 1 0 0 | 3 * * * * * | 2 0 0 0
xo .. ox ..&#x ♦ 2 2 | 1 4 0 1 | 0 2 0 2 0 0 0 | * 9 * * * * | 1 1 0 0
.. ox ox ..&#x ♦ 1 4 | 0 4 2 2 | 0 0 2 2 0 1 0 | * * 9 * * * | 1 0 1 0
.. .. ox3oo&#x ♦ 1 3 | 0 3 0 3 | 0 0 0 3 0 0 1 | * * * 6 * * | 0 1 1 0
.o3.x .x .. ♦ 0 6 | 0 0 6 3 | 0 0 0 0 2 3 0 | * * * * 3 * | 1 0 0 1
.. .x .x3.o ♦ 0 6 | 0 0 3 6 | 0 0 0 0 0 3 2 | * * * * * 3 | 0 0 1 1
---------------+-----+----------+----------------+-------------+--------
xo3ox ox ..&#x ♦ 3 6 | 3 12 6 3 | 1 6 6 6 2 3 0 | 2 3 3 0 1 0 | 3 * * *
xo .. ox3oo&#x ♦ 2 3 | 1 6 0 3 | 0 3 0 6 0 0 1 | 0 3 0 2 0 0 | * 3 * *
.. ox ox3oo&#x ♦ 1 6 | 0 6 3 6 | 0 0 3 6 0 3 2 | 0 0 3 2 0 1 | * * 3 *
.o3.x .x3.o ♦ 0 9 | 0 0 9 9 | 0 0 0 0 3 9 3 | 0 0 0 0 3 3 | * * * 1
xoo xox3oxo&#x → height(1,2) = height(1,3) = 1/sqrt(3) = 0.577350 height(2,3) = sqrt(2/3) = 0.816497 o.. o..3o.. | 6 * * | 1 2 1 2 0 0 0 | 2 1 2 2 1 1 2 2 0 0 0 0 | 1 2 1 1 2 1 2 2 1 0 | 1 1 2 1 1 .o. .o.3.o. | * 3 * ♦ 0 0 4 0 2 2 0 | 0 0 2 2 4 0 0 4 1 1 2 0 | 0 1 2 2 0 0 2 2 4 1 | 1 0 1 2 2 ..o ..o3..o | * * 3 | 0 0 0 2 0 2 2 | 0 0 0 0 0 1 4 4 0 2 1 1 | 0 0 0 0 2 2 2 4 2 1 | 0 1 2 1 2 ---------------+-------+----------------+--------------------------+---------------------+---------- x.. ... ... | 2 0 0 | 3 * * * * * * | 2 0 2 0 0 1 0 0 0 0 0 0 | 1 2 1 0 2 0 2 0 0 0 | 1 1 2 1 0 ... x.. ... | 2 0 0 | * 6 * * * * * | 1 1 0 1 0 0 1 0 0 0 0 0 | 1 1 0 1 1 1 0 1 0 0 | 1 1 1 0 1 oo. oo.3oo.&#x | 1 1 0 | * * 12 * * * * | 0 0 1 1 1 0 0 1 0 0 0 0 | 0 1 1 1 0 0 1 1 1 0 | 1 0 1 1 1 o.o o.o3o.o&#x | 1 0 1 | * * * 6 * * * | 0 0 0 0 0 1 2 2 0 0 0 0 | 0 0 0 0 2 1 2 2 1 0 | 0 1 2 1 1 ... ... .x. | 0 2 0 | * * * * 3 * * | 0 0 0 0 2 0 0 0 1 0 1 0 | 0 0 1 2 0 0 0 0 2 1 | 1 0 0 1 2 .oo .oo3.oo&#x | 0 1 1 | * * * * * 6 * | 0 0 0 0 0 0 0 2 0 1 1 0 | 0 0 0 0 0 0 1 2 2 1 | 0 0 1 1 2 ... ..x ... | 0 0 2 | * * * * * * 3 | 0 0 0 0 0 0 2 0 0 1 0 1 | 0 0 0 0 1 2 0 2 0 1 | 0 1 1 0 2 ---------------+-------+----------------+--------------------------+---------------------+---------- x.. x.. ... | 4 0 0 | 2 2 0 0 0 0 0 | 3 * * * * * * * * * * * | 1 1 0 0 1 0 0 0 0 0 | 1 1 1 0 0 ... x..3o.. | 3 0 0 | 0 3 0 0 0 0 0 | * 2 * * * * * * * * * * | 1 0 0 1 0 1 0 0 0 0 | 1 1 0 0 1 xo. ... ...&#x | 2 1 0 | 1 0 2 0 0 0 0 | * * 6 * * * * * * * * * | 0 1 1 0 0 0 1 0 0 0 | 1 0 1 1 0 ... xo. ...&#x | 2 1 0 | 0 1 2 0 0 0 0 | * * * 6 * * * * * * * * | 0 1 0 1 0 0 0 1 0 0 | 1 0 1 0 1 ... ... ox.&#x | 1 2 0 | 0 0 2 0 1 0 0 | * * * * 6 * * * * * * * | 0 0 1 1 0 0 0 0 1 0 | 1 0 0 1 1 x.o ... ...