Acronym tix Name truncated hexateron Circumradius sqrt(7)/2 = 1.322876 Vertex figure ` ©` Lace cityin approx. ASCII-art ``` o o with o = o3o3o x x x = x3o3o u u u = u3o3o t t = x3x3o ``` Externallinks

Incidence matrix according to Dynkin symbol

```x3x3o3o3o

. . . . . | 30 |  1  4 |  4  6 |  6  4 | 4 1
----------+----+-------+-------+-------+----
x . . . . |  2 | 15  * ♦  4  0 |  6  0 | 4 0
. x . . . |  2 |  * 60 |  1  3 |  3  3 | 3 1
----------+----+-------+-------+-------+----
x3x . . . |  6 |  3  3 | 20  * |  3  0 | 6 0
. x3o . . |  3 |  0  3 |  * 60 |  1  2 | 2 1
----------+----+-------+-------+-------+----
x3x3o . . ♦ 12 |  6 12 |  4  4 | 15  * | 2 0
. x3o3o . ♦  4 |  0  6 |  0  4 |  * 30 | 1 1
----------+----+-------+-------+-------+----
x3x3o3o . ♦ 20 | 10 30 | 10 20 |  5  5 | 6 *
. x3o3o3o ♦  5 |  0 10 |  0 10 |  0  5 | * 6
```

```xux3oox3ooo3ooo&#xt   → both heights = sqrt(3/5) = 0.774597
(pen || pseudo u-pen || tip)

o..3o..3o..3o..     | 5 *  * |  4 1  0  0  0 |  6  4  0  0  0 | 4  6  0 0 0 | 1 4 0 0
.o.3.o.3.o.3.o.     | * 5  * |  0 1  4  0  0 |  0  4  6  0  0 | 0  6  4 0 0 | 0 4 1 0
..o3..o3..o3..o     | * * 20 |  0 0  1  1  3 |  0  1  3  3  3 | 0  3  3 3 1 | 0 3 1 1
--------------------+--------+---------------+----------------+-------------+--------
x.. ... ... ...     | 2 0  0 | 10 *  *  *  * |  3  1  0  0  0 | 3  3  0 0 0 | 1 3 0 0
oo.3oo.3oo.3oo.&#x  | 1 1  0 |  * 5  *  *  * ♦  0  4  0  0  0 | 0  6  0 0 0 | 0 4 0 0
.oo3.oo3.oo3.oo&#x  | 0 1  1 |  * * 20  *  * |  0  1  3  0  0 | 0  3  3 0 0 | 0 3 1 0
..x ... ... ...     | 0 0  2 |  * *  * 10  * ♦  0  1  0  3  0 | 0  3  0 3 0 | 0 3 0 1
... ..x ... ...     | 0 0  2 |  * *  *  * 30 |  0  0  1  1  2 | 0  1  2 2 1 | 0 2 1 1
--------------------+--------+---------------+----------------+-------------+--------
x..3o.. ... ...     | 3 0  0 |  3 0  0  0  0 | 10  *  *  *  * | 2  1  0 0 0 | 1 2 0 0
xux ... ... ...&#xt | 2 2  2 |  1 2  2  1  0 |  * 10  *  *  * | 0  3  0 0 0 | 0 3 0 0
... .ox ... ...&#x  | 0 1  2 |  0 0  2  0  1 |  *  * 30  *  * | 0  1  2 0 0 | 0 2 1 0
..x3..x ... ...     | 0 0  6 |  0 0  0  3  3 |  *  *  * 10  * | 0  1  0 2 0 | 0 2 0 1
