Acronym hix Name hexateron,5D simplex (α5),pyroteron,pentachoral pyramid,regular tet-scalene,regular triangle-tettene,regular line-pennene,regular (point-)hixene Circumradius sqrt(5/12) = 0.645497 Inradius 1/sqrt(60) = 0.129099 Vertex figure ` ©` Dual (selfdual, in different orientation) Confer general polytopal classes: simplex Externallinks

Incidence matrix according to Dynkin symbol

```x3o3o3o3o

. . . . . | 6 ♦  5 | 10 | 10 | 5
----------+---+----+----+----+--
x . . . . | 2 | 15 ♦  4 |  6 | 4
----------+---+----+----+----+--
x3o . . . | 3 |  3 | 20 |  3 | 3
----------+---+----+----+----+--
x3o3o . . ♦ 4 |  6 |  4 | 15 | 2
----------+---+----+----+----+--
x3o3o3o . ♦ 5 | 10 | 10 |  5 | 6
```

```ox3oo3oo3oo&#x   → height = sqrt(3/5) = 0.774597
(pt || pen)

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

```xo ox3oo3oo&#x   → height = sqrt(3/8) = 0.612372
(line || ortho tet)

o. o.3o.3o.    | 2 * ♦ 1 4 0 | 4  6 0 | 6 4 0 | 4 1
.o .o3.o3.o    | * 4 ♦ 0 2 3 | 1  6 3 | 3 6 1 | 3 2
---------------+-----+-------+--------+-------+----
x. .. .. ..    | 2 0 | 1 * * ♦ 4  0 0 | 6 0 0 | 4 0
oo oo3oo3oo&#x | 1 1 | * 8 * ♦ 1  3 0 | 3 3 0 | 3 1
.. .x .. ..    | 0 2 | * * 6 ♦ 0  2 2 | 1 4 1 | 2 2
---------------+-----+-------+--------+-------+----
xo .. .. ..&#x | 2 1 | 1 2 0 | 4  * * | 3 0 0 | 3 0
.. ox .. ..&#x | 1 2 | 0 2 1 | * 12 * | 1 2 0 | 2 1
.. .x3.o ..    | 0 3 | 0 0 3 | *  * 4 | 0 2 1 | 1 2
---------------+-----+-------+--------+-------+----
xo ox .. ..&#x ♦ 2 2 | 1 4 1 | 2  2 0 | 6 * * | 2 0
.. ox3oo ..&#x ♦ 1 3 | 0 3 3 | 0  3 1 | * 8 * | 1 1
.. .x3.o3.o    ♦ 0 4 | 0 0 6 | 0  0 4 | * * 1 | 0 2
---------------+-----+-------+--------+-------+----
xo ox3oo ..&#x ♦ 2 3 | 1 6 3 | 3  6 1 | 3 2 0 | 4 *
.. ox3oo3oo&#x ♦ 1 4 | 0 4 6 | 0  6 4 | 0 4 1 | * 2
```

```xo3oo ox3oo&#x   → height = 1/sqrt(3) = 0.577350
({3} || ortho {3})

o.3o. o.3o.    | 3 * ♦ 2 3 0 | 1 6 3 0 | 3 6 1 | 3 2
.o3.o .o3.o    | * 3 ♦ 0 3 2 | 0 3 6 1 | 1 6 3 | 2 3
---------------+-----+-------+---------+-------+----
x. .. .. ..    | 2 0 | 3 * * ♦ 1 3 0 0 | 3 3 0 | 3 1
oo3oo oo3oo&#x | 1 1 | * 9 * ♦ 0 2 2 0 | 1 4 1 | 2 2
.. .. .x ..    | 0 2 | * * 3 ♦ 0 0 3 1 | 0 3 3 | 1 3
---------------+-----+-------+---------+-------+----
x.3o. .. ..    | 3 0 | 3 0 0 | 1 * * * | 3 0 0 | 3 0
xo .. .. ..&#x | 2 1 | 1 2 0 | * 9 * * | 1 2 0 | 2 1
.. .. ox ..&#x | 1 2 | 0 2 1 | * * 9 * | 0 2 1 | 1 2
.. .. .x3.o    | 0 3 | 0 0 3 | * * * 1 | 0 0 3 | 0 3
---------------+-----+-------+---------+-------+----
xo3oo .. ..&#x ♦ 3 1 | 3 3 0 | 1 3 0 0 | 3 * * | 2 0
xo .. ox ..&#x ♦ 2 2 | 1 4 1 | 0 2 2 0 | * 9 * | 1 1
.. .. ox3oo&#x ♦ 1 3 | 0 3 3 | 0 0 3 1 | * * 3 | 0 2
---------------+-----+-------+---------+-------+----
xo3oo ox ..&#x ♦ 3 2 | 3 6 1 | 1 6 3 0 | 2 3 0 | 3 *
xo .. ox3oo&#x ♦ 2 3 | 1 6 3 | 0 3 6 1 | 0 3 2 | * 3