&#x | 2 0 1 | 1 0 0 2 0 0 0 | * * * * * 3 * * * * * * | 0 0 0 0 2 0 2 0 0 0 | 0 1 2 1 0 ... x.x ...&#x | 2 0 2 | 0 1 0 2 0 0 1 | * * * * * * 6 * * * * * | 0 0 0 0 1 1 0 1 0 0 | 0 1 1 0 1 ooo ooo3ooo&#x | 1 1 1 | 0 0 1 1 0 1 0 | * * * * * * * 12 * * * * | 0 0 0 0 0 0 1 1 1 0 | 0 0 1 1 1 ... .o.3.x. | 0 3 0 | 0 0 0 0 3 0 0 | * * * * * * * * 1 * * * | 0 0 0 2 0 0 0 0 0 1 | 1 0 0 0 2 ... .ox ...&#x | 0 1 2 | 0 0 0 0 0 2 1 | * * * * * * * * * 3 * * | 0 0 0 0 0 0 0 2 0 1 | 0 0 1 0 2 ... ... .xo&#x | 0 2 1 | 0 0 0 0 1 2 0 | * * * * * * * * * * 3 * | 0 0 0 0 0 0 0 0 2 1 | 0 0 0 1 2 ... ..x3..o | 0 0 3 | 0 0 0 0 0 0 3 | * * * * * * * * * * * 1 | 0 0 0 0 0 2 0 0 0 1 | 0 1 0 0 2 ---------------+-------+----------------+--------------------------+---------------------+---------- x.. x..3o.. ♦ 6 0 0 | 3 6 0 0 0 0 0 | 3 2 0 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * | 1 1 0 0 0 xo. xo. ...&#x ♦ 4 1 0 | 2 2 4 0 0 0 0 | 1 0 2 2 0 0 0 0 0 0 0 0 | * 3 * * * * * * * * | 1 0 1 0 0 xo. ... ox.&#x ♦ 2 2 0 | 1 0 4 0 1 0 0 | 0 0 2 0 2 0 0 0 0 0 0 0 | * * 3 * * * * * * * | 1 0 0 1 0 ... xo.3ox.&#x ♦ 3 3 0 | 0 3 6 0 3 0 0 | 0 1 0 3 3 0 0 0 1 0 0 0 | * * * 2 * * * * * * | 1 0 0 0 1 x.o x.x ...&#x ♦ 4 0 2 | 2 2 0 4 0 0 1 | 1 0 0 0 0 2 2 0 0 0 0 0 | * * * * 3 * * * * * | 0 1 1 0 0 ... x.x3o.o&#x ♦ 3 0 3 | 0 3 0 3 0 0 3 | 0 1 0 0 0 0 3 0 0 0 0 1 | * * * * * 2 * * * * | 0 1 0 0 1 xoo ... ...&#x ♦ 2 1 1 | 1 0 2 2 0 1 0 | 0 0 1 0 0 1 0 2 0 0 0 0 | * * * * * * 6 * * * | 0 0 1 1 0 ... xox ...&#x ♦ 2 1 2 | 0 1 2 2 0 2 1 | 0 0 0 1 0 0 1 2 0 1 0 0 | * * * * * * * 6 * * | 0 0 1 0 1 ... ... oxo&#x ♦ 1 2 1 | 0 0 2 1 1 2 0 | 0 0 0 0 1 0 0 2 0 0 1 0 | * * * * * * * * 6 * | 0 0 0 1 1 ... .ox3.xo&#x ♦ 0 3 3 | 0 0 0 0 3 6 3 | 0 0 0 0 0 0 0 0 1 3 3 1 | * * * * * * * * * 1 | 0 0 0 0 2 ---------------+-------+----------------+--------------------------+---------------------+---------- xo. xo.3ox.&#x ♦ 6 3 0 | 3 6 12 0 3 0 0 | 3 2 6 6 6 0 0 0 1 0 0 0 | 1 3 3 2 0 0 0 0 0 0 | 1 * * * * x.o x.x3o.o&#x ♦ 6 0 3 | 3 6 0 6 0 0 3 | 3 2 0 0 0 3 6 0 0 0 0 1 | 1 0 0 0 3 2 0 0 0 0 | * 1 * * * xoo xox ...&#x ♦ 4 1 2 | 2 2 4 4 0 2 1 | 1 0 2 2 0 2 2 4 0 1 0 0 | 0 1 0 0 1 0 2 2 0 0 | * * 3 * * xoo ... oxo&#x ♦ 2 2 1 | 1 0 4 2 1 2 0 | 0 0 2 0 2 1 0 4 0 0 1 0 | 0 0 1 0 0 0 2 0 2 0 | * * * 3 * ... xox3oxo&#x ♦ 3 3 3 | 0 3 6 3 3 6 3 | 0 1 0 3 3 0 3 6 1 3 3 1 | 0 0 0 1 0 1 0 3 3 1 | * * * * 2