... ..x3..o ...     | 0 0  3 |  0 0  0  0  3 |  *  *  *  * 20 | 0  0  1 1 1 | 0 1 1 1
--------------------+--------+---------------+----------------+-------------+--------
x..3o..3o.. ...     ♦ 4 0  0 |  6 0  0  0  0 |  4  0  0  0  0 | 5  *  * * * | 1 1 0 0
xux3oox ... ...&#xt ♦ 3 3  6 |  3 3  6  3  3 |  1  3  3  1  0 | * 10  * * * | 0 2 0 0
... .ox3.oo ...&#x  ♦ 0 1  3 |  0 0  3  0  3 |  0  0  3  0  1 | *  * 20 * * | 0 1 1 0
..x3..x3..o ...     ♦ 0 0 12 |  0 0  0  6 12 |  0  0  0  4  4 | *  *  * 5 * | 0 1 0 1
... ..x3..o3..o     ♦ 0 0  4 |  0 0  0  0  6 |  0  0  0  0  4 | *  *  * * 5 | 0 0 1 1
--------------------+--------+---------------+----------------+-------------+--------
x..3o..3o..3o..     ♦ 5 0  0 | 10 0  0  0  0 | 10  0  0  0  0 | 5  0  0 0 0 | 1 * * *
xux3oox3ooo ...&#xt ♦ 4 4 12 |  6 4 12  6 12 |  4  6 12  4  4 | 1  4  4 1 0 | * 5 * *
... .ox3.oo3.oo&#x  ♦ 0 1  4 |  0 0  4  0  6 |  0  0  6  0  4 | 0  0  4 0 1 | * * 5 *
..x3..x3..o3..o     ♦ 0 0 20 |  0 0  0 10 30 |  0  0  0 10 20 | 0  0  0 5 5 | * * * 1
```

```xuxo oxux3ooox3oooo&#xt   → all heights = sqrt(3/8) = 0.612372
(line || pseudo (x,u)-tepe|| pseudo (u,x)-tepe|| tut)

o... o...3o...3o...     | 2 * *  * | 1 4  0 0 0  0 0  0 |  6 4 0  0  0  0 0 0 | 4 6 0 0  0 0 0 | 1 4 0 0
.o.. .o..3.o..3.o..     | * 8 *  * | 0 1  3 1 0  0 0  0 |  3 1 3  3  0  0 0 0 | 3 3 1 3  0 0 0 | 1 3 1 0
..o. ..o.3..o.3..o.     | * * 8  * | 0 0  0 1 1  3 0  0 |  0 1 0  3  3  3 0 0 | 0 3 0 3  3 1 0 | 0 3 1 1
...o ...o3...o3...o     | * * * 12 | 0 0  0 0 0  2 1  2 |  0 0 0  2  1  4 2 1 | 0 1 0 4  2 2 1 | 0 2 2 1
------------------------+----------+--------------------+---------------------+----------------+--------
x... .... .... ....     | 2 0 0  0 | 1 *  * * *  * *  * ♦  0 4 0  0  0  0 0 0 | 0 6 0 0  0 0 0 | 0 4 0 0
oo.. oo..3oo..3oo..&#x  | 1 1 0  0 | * 8  * * *  * *  * |  3 1 0  0  0  0 0 0 | 3 3 0 0  0 0 0 | 1 3 0 0
.... .x.. .... ....     | 0 2 0  0 | * * 12 * *  * *  * |  1 0 2  1  0  0 0 0 | 2 1 1 2  0 0 0 | 1 2 1 0
.oo. .oo.3.oo.3.oo.&#x  | 0 1 1  0 | * *  * 8 *  * *  * ♦  0 1 0  3  0  0 0 0 | 0 3 0 3  0 0 0 | 0 3 1 0
..x. .... .... ....     | 0 0 2  0 | * *  * * 4  * *  * |  0 1 0  0  3  0 0 0 | 0 3 0 0  3 0 0 | 0 3 0 1
