Acronym rix Name rectified hexateron Circumradius sqrt(2/3) = 0.816497 Vertex figure ` ©` Lace cityin approx. ASCII-art ```P T P=point T=tet T O O=oct ``` Externallinks

Incidence matrix according to Dynkin symbol

```o3x3o3o3o

. . . . . | 15 ♦  8 |  4 12 |  6  8 | 4 2
----------+----+----+-------+-------+----
. x . . . |  2 | 60 |  1  3 |  3  3 | 3 1
----------+----+----+-------+-------+----
o3x . . . |  3 |  3 | 20  * |  3  0 | 3 0
. x3o . . |  3 |  3 |  * 60 |  1  2 | 2 1
----------+----+----+-------+-------+----
o3x3o . . ♦  6 | 12 |  4  4 | 15  * | 2 0
. x3o3o . ♦  4 |  6 |  0  4 |  * 30 | 1 1
----------+----+----+-------+-------+----
o3x3o3o . ♦ 10 | 30 | 10 20 |  5  5 | 6 *
. x3o3o3o ♦  5 | 10 |  0 10 |  0  5 | * 6
```

```xo3ox3oo3oo&#x   → height = sqrt(3/5) = 0.774597
(pen || rap)

o.3o.3o.3o.    | 5  * ♦  4  4  0 |  6  4  6  0  0 | 4  6  4 0 0 | 1 4 1 0
.o3.o3.o3.o    | * 10 ♦  0  2  6 |  0  1  6  3  6 | 0  3  6 3 2 | 0 3 2 1
---------------+------+----------+----------------+-------------+--------
x. .. .. ..    | 2  0 | 10  *  * |  3  1  0  0  0 | 3  3  0 0 0 | 1 3 0 0
oo3oo3oo3oo&#x | 1  1 |  * 20  * |  0  1  3  0  0 | 0  3  3 0 0 | 0 3 1 0
.. .x .. ..    | 0  2 |  *  * 30 |  0  0  1  1  2 | 0  1  2 2 1 | 0 2 1 1
---------------+------+----------+----------------+-------------+--------
x.3o. .. ..    | 3  0 |  3  0  0 | 10  *  *  *  * | 2  1  0 0 0 | 1 2 0 0
xo .. .. ..&#x | 2  1 |  1  2  0 |  * 10  *  *  * | 0  3  0 0 0 | 0 3 0 0
.. ox .. ..&#x | 1  2 |  0  2  1 |  *  * 30  *  * | 0  1  2 0 0 | 0 2 1 0
.o3.x .. ..    | 0  3 |  0  0  3 |  *  *  * 10  * | 0  1  0 2 0 | 0 2 0 1
.. .x3.o ..    | 0  3 |  0  0  3 |  *  *  *  * 20 | 0  0  1 1 1 | 0 1 1 1
---------------+------+----------+----------------+-------------+--------
x.3o.3o. ..    ♦ 4  0 |  6  0  0 |  4  0  0  0  0 | 5  *  * * * | 1 1 0 0
xo3ox .. ..&#x ♦ 3  3 |  3  6  3 |  1  3  3  1  0 | * 10  * * * | 0 2 0 0
.. ox3oo ..&#x ♦ 1  3 |  0  3  3 |  0  0  3  0  1 | *  * 20 * * | 0 1 1 0
.o3.x3.o ..    ♦ 0  6 |  0  0 12 |  0  0  0  4  4 | *  *  * 5 * | 0 1 0 1
.. .x3.o3.o    ♦ 0  4 |  0  0  6 |  0  0  0  0  4 | *  *  * * 5 | 0 0 1 1
---------------+------+----------+----------------+-------------+--------
x.3o.3o.3o.    ♦ 5  0 | 10  0  0 | 10  0  0  0  0 | 5  0  0 0 0 | 1 * * *
xo3ox3oo ..&#x ♦ 4  6 |  6 12 12 |  4  6 12  4  4 | 1  4  4 1 0 | * 5 * *
.. ox3oo3oo&#x ♦ 1  4 |  0  4  6 |  0  0  6  0  4 | 0  0  4 0 1 | * * 5 *
.o3.x3.o3.o    ♦ 0 10 |  0  0 30 |  0  0  0 10 20 | 0  0  0 5 5 | * * * 1
```

```oxo oxo3oox3ooo&#xt   → both heights = sqrt(3/8) = 0.612372