ooox3xxxo&#x → height(1,2) = height(1,3) = height(2,3) = 1 height(1,4) = height(2,4) = height(3,4) = sqrt(2/3) = sqrt(2/3) = 0.816497 o...3o... | 3 * * * | 2 1 1 2 0 0 0 0 0 0 | 1 2 1 2 2 2 1 2 0 0 0 0 0 0 0 0 0 | 1 2 2 1 2 1 1 2 1 0 0 0 0 0 | 1 1 2 1 1 0 .o..3.o.. | * 3 * * | 0 1 0 0 2 1 2 0 0 0 | 0 2 1 2 0 0 0 0 1 2 2 1 2 0 0 0 0 | 1 2 2 1 2 0 0 0 0 1 1 2 1 0 | 1 1 2 1 0 1 ..o.3..o. | * * 3 * | 0 0 1 0 0 1 0 2 2 0 | 0 0 1 0 2 2 0 0 0 2 2 0 0 1 1 2 0 | 0 2 2 0 0 1 1 2 0 1 1 2 0 1 | 1 1 2 0 1 1 ...o3...o | * * * 3 ♦ 0 0 0 2 0 0 2 0 2 2 | 0 0 0 2 0 2 2 1 0 0 2 2 1 0 2 1 1 | 0 0 2 2 1 0 2 1 1 0 2 1 1 1 | 0 2 1 1 1 1 -------------+---------+---------------------+-----------------------------------+-----------------------------+------------ .... x... | 2 0 0 0 | 3 * * * * * * * * * | 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 | 1 1 0 0 1 1 0 1 1 0 0 0 0 0 | 1 0 1 1 1 0 oo..3oo..&#x | 1 1 0 0 | * 3 * * * * * * * * | 0 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 2 2 1 2 0 0 0 0 0 0 0 0 0 | 1 1 2 1 0 0 o.o.3o.o.&#x | 1 0 1 0 | * * 3 * * * * * * * | 0 0 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 | 0 2 2 0 0 1 1 2 0 0 0 0 0 0 | 1 1 2 0 1 0 o..o3o..o&#x | 1 0 0 1 | * * * 6 * * * * * * | 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 | 0 0 1 1 1 0 1 1 1 0 0 0 0 0 | 0 1 1 1 1 0 .... .x.. | 0 2 0 0 | * * * * 3 * * * * * | 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 | 1 1 0 0 1 0 0 0 0 1 0 1 1 0 | 1 0 1 1 0 1 .oo.3.oo.&#x | 0 1 1 0 | * * * * * 3 * * * * | 0 0 1 0 0 0 0 0 0 2 2 0 0 0 0 0 0 | 0 2 2 0 0 0 0 0 0 1 1 2 0 0 | 1 1 2 0 0 1 .o.o3.o.o&#x | 0 1 0 1 | * * * * * * 6 * * * | 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 | 0 0 1 1 1 0 0 0 0 0 1 1 1 0 | 0 1 1 1 0 1 .... ..x. | 0 0 2 0 | * * * * * * * 3 * * | 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 | 0 1 0 0 0 1 0 1 0 1 0 1 0 1 | 1 0 1 0 1 1 ..oo3..oo&#x | 0 0 1 1 | * * * * * * * * 6 * | 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 | 0 0 1 0 0 0 1 1 0 0 1 1 0 1 | 0 1 1 0 1 1 ...x .... | 0 0 0 2 | * * * * * * * * * 3 | 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 | 0 0 0 1 0 0 1 0 1 0 1 0 1 1 | 0 1 0 1 1 1 -------------+---------+---------------------+-----------------------------------+-----------------------------+------------ o...3x... | 3 0 0 0 | 3 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * * * * * | 1 0 0 0 0 1 0 0 1 0 0 0 0 0 | 1 0 0 1 1 0 .... xx..&#x | 2 2 0 0 | 1 2 0 0 1 0 0 0 0 0 | * 3 * * * * * * * * * * * * * * * | 1 1 0 0 1 0 0 0 0 0 0 0 0 0 | 1 0 1 1 0 0 ooo.3ooo.