..oo ..oo3..oo3..oo&#x  | 0 0 1  1 | * *  * * * 24 *  * |  0 0 0  1  1  2 0 0 | 0 1 0 2  2 1 0 | 0 2 1 1
.... ...x .... ....     | 0 0 0  2 | * *  * * *  * 6  * ♦  0 0 0  2  0  0 2 0 | 0 1 0 4  0 0 1 | 0 2 2 0
.... .... ...x ....     | 0 0 0  2 | * *  * * *  * * 12 |  0 0 0  0  0  2 1 1 | 0 0 0 2  1 2 1 | 0 1 2 1
------------------------+----------+--------------------+---------------------+----------------+--------
.... ox.. .... ....&#x  | 1 2 0  0 | 0 2  1 0 0  0 0  0 | 12 * *  *  *  * * * | 2 1 0 0  0 0 0 | 1 2 0 0
xux. .... .... ....&#xt | 2 2 2  0 | 1 2  0 2 1  0 0  0 |  * 4 *  *  *  * * * | 0 3 0 0  0 0 0 | 0 3 0 0
.... .x..3.o.. ....     | 0 3 0  0 | 0 0  3 0 0  0 0  0 |  * * 8  *  *  * * * | 1 0 1 1  0 0 0 | 1 1 1 0
.... .xux .... ....&#xt | 0 2 2  2 | 0 0  1 2 0  2 1  0 |  * * * 12  *  * * * | 0 1 0 2  0 0 0 | 0 2 1 0
..xo .... .... ....&#x  | 0 0 2  1 | 0 0  0 0 1  2 0  0 |  * * *  * 12  * * * | 0 1 0 0  2 0 0 | 0 2 0 1
.... .... ..ox ....&#x  | 0 0 1  2 | 0 0  0 0 0  2 0  1 |  * * *  *  * 24 * * | 0 0 0 1  1 1 0 | 0 1 1 1
.... ...x3...x ....     | 0 0 0  6 | 0 0  0 0 0  0 3  3 |  * * *  *  *  * 4 * | 0 0 0 2  0 0 1 | 0 1 2 0
.... .... ...x3...o     | 0 0 0  3 | 0 0  0 0 0  0 0  3 |  * * *  *  *  * * 4 | 0 0 0 0  0 2 1 | 0 0 2 1
------------------------+----------+--------------------+---------------------+----------------+--------
.... ox..3oo.. ....&#x  ♦ 1 3 0  0 | 0 3  3 0 0  0 0  0 |  3 0 1  0  0  0 0 0 | 8 * * *  * * * | 1 1 0 0
xuxo oxux .... ....&#xt ♦ 2 4 4  2 | 1 4  2 4 2  4 1  0 |  2 2 0  2  2  0 0 0 | * 6 * *  * * * | 0 2 0 0
.... .x..3.o..3.o..     ♦ 0 4 0  0 | 0 0  6 0 0  0 0  0 |  0 0 4  0  0  0 0 0 | * * 2 *  * * * | 1 0 1 0
.... .xux3.oox ....&#xt ♦ 0 3 3  6 | 0 0  3 3 0  6 3  3 |  0 0 1  3  0  3 1 0 | * * * 8  * * * | 0 1 1 0
..xo .... ..ox ....&#x  ♦ 0 0 2  2 | 0 0  0 0 1  4 0  1 |  0 0 0  0  2  2 0 0 | * * * * 12 * * | 0 1 0 1
.... .... ..ox3..oo&#x  ♦ 0 0 1  3 | 0 0  0 0 0  3 0  3 |  0 0 0  0  0  3 0 1 | * * * *  * 8 * | 0 0 1 1
.... ...x3...x3...o     ♦ 0 0 0 12 | 0 0  0 0 0  0 6 12 |  0 0 0  0  0  0 4 4 | * * * *  * * 1 | 0 0 2 0