or
o.3o. o.3o.    & | 6 ♦ 2 3 | 1  9 | 4 3 | 5
-----------------+---+-----+------+-----+--
x. .. .. ..    & | 2 | 6 * ♦ 1  3 | 3 3 | 4
oo3oo oo3oo&#x   | 2 | * 9 ♦ 0  4 | 2 4 | 4
-----------------+---+-----+------+-----+--
x.3o. .. ..    & | 3 | 3 0 | 2  * | 3 0 | 3
xo .. .. ..&#x & | 3 | 1 2 | * 18 | 1 2 | 3
-----------------+---+-----+------+-----+--
xo3oo .. ..&#x & ♦ 4 | 3 3 | 1  3 | 6 * | 2
xo .. ox ..&#x   ♦ 4 | 2 4 | 0  4 | * 9 | 2
-----------------+---+-----+------+-----+--
xo3oo ox ..&#x & ♦ 5 | 4 6 | 1  9 | 2 3 | 6
```

```oxo3ooo3ooo&#x   → height(1,2) = height(2,3) = sqrt(5/8) = 0.790569
height(1,3) = 1
( (pt || tet) || pt )

o..3o..3o..    | 1 * * ♦ 4 1 0 0 | 6 4 0 0 | 4 6 0 0 | 1 4 0
.o.3.o.3.o.    | * 4 * ♦ 1 0 3 1 | 3 1 3 3 | 3 3 1 3 | 1 3 1
..o3..o3..o    | * * 1 ♦ 0 1 0 4 | 0 4 0 6 | 0 6 0 4 | 0 4 1
---------------+-------+---------+---------+---------+------
oo.3oo.3oo.&#x | 1 1 0 | 4 * * * ♦ 3 1 0 0 | 3 3 0 0 | 1 3 0
o.o3o.o3o.o&#x | 1 0 1 | * 1 * * ♦ 0 4 0 0 | 0 6 0 0 | 0 4 0
.x. ... ...    | 0 2 0 | * * 6 * ♦ 1 0 2 1 | 2 1 1 2 | 1 2 1
.oo3.oo3.oo&#x | 0 1 1 | * * * 4 ♦ 0 1 0 3 | 0 3 0 3 | 0 3 1
---------------+-------+---------+---------+---------+------
ox. ... ...&#x | 1 2 0 | 2 0 1 0 | 6 * * * | 2 1 0 0 | 1 2 0
ooo3ooo3ooo&#x | 1 1 1 | 1 1 0 1 | * 4 * * | 0 3 0 0 | 0 3 0
.x.3.o. ...    | 0 3 0 | 0 0 3 0 | * * 4 * | 1 0 1 1 | 1 1 1
.xo ... ...&#x | 0 2 1 | 0 0 1 2 | * * * 6 | 0 1 0 2 | 0 2 1
---------------+-------+---------+---------+---------+------
ox.3oo. ...&#x ♦ 1 3 0 | 3 0 3 0 | 3 0 1 0 | 4 * * * | 1 1 0
oxo ... ...&#x ♦ 1 2 1 | 2 1 1 2 | 1 2 0 1 | * 6 * * | 0 2 0
.x.3.o.3.o.    ♦ 0 4 0 | 0 0 6 0 | 0 0 4 0 | * * 1 * | 1 0 1
.xo3.oo ...&#x ♦ 0 3 1 | 0 0 3 3 | 0 0 1 3 | * * * 4 | 0 1 1
---------------+-------+---------+---------+---------+------
ox.3oo.3oo.&#x ♦ 1 4 0 | 4 0 6 0 | 6 0 4 0 | 4 0 1 0 | 1 * *
oxo3ooo ...&#x ♦ 1 3 1 | 3 1 3 3 | 3 3 1 3 | 1 3 0 1 | * 4 *
.xo3.oo3.oo&#x ♦ 0 4 1 | 0 0 6 4 | 0 0 4 6 | 0 0 1 4 | * * 1
```