&#x | 1 1 1 0 | 0 1 1 0 0 1 0 0 0 0 | * * 3 * * * * * * * * * * * * * * | 0 2 2 0 0 0 0 0 0 0 0 0 0 0 | 1 1 2 0 0 0 oo.o3oo.o&#x | 1 1 0 1 | 0 1 0 1 0 0 1 0 0 0 | * * * 6 * * * * * * * * * * * * * | 0 0 1 1 1 0 0 0 0 0 0 0 0 0 | 0 1 1 1 0 0 .... x.x.&#x | 2 0 2 0 | 1 0 2 0 0 0 0 1 0 0 | * * * * 3 * * * * * * * * * * * * | 0 1 0 0 0 1 0 1 0 0 0 0 0 0 | 1 0 1 0 1 0 o.oo3o.oo&#x | 1 0 1 1 | 0 0 1 1 0 0 0 0 1 0 | * * * * * 6 * * * * * * * * * * * | 0 0 1 0 0 0 1 1 0 0 0 0 0 0 | 0 1 1 0 1 0 o..x ....&#x | 1 0 0 2 | 0 0 0 2 0 0 0 0 0 1 | * * * * * * 3 * * * * * * * * * * | 0 0 0 1 0 0 1 0 1 0 0 0 0 0 | 0 1 0 1 1 0 .... x..o&#x | 2 0 0 1 | 1 0 0 2 0 0 0 0 0 0 | * * * * * * * 3 * * * * * * * * * | 0 0 0 0 1 0 0 1 1 0 0 0 0 0 | 0 0 1 1 1 0 .o..3.x.. | 0 3 0 0 | 0 0 0 0 3 0 0 0 0 0 | * * * * * * * * 1 * * * * * * * * | 1 0 0 0 0 0 0 0 0 1 0 0 1 0 | 1 0 0 1 0 1 .... .xx.&#x | 0 2 2 0 | 0 0 0 0 1 2 0 1 0 0 | * * * * * * * * * 3 * * * * * * * | 0 1 0 0 0 0 0 0 0 1 0 1 0 0 | 1 0 1 0 0 1 .ooo3.ooo&#x | 0 1 1 1 | 0 0 0 0 0 1 1 0 1 0 | * * * * * * * * * * 6 * * * * * * | 0 0 1 0 0 0 0 0 0 0 1 1 0 0 | 0 1 1 0 0 1 .o.x ....&#x | 0 1 0 2 | 0 0 0 0 0 0 2 0 0 1 | * * * * * * * * * * * 3 * * * * * | 0 0 0 1 0 0 0 0 0 0 1 0 1 0 | 0 1 0 1 0 1 .... .x.o&#x | 0 2 0 1 | 0 0 0 0 1 0 2 0 0 0 | * * * * * * * * * * * * 3 * * * * | 0 0 0 0 1 0 0 0 0 0 0 1 1 0 | 0 0 1 1 0 1 ..o.3..x. | 0 0 3 0 | 0 0 0 0 0 0 0 3 0 0 | * * * * * * * * * * * * * 1 * * * | 0 0 0 0 0 1 0 0 0 1 0 0 0 1 | 1 0 0 0 1 1 ..ox ....&#x | 0 0 1 2 | 0 0 0 0 0 0 0 0 2 1 | * * * * * * * * * * * * * * 3 * * | 0 0 0 0 0 0 1 0 0 0 1 0 0 1 | 0 1 0 0 1 1 .... ..xo&#x | 0 0 2 1 | 0 0 0 0 0 0 0 1 2 0 | * * * * * * * * * * * * * * * 3 * | 0 0 0 0 0 0 0 1 0 0 0 1 0 1 | 0 0 1 0 1 1 ...x3...o | 0 0 0 3 | 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * * * * * * 1 | 0 0 0 0 0 0 0 0 1 0 0 0 1 1 | 0 0 0 1 1 1 -------------+---------+---------------------+-----------------------------------+-----------------------------+------------ oo..3xx..&#x ♦ 3 3 0 0 | 3 3 0 0 3 0 0 0 0 0 | 1 3 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * * | 1 0 0 1 0 0 .... xxx.&#x ♦ 2 2 2 0 | 1 2 2 0 1 2 0 1 0 0 | 0 1 2 0 1 0 0 0 0 1 0 0 0 0 0 0 0 | * 3 * * * * * * * * * * * * | 1 0 1 0 0 0 oooo3oooo&#x ♦ 1 1 1 1 | 0 1 1 1 0 1 1 0 1 0 | 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 | * * 6 * * * * * * * * * * * | 0 1 1 0 0 0 oo.x ....