------------------------+----------+--------------------+---------------------+----------------+--------
.... ox..3oo..3oo..&#x  ♦ 1 4 0  0 | 0 4  6 0 0  0 0  0 |  6 0 4  0  0  0 0 0 | 4 0 1 0  0 0 0 | 2 * * *
xuxo oxux3ooox ....&#xt ♦ 2 6 6  6 | 1 6  6 6 3 12 3  3 |  6 3 2  6  6  6 1 0 | 2 3 0 2  3 0 0 | * 4 * *
.... .xux3.oox3.ooo&#xt ♦ 0 4 4 12 | 0 0  6 4 0 12 6 12 |  0 0 4  6  0 12 4 4 | 0 0 1 4  0 4 1 | * * 2 *
..xo .... ..ox3..oo&#x  ♦ 0 0 2  3 | 0 0  0 0 1  6 0  3 |  0 0 0  0  3  6 0 1 | 0 0 0 0  3 2 0 | * * * 4
```

```xuxo3xooo oxux3ooox&#xt   → all heights = 1/sqrt(3) = 0.577350
({6} || pseudo (u,x)-triddip || pseudo (x,u)-triddip || ortho {6})

o...3o... o...3o...     | 6 * * * | 1 1  3 0 0 0  0 0 0 | 1 3  3 3 0 0 0  0 0 0 | 3 1 3 3 0 0 0 | 1 3 1 0
.o..3.o.. .o..3.o..     | * 9 * * | 0 0  2 2 1 0  0 0 0 | 0 1  4 2 1 2 0  0 0 0 | 2 2 1 4 1 0 0 | 1 2 2 0
..o.3..o. ..o.3..o.     | * * 9 * | 0 0  0 0 1 2  2 0 0 | 0 0  0 2 0 2 1  4 1 0 | 0 0 1 4 1 2 2 | 0 2 2 1
...o3...o ...o3...o     | * * * 6 | 0 0  0 0 0 0  3 1 1 | 0 0  0 0 0 3 0  3 3 1 | 0 0 0 3 3 1 3 | 0 1 3 1
------------------------+---------+---------------------+-----------------------+---------------+--------
x... .... .... ....     | 2 0 0 0 | 3 *  * * * *  * * * ♦ 1 0  0 3 0 0 0  0 0 0 | 0 0 3 3 0 0 0 | 0 3 1 0
.... x... .... ....     | 2 0 0 0 | * 3  * * * *  * * * | 1 3  0 0 0 0 0  0 0 0 | 3 0 3 0 0 0 0 | 1 3 0 0
oo..3oo.. oo..3oo..&#x  | 1 1 0 0 | * * 18 * * *  * * * | 0 1  2 1 0 0 0  0 0 0 | 2 1 1 2 0 0 0 | 1 2 1 0
.... .... .x.. ....     | 0 2 0 0 | * *  * 9 * *  * * * | 0 0  2 0 1 1 0  0 0 0 | 1 2 0 2 1 0 0 | 1 1 2 0
.oo.3.oo. .oo.3.oo.&#x  | 0 1 1 0 | * *  * * 9 *  * * * ♦ 0 0  0 2 0 2 0  0 0 0 | 0 0 1 4 1 0 0 | 0 2 2 0
..x. .... .... ....     | 0 0 2 0 | * *  * * * 9  * * * | 0 0  0 1 0 0 1  2 0 0 | 0 0 1 2 0 2 1 | 0 2 1 1
..oo3..oo ..oo3..oo&#x  | 0 0 1 1 | * *  * * * * 18 * * | 0 0  0 0 0 1 0  2 1 0 | 0 0 0 2 1 1 2 | 0 1 2 1
.... .... ...x ....     | 0 0 0 2 | * *  * * * *  * 3 * ♦ 0 0  0 0 0 3 0  0 0 1 | 0 0 0 3 3 0 0 | 0 1 3 0
.... .... .... ...x     | 0 0 0 2 | * *  * * * *  * * 3 | 0 0  0 0 0 0 0  0 3 1 | 0 0 0 0 3 0 3 | 0 0 3 1
------------------------+---------+---------------------+-----------------------+---------------+--------