```oxo3ooo oox&#x   → height(1,2) = sqrt(2/3) = 0.816497
height(1,3) = sqrt(3)/2 = 0.866025
height(2,3) = sqrt(5/12) = 0.645497
( (pt || {3}) || line )

o..3o.. o..    | 1 * * ♦ 3 2 0 0 0 | 3 1 6 0 0 0 | 1 6 3 0 0 | 2 3 0
.o.3.o. .o.    | * 3 * ♦ 1 0 2 2 0 | 2 0 2 1 4 1 | 1 4 1 2 2 | 2 2 1
..o3..o ..o    | * * 2 ♦ 0 1 0 3 1 | 0 1 3 0 3 3 | 0 3 3 1 3 | 1 3 1
---------------+-------+-----------+-------------+-----------+------
oo.3oo. oo.&#x | 1 1 0 | 3 * * * * ♦ 2 0 2 0 0 0 | 1 4 1 0 0 | 2 2 0
o.o3o.o o.o&#x | 1 0 1 | * 2 * * * ♦ 0 1 3 0 0 0 | 0 3 3 0 0 | 1 3 0
.x. ... ...    | 0 2 0 | * * 3 * * ♦ 1 0 0 1 2 0 | 1 2 0 2 1 | 2 1 1
.oo3.oo .oo&#x | 0 1 1 | * * * 6 * ♦ 0 0 1 0 2 1 | 0 2 1 1 2 | 1 2 1
... ... ..x    | 0 0 2 | * * * * 1 ♦ 0 1 0 0 0 3 | 0 0 3 0 3 | 0 3 1
---------------+-------+-----------+-------------+-----------+------
ox. ... ...&#x | 1 2 0 | 2 0 1 0 0 | 3 * * * * * | 1 2 0 0 0 | 2 1 0
... ... o.x&#x | 1 0 2 | 0 2 0 0 1 | * 1 * * * * | 0 0 3 0 0 | 0 3 0
ooo3ooo ooo&#x | 1 1 1 | 1 1 0 1 0 | * * 6 * * * | 0 2 1 0 0 | 1 2 0
.x.3.o. ...    | 0 3 0 | 0 0 3 0 0 | * * * 1 * * | 1 0 0 2 0 | 2 0 1
.xo ... ...&#x | 0 2 1 | 0 0 1 2 0 | * * * * 6 * | 0 1 0 1 1 | 1 1 1
... ... .ox&#x | 0 1 2 | 0 0 0 2 1 | * * * * * 3 | 0 0 1 0 2 | 0 2 1
---------------+-------+-----------+-------------+-----------+------
ox.3oo. ...&#x ♦ 1 3 0 | 3 0 3 0 0 | 3 0 0 1 0 0 | 1 * * * * | 2 0 0
oxo ... ...&#x ♦ 1 2 1 | 2 1 1 2 0 | 1 0 2 0 1 0 | * 6 * * * | 1 1 0
... ... oox&#x ♦ 1 1 2 | 1 2 0 2 1 | 0 1 2 0 0 1 | * * 3 * * | 0 2 0
.xo3.oo ...&#x ♦ 0 3 1 | 0 0 3 3 0 | 0 0 0 1 3 0 | * * * 2 * | 1 0 1
.xo ... .ox&#x ♦ 0 2 2 | 0 0 1 4 1 | 0 0 0 0 2 2 | * * * * 3 | 0 1 1
---------------+-------+-----------+-------------+-----------+------
oxo3ooo ...&#x ♦ 1 3 1 | 3 1 3 3 0 | 3 0 3 1 3 0 | 1 3 0 1 0 | 2 * *
oxo ... oox&#x ♦ 1 2 2 | 2 2 1 4 1 | 1 1 4 0 2 2 | 0 2 2 0 1 | * 3 *
.xo3.oo .ox&#x ♦ 0 3 2 | 0 0 3 6 1 | 0 0 0 1 6 3 | 0 0 0 2 3 | * * 1
```