&#x ♦ 1 1 0 2 | 0 1 0 2 0 0 2 0 0 1 | 0 0 0 2 0 0 1 0 0 0 0 1 0 0 0 0 0 | * * * 3 * * * * * * * * * * | 0 1 0 1 0 0 .... xx.o&#x ♦ 2 2 0 1 | 1 2 0 2 1 0 2 0 0 0 | 0 1 0 2 0 0 0 1 0 0 0 0 1 0 0 0 0 | * * * * 3 * * * * * * * * * | 0 0 1 1 0 0 o.o.3x.x.&#x ♦ 3 0 3 0 | 3 0 3 0 0 0 0 3 0 0 | 1 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 | * * * * * 1 * * * * * * * * | 1 0 0 0 1 0 o.ox ....&#x ♦ 1 0 1 2 | 0 0 1 2 0 0 0 0 2 1 | 0 0 0 0 0 2 1 0 0 0 0 0 0 0 1 0 0 | * * * * * * 3 * * * * * * * | 0 1 0 0 1 0 .... x.xo&#x ♦ 2 0 2 1 | 1 0 2 2 0 0 0 1 2 0 | 0 0 0 0 1 2 0 1 0 0 0 0 0 0 0 1 0 | * * * * * * * 3 * * * * * * | 0 0 1 0 1 0 o..x3x..o&#x ♦ 3 0 0 3 | 3 0 0 6 0 0 0 0 0 3 | 1 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 1 | * * * * * * * * 1 * * * * * | 0 0 0 1 1 0 .oo.3.xx.&#x ♦ 0 3 3 0 | 0 0 0 0 3 3 0 3 0 0 | 0 0 0 0 0 0 0 0 1 3 0 0 0 1 0 0 0 | * * * * * * * * * 1 * * * * | 1 0 0 0 0 1 .oox ....&#x ♦ 0 1 1 2 | 0 0 0 0 0 1 2 0 2 1 | 0 0 0 0 0 0 0 0 0 0 2 1 0 0 1 0 0 | * * * * * * * * * * 3 * * * | 0 1 0 0 0 1 .... .xxo&#x ♦ 0 2 2 1 | 0 0 0 0 1 2 2 1 2 0 | 0 0 0 0 0 0 0 0 0 1 2 0 1 0 0 1 0 | * * * * * * * * * * * 3 * * | 0 0 1 0 0 1 .o.x3.x.o&#x ♦ 0 3 0 3 | 0 0 0 0 3 0 6 0 0 3 | 0 0 0 0 0 0 0 0 1 0 0 3 3 0 0 0 1 | * * * * * * * * * * * * 1 * | 0 0 0 1 0 1 ..ox3..xo&#x ♦ 0 0 3 3 | 0 0 0 0 0 0 0 3 6 3 | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 3 1 | * * * * * * * * * * * * * 1 | 0 0 0 0 1 1 -------------+---------+---------------------+-----------------------------------+-----------------------------+------------ ooo.3xxx.&#x ♦ 3 3 3 0 | 3 3 3 0 3 3 0 3 0 0 | 1 3 3 0 3 0 0 0 1 3 0 0 0 1 0 0 0 | 1 3 0 0 0 1 0 0 0 1 0 0 0 0 | 1 * * * * * ooox ....&#x ♦ 1 1 1 2 | 0 1 1 2 0 1 2 0 2 1 | 0 0 1 2 0 2 1 0 0 0 2 1 0 0 1 0 0 | 0 0 2 1 0 0 1 0 0 0 1 0 0 0 | * 3 * * * * .... xxxo&#x ♦ 2 2 2 1 | 1 2 2 2 1 2 2 1 2 0 | 0 1 2 2 1 2 0 1 0 1 2 0 1 0 0 1 0 | 0 1 2 0 1 0 0 1 0 0 0 1 0 0 | * * 3 * * * oo.x3xx.o&#x ♦ 3 3 0 3 | 3 3 0 6 3 0 6 0 0 3 | 1 3 0 6 0 0 3 3 1 0 0 3 3 0 0 0 1 | 1 0 0 3 3 0 0 0 1 0 0 0 1 0 | * * * 1 * * o.ox3x.xo&#x ♦ 3 0 3 3 | 3 0 3 6 0 0 0 3 6 3 | 1 0 0 0 3 6 3 3 0 0 0 0 0 1 3 3 1 | 0 0 0 0 0 1 3 3 1 0 0 0 0 1 | * * * * 1 * .oox3.xxo&#x ♦ 0 3 3 3 | 0 0 0 0 3 3 6 3 6 3 | 0 0 0 0 0 0 0 0 1 3 6 3 3 1 3 3 1 | 0 0 0 0 0 0 0 0 0 1 3 3 1 1 | * * * * * 1