x...3x... .... ....     | 6 0 0 0 | 3 3  0 0 0 0  0 0 0 | 1 *  * * * * *  * * * | 0 0 3 0 0 0 0 | 0 3 0 0
.... xo.. .... ....&#x  | 2 1 0 0 | 0 1  2 0 0 0  0 0 0 | * 9  * * * * *  * * * | 2 0 1 0 0 0 0 | 1 2 0 0
.... .... ox.. ....&#x  | 1 2 0 0 | 0 0  2 1 0 0  0 0 0 | * * 18 * * * *  * * * | 1 1 0 1 0 0 0 | 1 1 1 0
xux. .... .... ....&#xt | 2 2 2 0 | 1 0  2 0 2 1  0 0 0 | * *  * 9 * * *  * * * | 0 0 1 2 0 0 0 | 0 2 1 0
.... .... .x..3.o..     | 0 3 0 0 | 0 0  0 3 0 0  0 0 0 | * *  * * 3 * *  * * * | 0 2 0 0 1 0 0 | 1 0 2 0
.... .... .xux ....&#xt | 0 2 2 2 | 0 0  0 1 2 0  2 1 0 | * *  * * * 9 *  * * * | 0 0 0 2 1 0 0 | 0 1 2 0
..x.3..o. .... ....     | 0 0 3 0 | 0 0  0 0 0 3  0 0 0 | * *  * * * * 3  * * * | 0 0 1 0 0 2 0 | 0 2 0 1
..xo .... .... ....&#x  | 0 0 2 1 | 0 0  0 0 0 1  2 0 0 | * *  * * * * * 18 * * | 0 0 0 1 0 1 1 | 0 1 1 1
.... .... .... ..ox&#x  | 0 0 1 2 | 0 0  0 0 0 0  2 0 1 | * *  * * * * *  * 9 * | 0 0 0 0 1 0 2 | 0 0 2 1
.... .... ...x3...x     | 0 0 0 6 | 0 0  0 0 0 0  0 3 3 | * *  * * * * *  * * 1 | 0 0 0 0 3 0 0 | 0 0 3 0
------------------------+---------+---------------------+-----------------------+---------------+--------
.... xo.. ox.. ....&#x  ♦ 2 2 0 0 | 0 1  4 1 0 0  0 0 0 | 0 2  2 0 0 0 0  0 0 0 | 9 * * * * * * | 1 1 0 0
.... .... ox..3oo..&#x  ♦ 1 3 0 0 | 0 0  3 3 0 0  0 0 0 | 0 0  3 0 1 0 0  0 0 0 | * 6 * * * * * | 1 0 1 0
xux.3xoo. .... ....&#xt ♦ 6 3 3 0 | 3 3  6 0 3 3  0 0 0 | 1 3  0 3 0 0 1  0 0 0 | * * 3 * * * * | 0 2 0 0
xuxo .... oxux ....&#xt ♦ 2 4 4 2 | 1 0  4 2 4 2  4 1 0 | 0 0  2 2 0 2 0  2 0 0 | * * * 9 * * * | 0 1 1 0
.... .... .xux3.oox&#xt ♦ 0 3 3 6 | 0 0  0 3 3 0  6 3 3 | 0 0  0 0 1 3 0  0 3 1 | * * * * 3 * * | 0 0 2 0
..xo3..oo .... ....&#x  ♦ 0 0 3 1 | 0 0  0 0 0 3  3 0 0 | 0 0  0 0 0 0 1  3 0 0 | * * * * * 6 * | 0 1 0 1
..xo .... .... ..ox&#x  ♦ 0 0 2 2 | 0 0  0 0 0 1  4 0 1 | 0 0  0 0 0 0 0  2 2 0 | * * * * * * 9 | 0 0 1 1
------------------------+---------+---------------------+-----------------------+---------------+--------
.... xo.. ox..3oo..&#x  ♦ 2 3 0 0 | 0 1  6 3 0 0  0 0 0 | 0 3  6 0 1 0 0  0 0 0 | 3 2 0 0 0 0 0 | 3 * * *