(pt || pseudo tepe || oct)

o.. o..3o..3o..     | 1 * * ♦ 8 0  0  0  0 | 4 12 0  0  0  0 0 0 | 6 8 0  0 0 0 0 | 4 2 0 0
.o. .o.3.o.3.o.     | * 8 * ♦ 1 1  3  3  0 | 1  3 3  3  3  3 0 0 | 3 3 1  3 3 1 0 | 3 1 1 1
..o ..o3..o3..o     | * * 6 ♦ 0 0  0  4  4 | 0  0 0  2  2  8 2 2 | 1 0 0  4 4 4 1 | 2 0 2 2
--------------------+-------+--------------+---------------------+----------------+--------
oo. oo.3oo.3oo.&#x  | 1 1 0 | 8 *  *  *  * | 1  3 0  0  0  0 0 0 | 3 3 0  0 0 0 0 | 3 1 0 0
.x. ... ... ...     | 0 2 0 | * 4  *  *  * | 1  0 0  3  0  0 0 0 | 3 0 0  3 0 0 0 | 3 0 1 0
... .x. ... ...     | 0 2 0 | * * 12  *  * | 0  1 2  0  1  0 0 0 | 1 2 1  0 2 0 0 | 2 1 0 1
.oo .oo3.oo3.oo&#x  | 0 1 1 | * *  * 24  * | 0  0 0  1  1  2 0 0 | 1 0 0  2 2 1 0 | 2 0 1 1
... ... ..x ...     | 0 0 2 | * *  *  * 12 | 0  0 0  0  0  2 1 1 | 0 0 0  1 2 2 1 | 1 0 1 2
--------------------+-------+--------------+---------------------+----------------+--------
ox. ... ... ...&#x  | 1 2 0 | 2 1  0  0  0 | 4  * *  *  *  * * * | 3 0 0  0 0 0 0 | 3 0 0 0
... ox. ... ...&#x  | 1 2 0 | 2 0  1  0  0 | * 12 *  *  *  * * * | 1 2 0  0 0 0 0 | 2 1 0 0
... .x.3.o. ...     | 0 3 0 | 0 0  3  0  0 | *  * 8  *  *  * * * | 0 1 1  0 1 0 0 | 1 1 0 1
.xo ... ... ...&#x  | 0 2 1 | 0 1  0  2  0 | *  * * 12  *  * * * | 1 0 0  2 0 0 0 | 2 0 1 0
... .xo ... ...&#x  | 0 2 1 | 0 0  1  2  0 | *  * *  * 12  * * * | 1 0 0  0 2 0 0 | 2 0 0 1
... ... .ox ...&#x  | 0 1 2 | 0 0  0  2  1 | *  * *  *  * 24 * * | 0 0 0  1 1 1 0 | 1 0 1 1
... ..o3..x ...     | 0 0 3 | 0 0  0  0  3 | *  * *  *  *  * 4 * | 0 0 0  0 2 0 1 | 1 0 0 2
... ... ..x3..o     | 0 0 3 | 0 0  0  0  3 | *  * *  *  *  * * 4 | 0 0 0  0 0 2 1 | 0 0 1 2
--------------------+-------+--------------+---------------------+----------------+--------
oxo oxo ... ...&#xt ♦ 1 4 1 | 4 2  2  4  0 | 2  2 0  2  2  0 0 0 | 6 * *  * * * * | 2 0 0 0
... ox.3oo. ...&#x  ♦ 1 3 0 | 3 0  3  0  0 | 0  3 1  0  0  0 0 0 | * 8 *  * * * * | 1 1 0 0
... .x.3.o.3.o.     ♦ 0 4 0 | 0 0  6  0  0 | 0  0 4  0  0  0 0 0 | * * 2  * * * * | 0 1 0 1
.xo ... .ox ...&#x  ♦ 0 2 2 | 0 1  0  4  1 | 0  0 0  2  0  2 0 0 | * * * 12 * * * | 1 0 1 0
... .xo3.ox ...&#x  ♦ 0 3 3 | 0 0  3  6  3 | 0  0 1  0  3  3 1 0 | * * *  * 8 * * | 1 0 0 1
... ... .ox3.oo&#x  ♦ 0 1 3 | 0 0  0  3  3 | 0  0 0  0  0  3 0 1 | * * *  * * 8 * | 0 0 1 1
... ..o3..x3..o     ♦ 0 0 6 | 0 0  0  0 12 | 0  0 0  0  0  0 4 4 | * * *  * * * 1 | 0 0 0 2
--------------------+-------+--------------+---------------------+----------------+--------