oxox ooxx3oooo&#xr → height(1,2) = height(3,4) = sqrt(3)/2 = 0.866025 height(1,4) = height(2,3) = sqrt(5/12) = 0.645497 o... o...3o... | 1 * * * ♦ 2 6 0 0 0 0 0 0 0 | 1 6 3 6 0 0 0 0 0 0 0 0 0 | 3 6 3 2 0 0 0 0 0 0 0 | 3 2 1 0 0 0 .o.. .o..3.o.. | * 2 * * ♦ 1 0 1 3 3 0 0 0 0 | 1 3 0 0 3 3 3 3 0 0 0 0 0 | 3 3 0 0 3 1 3 1 0 0 0 | 3 1 0 1 1 0 ..o. ..o.3..o. | * * 3 * | 0 0 0 2 0 2 2 0 0 | 0 0 0 0 1 4 2 0 1 1 4 0 0 | 1 0 0 0 2 2 4 0 2 2 0 | 2 0 0 1 2 1 ...o ...o3...o | * * * 6 | 0 1 0 0 1 0 1 1 2 | 0 1 1 2 0 0 1 2 0 1 2 2 1 | 1 2 2 1 0 0 2 1 2 1 1 | 2 1 1 0 1 1 -------------------+---------+-------------------+---------------------------+-----------------------+------------ oo.. oo..3oo..&#x | 1 1 0 0 | 2 * * * * * * * * | 1 3 0 0 0 0 0 0 0 0 0 0 0 | 3 3 0 0 0 0 0 0 0 0 0 | 3 1 0 0 0 0 o..o o..o3o..o&#x | 1 0 0 1 | * 6 * * * * * * * | 0 1 1 2 0 0 0 0 0 0 0 0 0 | 1 2 2 1 0 0 0 0 0 0 0 | 2 1 1 0 0 0 .x.. .... .... | 0 2 0 0 | * * 1 * * * * * * | 1 0 0 0 3 0 0 0 0 0 0 0 0 | 3 0 0 0 3 0 0 0 0 0 0 | 3 0 0 1 0 0 .oo. .oo.3.oo.&#x | 0 1 1 0 | * * * 6 * * * * * | 0 0 0 0 1 2 1 0 0 0 0 0 0 | 1 0 0 0 2 1 2 0 0 0 0 | 2 0 0 1 1 0 .o.o .o.o3.o.o&#x | 0 1 0 1 | * * * * 6 * * * * | 0 1 0 0 0 0 1 2 0 0 0 0 0 | 1 2 0 0 0 0 2 1 0 0 0 | 2 1 0 0 1 0 .... ..x. .... | 0 0 2 0 | * * * * * 3 * * * | 0 0 0 0 0 2 0 0 1 0 2 0 0 | 0 0 0 0 1 2 2 0 1 2 0 | 1 0 0 1 2 1 ..oo ..oo3..oo&#x | 0 0 1 1 | * * * * * * 6 * * | 0 0 0 0 0 0 1 0 0 1 2 0 0 | 1 0 0 0 0 0 2 0 2 1 0 | 2 0 0 0 1 1 ...x .... .... | 0 0 0 2 | * * * * * * * 3 * | 0 0 1 0 0 0 0 0 0 1 0 2 0 | 1 0 2 0 0 0 0 0 2 0 1 | 2 0 1 0 0 1 .... ...x .... | 0 0 0 2 | * * * * * * * * 6 | 0 0 0 1 0 0 0 1 0 0 1 1 1 | 0 1 1 1 0 0 1 1 1 1 1 | 1 1 1 0 1 1 -------------------+---------+-------------------+---------------------------+-----------------------+------------ ox.. .... ....&#x | 1 2 0 0 | 2 0 1 0 0 0 0 0 0 | 1 * * * * * * * * * * * * | 3 0 0 0 0 0 0 0 0 0 0 | 3 0 0 0 0 0 oo.o oo.o3oo.o&#x | 1 1 0 1 | 1 1 0 0 1 0 0 0 0 | * 6 * * * * * * * * * * * | 1 2 0 0 0 0 0 0 0 0 0 | 2 1 0 0 0 0 o..x .... ....&#x | 1 0 0 2 | 0 2 0 0 0 0 0 1 0 | * * 3 * * * * * * * * * * | 1 0 2 0 0 0 0 0 0 0 0 | 2 0 1 0 0 0 .... o..x ....&#x | 1 0 0 2 | 0 2 0 0 0 0 0 0 1 | * * * 6 * * * * * * * * * | 0 1 1 1 0 0 0 0 0 0 0 | 1 1 1 0 0 0 .xo. .... ....&#x | 0 2 1 0 | 0 0 1 2 0 0 0 0 0 | * * * * 3 * * * * * * * * | 1 0 0 0 2 0 0 0 0 0 0 | 2 0 0 1 0 0 .... .ox. ....&#x | 0 1 2 0 | 0 0 0 2 0 1 0 0 0 | * * * * * 6 * * * * * * * | 0 0 0 0 1 1 1 0 0 0 0 | 1 0 0 1 1 0 .ooo .ooo3.ooo&#x | 0 1 1 1 | 0 0 0 1 1 0 1 0 0 | * * * * * * 6 * * * * * * | 1 0 0 0 0 0 2 0 0 0 0 | 2 0 0 0 1 0 .... .o.x ....&#x | 0 1 0 2 | 0 0 0 0 2 0 0 0 1 | * * * * * * * 6 * * * * * | 0 1 0 0 0 0 1 1 0 0 0 | 1 1 0 0 1 0 .... ..x.3..o. | 0 0 3 0 | 0 0 0 0 0 3 0 0 0 | * * * * * * * * 1 * * * * | 0 0 0 0 0 2 0 0 0 2 0 | 0 0 0 1 2 1 ..ox .... ....