xuxo3xooo oxux ....&#xt ♦ 6 6 6 2 | 3 3 12 3 6 6  6 1 0 | 1 6  6 6 0 3 2  6 0 0 | 3 0 2 3 0 2 0 | * 3 * *
xuxo .... oxux3ooox&#xt ♦ 2 6 6 6 | 1 0  6 6 6 3 12 3 3 | 0 0  6 3 2 6 0  6 6 1 | 0 2 0 3 2 0 3 | * * 3 *
..xo3..oo .... ..ox&#x  ♦ 0 0 3 2 | 0 0  0 0 0 3  6 0 1 | 0 0  0 0 0 0 1  6 3 0 | 0 0 0 0 0 2 3 | * * * 3

or
o...3o... o...3o...      & | 12  * | 1 1  3  0 0 | 1  3  3  3 0 |  3  1 3 3 | 1 4
.o..3.o.. .o..3.o..      & |  * 18 | 0 0  2  2 1 | 0  1  4  4 1 |  2  2 2 4 | 1 4
---------------------------+-------+-------------+--------------+-----------+----
x... .... .... ....      & |  2  0 | 6 *  *  * * ♦ 1  0  0  3 0 |  0  0 3 3 | 0 4
.... x... .... ....      & |  2  0 | * 6  *  * * | 1  3  0  0 0 |  3  0 3 0 | 1 3
oo..3oo.. oo..3oo..&#x   & |  1  1 | * * 36  * * | 0  1  2  1 0 |  2  1 1 2 | 1 3
.... .... .x.. ....      & |  0  2 | * *  * 18 * | 0  0  2  1 1 |  1  2 1 2 | 1 3
.oo.3.oo. .oo.3.oo.&#x     |  0  2 | * *  *  * 9 ♦ 0  0  0  4 0 |  0  0 2 4 | 0 4
---------------------------+-------+-------------+--------------+-----------+----
x...3x... .... ....      & |  6  0 | 3 3  0  0 0 | 2  *  *  * * |  0  0 3 0 | 0 3
.... xo.. .... ....&#x   & |  2  1 | 0 1  2  0 0 | * 18  *  * * |  2  0 1 0 | 1 2
.... .... ox.. ....&#x   & |  1  2 | 0 0  2  1 0 | *  * 36  * * |  1  1 0 1 | 1 2
xux. .... .... ....&#xt  & |  2  4 | 1 0  2  1 2 | *  *  * 18 * |  0  0 1 2 | 0 3
.... .... .x..3.o..      & |  0  3 | 0 0  0  3 0 | *  *  *  * 6 |  0  2 1 0 | 1 2
---------------------------+-------+-------------+--------------+-----------+----
.... xo.. ox.. ....&#x   & ♦  2  2 | 0 1  4  1 0 | 0  2  2  0 0 | 18  * * * | 1 1
.... .... ox..3oo..&#x   & ♦  1  3 | 0 0  3  3 0 | 0  0  3  0 1 |  * 12 * * | 1 1
xux.3xoo. .... ....&#xt  & ♦  6  6 | 3 3  6  3 3 | 1  3  0  3 1 |  *  * 6 * | 0 2
xuxo .... oxux ....&#xt    ♦  4  8 | 2 0  8  4 4 | 0  0  4  4 0 |  *  * * 9 | 0 2
---------------------------+-------+-------------+--------------+-----------+----
.... xo.. ox..3oo..&#x   & ♦  2  3 | 0 1  6  3 0 | 0  3  6  0 1 |  3  2 0 0 | 6 *
xuxo3xooo oxux ....&#xt  & ♦  8 12 | 4 3 18  9 6 | 1  6 12  9 2 |  3  2 2 3 | * 6
```

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