oxo oxo3oox ...&#xt ♦ 1 6 3 | 6 3  6 12  3 | 3  6 2  6  6  6 1 0 | 3 2 0  3 2 0 0 | 4 * * *
... ox.3oo.3oo.&#x  ♦ 1 4 0 | 4 0  6  0  0 | 0  6 4  0  0  0 0 0 | 0 4 1  0 0 0 0 | * 2 * *
.xo ... .ox3.oo&#x  ♦ 0 2 3 | 0 1  0  6  3 | 0  0 0  3  0  6 0 1 | 0 0 0  3 0 2 0 | * * 4 *
... .xo3.ox3.oo&#x  ♦ 0 4 6 | 0 0  6 12 12 | 0  0 4  0  6 12 4 4 | 0 0 1  0 4 4 1 | * * * 2
```

```xoo3oxo oxo3oox&#xt   → both heights = 1/sqrt(3) = 0.577350
({3} || pseudo triddip || perp {3})

o..3o.. o..3o..     | 3 * * ♦ 2  6 0 0  0 0 | 1 6 3  6 0 0  0 0 0 0 | 3 6 3 2 0 0 0 | 3 2 1 0
.o.3.o. .o.3.o.     | * 9 * ♦ 0  2 2 2  2 0 | 0 1 2  4 1 1  4 2 1 0 | 1 2 4 2 2 2 1 | 2 1 2 1
..o3..o ..o3..o     | * * 3 ♦ 0  0 0 0  6 2 | 0 0 0  0 0 0  6 3 6 1 | 0 0 3 0 2 6 3 | 1 0 3 2
--------------------+-------+---------------+-----------------------+---------------+--------
x.. ... ... ...     | 2 0 0 | 3  * * *  * * | 1 3 0  0 0 0  0 0 0 0 | 3 3 0 0 0 0 0 | 3 1 0 0
oo.3oo. oo.3oo.&#x  | 1 1 0 | * 18 * *  * * | 0 1 1  2 0 0  0 0 0 0 | 1 2 2 1 0 0 0 | 2 1 1 0
... .x. ... ...     | 0 2 0 | *  * 9 *  * * | 0 0 1  0 1 0  2 0 0 0 | 1 0 2 0 2 1 0 | 2 0 1 1
... ... .x. ...     | 0 2 0 | *  * * 9  * * | 0 0 0  2 0 1  0 1 0 0 | 0 1 2 2 0 0 1 | 1 1 2 0
.oo3.oo .oo3.oo&#x  | 0 1 1 | *  * * * 18 * | 0 0 0  0 0 0  2 1 1 0 | 0 0 2 0 1 2 1 | 1 0 2 1
... ... ... ..x     | 0 0 2 | *  * * *  * 3 | 0 0 0  0 0 0  0 0 3 1 | 0 0 0 0 0 3 3 | 0 0 3 1
--------------------+-------+---------------+-----------------------+---------------+--------
x..3o.. ... ...     | 3 0 0 | 3  0 0 0  0 0 | 1 * *  * * *  * * * * | 3 0 0 0 0 0 0 | 3 0 0 0
xo. ... ... ...&#x  | 2 1 0 | 1  2 0 0  0 0 | * 9 *  * * *  * * * * | 1 2 0 0 0 0 0 | 2 1 0 0
... ox. ... ...&#x  | 1 2 0 | 0  2 1 0  0 0 | * * 9  * * *  * * * * | 1 0 2 0 0 0 0 | 2 0 1 0
... ... ox. ...&#x  | 1 2 0 | 0  2 0 1  0 0 | * * * 18 * *  * * * * | 0 1 1 1 0 0 0 | 1 1 1 0
.o.3.x. ... ...     | 0 3 0 | 0  0 3 0  0 0 | * * *  * 3 *  * * * * | 1 0 0 0 2 0 0 | 2 0 0 1
... ... .x.3.o.     | 0 3 0 | 0  0 0 3  0 0 | * * *  * * 3  * * * * | 0 0 0 2 0 0 1 | 0 1 2 0
... .xo ... ...&#x  | 0 2 1 | 0  0 1 0  2 0 | * * *  * * * 18 * * * | 0 0 1 0 1 1 0 | 1 0 1 1
... ... .xo ...&#x  | 0 2 1 | 0  0 0 1  2 0 | * * *  * * *  * 9 * * | 0 0 2 0 0 0 1 | 1 0 2 0
... ... ... .ox&#x  | 0 1 2 | 0  0 0 0  2 1 | * * *  * * *  * * 9 * | 0 0 0 0 0 2 1 | 0 0 2 1
... ... ..o3..x     | 0 0 3 | 0  0 0 0  0 3 | * * *  * * *  * * * 1 | 0 0 0 0 0 0 3 | 0 0 3 0
--------------------+-------+---------------+-----------------------+---------------+--------
xo.3ox. ... ...&#x  ♦ 3 3 0 | 3  6 3 0  0 0 | 1 3 3  0 1 0  0 0 0 0 | 3 * * * * * * | 2 0 0 0