&#x | 0 0 1 2 | 0 0 0 0 0 0 2 1 0 | * * * * * * * * * 3 * * * | 1 0 0 0 0 0 0 0 2 0 0 | 2 0 0 0 0 1 .... ..xx ....&#x | 0 0 2 2 | 0 0 0 0 0 1 2 0 1 | * * * * * * * * * * 6 * * | 0 0 0 0 0 0 1 0 1 1 0 | 1 0 0 0 1 1 ...x ...x .... | 0 0 0 4 | 0 0 0 0 0 0 0 2 2 | * * * * * * * * * * * 3 * | 0 0 1 0 0 0 0 0 1 0 1 | 1 0 1 0 0 1 .... ...x3...o | 0 0 0 3 | 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * * 2 | 0 0 0 1 0 0 0 1 0 1 1 | 0 1 1 0 1 1 -------------------+---------+-------------------+---------------------------+-----------------------+------------ oxox .... ....&#xr ♦ 1 2 1 2 | 2 2 1 2 2 0 2 1 0 | 1 2 1 0 1 0 2 0 0 1 0 0 0 | 3 * * * * * * * * * * | 2 0 0 0 0 0 .... oo.x ....&#x ♦ 1 1 0 2 | 1 2 0 0 2 0 0 0 1 | 0 2 0 1 0 0 0 1 0 0 0 0 0 | * 6 * * * * * * * * * | 1 1 0 0 0 0 o..x o..x ....&#x ♦ 1 0 0 4 | 0 4 0 0 0 0 0 2 2 | 0 0 2 2 0 0 0 0 0 0 0 1 0 | * * 3 * * * * * * * * | 1 0 1 0 0 0 .... o..x3o..o&#x ♦ 1 0 0 3 | 0 3 0 0 0 0 0 0 3 | 0 0 0 3 0 0 0 0 0 0 0 0 1 | * * * 2 * * * * * * * | 0 1 1 0 0 0 .xo. .ox. ....&#x ♦ 0 2 2 0 | 0 0 1 4 0 1 0 0 0 | 0 0 0 0 2 2 0 0 0 0 0 0 0 | * * * * 3 * * * * * * | 1 0 0 1 0 0 .... .ox.3.oo.&#x ♦ 0 1 3 0 | 0 0 0 3 0 3 0 0 0 | 0 0 0 0 0 3 0 0 1 0 0 0 0 | * * * * * 2 * * * * * | 0 0 0 1 1 0 .... .oxx ....&#x ♦ 0 1 2 2 | 0 0 0 2 2 1 2 0 1 | 0 0 0 0 0 1 2 1 0 0 1 0 0 | * * * * * * 6 * * * * | 1 0 0 0 1 0 .... .o.x3.o.o&#x ♦ 0 1 0 3 | 0 0 0 0 3 0 0 0 3 | 0 0 0 0 0 0 0 3 0 0 0 0 1 | * * * * * * * 2 * * * | 0 1 0 0 1 0 ..ox ..xx ....&#x ♦ 0 0 2 4 | 0 0 0 0 0 1 4 2 2 | 0 0 0 0 0 0 0 0 0 2 2 1 0 | * * * * * * * * 3 * * | 1 0 0 0 0 1 .... ..xx3..oo&#x ♦ 0 0 3 3 | 0 0 0 0 0 3 3 0 3 | 0 0 0 0 0 0 0 0 1 0 3 0 1 | * * * * * * * * * 2 * | 0 0 0 0 1 1 ...x ...x3...o ♦ 0 0 0 6 | 0 0 0 0 0 0 0 3 6 | 0 0 0 0 0 0 0 0 0 0 0 3 2 | * * * * * * * * * * 1 | 0 0 1 0 0 1 -------------------+---------+-------------------+---------------------------+-----------------------+------------ oxox ooxx ....&#xr ♦ 1 2 2 4 | 2 4 1 4 4 1 4 2 2 | 1 4 2 2 2 2 4 2 0 2 2 1 0 | 2 2 1 0 1 0 2 0 1 0 0 | 3 * * * * * .... oo.x3oo.o&#x ♦ 1 1 0 3 | 1 3 0 0 3 0 0 0 3 | 0 3 0 3 0 0 0 3 0 0 0 0 1 | 0 3 0 1 0 0 0 1 0 0 0 | * 2 * * * * o..x o..x3o..o&#x ♦ 1 0 0 6 | 0 6 0 0 0 0 0 3 6 | 0 0 3 6 0 0 0 0 0 0 0 3 2 | 0 0 3 2 0 0 0 0 0 0 1 | * * 1 * * * .xo. .ox.3.oo.&#x ♦ 0 2 3 0 | 0 0 1 6 0 3 0 0 0 | 0 0 0 0 3 6 0 0 1 0 0 0 0 | 0 0 0 0 3 2 0 0 0 0 0 | * * * 1 * * .... .oxx3.ooo&#x ♦ 0 1 3 3 | 0 0 0 3 3 3 3 0 3 | 0 0 0 0 0 3 3 3 1 0 3 0 1 | 0 0 0 0 0 1 3 1 0 1 0 | * * * * 2 * ..ox ..xx3..oo&#x ♦ 0 0 3 6 | 0 0 0 0 0 3 6 3 6 | 0 0 0 0 0 0 0 0 1 3 6 3 2 | 0 0 0 0 0 0 0 0 3 2 1 | * * * * * 1
xoxoxo3oooooo&#xr → all consecutive pairwise heights = sqrt(2/3) = 0.816497 all alternating pairwise heights = 1 Note: these lengths show that this cycle is not flat, rather it is wobbling up and down! o.....3o..... & | 9 * | 2 2 2 0 | 1 4 2 4 1 1 0 | 2 4 1 2 2 2 | 2 2 1 1 .o....3.o.... & | * 3 ♦ 0 6 0 2 | 0 6 3 0 0 6 1 | 2 3 3 0 0 6 | 1 3 0 2 --------------------+-----+----------+----------------+-------------+-------- x..... ...... & | 2 0 | 9 * * * | 1 2 0 2 0 0 0 | 2 2 0 2 1 1 | 2 1 1 1 oo....3oo....&#x & | 1 1 | * 18 * * | 0 2 1 0 0 1 0 | 1 2 1 0 0 2 | 1 2 0 1 o.o...3o.o...&#x & | 2 0 | * * 9 * | 0 0 1 2 1 0 0 | 0 2 1 1 2 0 | 1 2 1 0 .o.o..3.o.o..&#x & | 0 2 | * * * 3 | 0 0 0 0 0 3 1 | 0 0 3 0 0 3 | 0 3 0 1 --------------------+-----+----------+----------------+-------------+-------- x.....3o..... & | 3 0 | 3 0 0 0 | 3 * * * * * * | 2 0 0 2 0 0 | 2 0 1 1 xo.... ......&#x & | 2 1 | 1 2 0 0 | * 18 * * * * * | 1 1 0 0 0 1 | 1 1 0 1 ooo...3ooo...&#x & | 2 1 | 0 2 1 0 | * * 9 * * * * | 0 2 1 0 0 0 | 1 2 0 0 x.x... ......&#x & | 4 0 | 2 0 2 0 | * * * 9 * * * | 0 1 0 1 1 0 | 1 1 1 0 o.o.o.3o.o.o.&#x | 3 0 | 0 0 3 0 | * * * * 3 * * | 0 0 1 0 2 0 | 0 2 1 0 .ooo..3.ooo..&#x & | 1 2 | 0 2 0 1 | * * * * * 9 * | 0 0 1 0 0 2 | 0 2 0 1 .o.o.o3.o.o.o&#x | 0 3 | 0 0 0 3 | * * * * * * 1 | 0 0 3 0 0 0 | 0 3 0 0 --------------------+-----+----------+----------------+-------------+-------- xo....3oo....&#x & ♦ 3 1 | 3 3 0 0 | 1 3 0 0 0 0 0 | 6 * * * * * | 1 0 0 1 xox... ......&#x & ♦ 4 1 | 2 4 2 0 | 0 2 2 1 0 0 0 | * 9 * * * * | 1 1 0 0 oooooo3oooooo&#xr ♦ 3 3 | 0 6 3 3 | 0 0 3 0 1 3 1 | * * 3 * * * | 0 2 0 0 x.x...3o.o...&#x & ♦ 6 0 | 6 0 3 0 | 2 0 0 3 0 0 0 | * * * 3 * * | 1 0 1 0 x.x.x. ......&#x ♦ 6 0 | 3 0 6 0 | 0 0 0 3 2 0 0 | * * * * 3 * | 0 1 1 0 .oxo.. ......&#x & ♦ 2 2 | 1 4 0 1 | 0 2 0 0 0 2 0 | * * * * * 9 | 0 1 0 1 --------------------+-----+----------+----------------+-------------+-------- xox...3ooo...&#x & ♦ 6 1 | 6 6 3 0 | 2 6 3 3 0 0 0 | 2 3 0 1 0 0 | 3 * * * xoxoxo ......&#x ♦ 6 3 | 3 12 6 3 | 0 6 6 3 2 6 1 | 0 3 2 0 1 3 | * 3 * * x.x.x.3o.o.o.&#x ♦ 9 0 | 9 0 9 0 | 3 0 0 9 3 0 0 | 0 0 0 3 3 0 | * * 1 * .oxo..3.ooo..&#x & ♦ 3 2 | 3 6 0 1 | 1 6 0 0 0 3 0 | 2 0 0 0 0 3 | * * * 3
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