xo. ... ox. ...&#x  ♦ 2 2 0 | 1  4 0 1  0 0 | 0 2 0  2 0 0  0 0 0 0 | * 9 * * * * * | 1 1 0 0
... oxo oxo ...&#x  ♦ 1 4 1 | 0  4 2 2  4 0 | 0 0 2  2 0 0  2 2 0 0 | * * 9 * * * * | 1 0 1 0
... ... ox.3oo.&#x  ♦ 1 3 0 | 0  3 0 3  0 0 | 0 0 0  3 0 1  0 0 0 0 | * * * 6 * * * | 0 1 1 0
.oo3.xo ... ...&#x  ♦ 0 3 1 | 0  0 3 0  3 0 | 0 0 0  0 1 0  3 0 0 0 | * * * * 6 * * | 1 0 0 1
... .xo ... .ox&#x  ♦ 0 2 2 | 0  0 1 0  4 1 | 0 0 0  0 0 0  2 0 2 0 | * * * * * 9 * | 0 0 1 1
... ... .xo3.ox&#x  ♦ 0 3 3 | 0  0 0 3  6 3 | 0 0 0  0 0 1  0 3 3 1 | * * * * * * 3 | 0 0 2 0
--------------------+-------+---------------+-----------------------+---------------+--------
xoo3oxo oxo ...&#xt ♦ 3 6 1 | 3 12 6 3  6 0 | 1 6 6  6 2 0  6 3 0 0 | 2 3 3 0 2 0 0 | 3 * * *
xo. ... ox.3oo.&#x  ♦ 2 3 0 | 1  6 0 3  0 0 | 0 3 0  6 0 1  0 0 0 0 | 0 3 0 2 0 0 0 | * 3 * *
... oxo oxo3oox&#xt ♦ 1 6 3 | 0  6 3 6 12 3 | 0 0 3  6 0 2  6 6 6 1 | 0 0 3 2 0 3 2 | * * 3 *
.oo3.xo ... .ox&#x  ♦ 0 3 2 | 0  0 3 0  6 1 | 0 0 0  0 1 0  6 0 3 0 | 0 0 0 0 2 3 0 | * * * 3

or
o..3o.. o..3o..     & | 6 * ♦ 2  6  0 | 1  6  3  6 0 | 3  6 3  2 | 4 2
.o.3.o. .o.3.o.       | * 9 ♦ 0  4  4 | 0  2  4  8 2 | 2  4 4  4 | 4 2
----------------------+-----+---------+--------------+-----------+----
x.. ... ... ...     & | 2 0 | 6  *  * | 1  3  0  0 0 | 3  3 0  0 | 3 1
oo.3oo. oo.3oo.&#x  & | 1 1 | * 36  * | 0  1  1  2 0 | 1  2 2  1 | 3 1
... .x. ... ...     & | 0 2 | *  * 18 | 0  0  1  2 1 | 1  1 2  2 | 3 1
----------------------+-----+---------+--------------+-----------+----
x..3o.. ... ...     & | 3 0 | 3  0  0 | 2  *  *  * * | 3  0 0  0 | 3 0
xo. ... ... ...&#x  & | 2 1 | 1  2  0 | * 18  *  * * | 1  2 0  0 | 2 1
... ox. ... ...&#x  & | 1 2 | 0  2  1 | *  * 18  * * | 1  0 2  0 | 3 0
... ... ox. ...&#x  & | 1 2 | 0  2  1 | *  *  * 36 * | 0  1 1  1 | 2 1
.o.3.x. ... ...     & | 0 3 | 0  0  3 | *  *  *  * 6 | 1  0 0  2 | 2 1
----------------------+-----+---------+--------------+-----------+----
xo.3ox. ... ...&#x  & ♦ 3 3 | 3  6  3 | 1  3  3  0 1 | 6  * *  * | 2 0
xo. ... ox. ...&#x  & ♦ 2 2 | 1  4  1 | 0  2  0  2 0 | * 18 *  * | 1 1
... oxo oxo ...&#x    ♦ 2 4 | 0  8  4 | 0  0  4  4 0 | *  * 9  * | 2 0
... ... ox.3oo.&#x  & ♦ 1 3 | 0  3  3 | 0  0  0  3 1 | *  * * 12 | 1 1
----------------------+-----+---------+--------------+-----------+----
xoo3oxo oxo ...&#xt & ♦ 4 6 | 3 18  9 | 1  6  9 12 2 | 2  3 3  2 | 6 *
xo. ... ox.3oo.&#x  & ♦ 2 3 | 1  6  3 | 0  3  0  6 1 | 0  3 0  2 | * 6
```

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