Acronym dihin (alt.: penalrap)
Name (vertex-)diminished hemipenteract,
pentachoron atop alternate rectified pentachoron
Lace city
in approx. ASCII-art
```      +----- oo3ox3oo&#x (octpy)
/   +-- xo3oo3ox&#x (alt. hex)
/   /

O   t	-- o3o3x3o (rap)

o   T  	-- x3o3o3o (inv pen)

\
+-- o3x3o3o (inv rap)

where:
o = o3o3o (point)
T = x3o3o (tet)
O = o3x3o (oct)
t = o3o3x (dual tet)
```
Lace hyper city
in approx. ASCII-art
 ``` o o o o x x o o ``` ``` x o o x o x x o ```
(octpy || alternate hex)
Confer
uniform relative:
hin
segmentotera:
bidhin   tedhin   squasquasc   rappy
general polytopal classes:
segmentotera   fundamental lace prisms   lace simplices
External

Incidence matrix according to Dynkin symbol

```oo3xo3oo3ox&#x   → height = sqrt(2/5) = 0.632456
(rap || inv pen)

o.3o.3o.3o.    | 10 * ♦  6  3  0 |  3  6 12  3  0 | 3 2  6  6  6  1 0 | 1 3  3 2 0
.o3.o3.o3.o    |  * 5 ♦  0  6  4 |  0  0 12 12  6 | 0 0  4  4 12  6 4 | 0 1  4 4 1
---------------+------+----------+----------------+-------------------+-----------
.. x. .. ..    |  2 0 | 30  *  * |  1  2  2  0  0 | 2 1  2  2  1  0 0 | 1 2  1 1 0
oo3oo3oo3oo&#x |  1 1 |  * 30  * ♦  0  0  4  2  0 | 0 0  2  2  4  1 0 | 0 1  2 2 0
.. .. .. .x    |  0 2 |  *  * 10 ♦  0  0  0  3  3 | 0 0  0  0  3  3 3 | 0 0  1 3 1
---------------+------+----------+----------------+-------------------+-----------
o.3x. .. ..    |  3 0 |  3  0  0 | 10  *  *  *  * | 2 2  0  0  0  0 0 | 1 2  1 0 0
.. x.3o. ..    |  3 0 |  3  0  0 |  * 20  *  *  * | 1 1  0  1  0  0 0 | 1 1  0 1 0
.. xo .. ..&#x |  2 1 |  1  2  0 |  *  * 60  *  * | 0 0  1  1  1  0 0 | 0 1  1 1 0
.. .. .. ox&#x |  1 2 |  0  2  1 |  *  *  * 30  * | 0 0  0  0  2  1 0 | 0 0  1 2 0
.. .. .o3.x    |  0 3 |  0  0  3 |  *  *  *  * 10 | 0 0  0  0  0  1 2 | 0 0  0 2 1
---------------+------+----------+----------------+-------------------+-----------
o.3x.3o. ..&#x ♦  6 0 | 12  0  0 |  4  4  0  0  0 | 5 *  *  *  *  * * | 1 1  0 0 0
.. x.3o.3o.    ♦  4 0 |  6  0  0 |  0  4  0  0  0 | * 5  *  *  *  * * | 1 0  0 1 0
oo3xo .. ..&#x ♦  3 1 |  3  3  0 |  1  0  3  0  0 | * * 20  *  *  * * | 0 1  1 0 0
.. xo3oo ..&#x ♦  3 1 |  3  3  0 |  0  1  3  0  0 | * *  * 20  *  * * | 0 1  0 1 0
.. xo .. ox&#x ♦  2 2 |  1  4  1 |  0  0  2  2  0 | * *  *  * 30  * * | 0 0  1 1 0
.. .. oo3ox&#x ♦  1 3 |  0  3  3 |  0  0  0  3  1 | * *  *  *  * 10 * | 0 0  0 2 0
.. .o3.o3.x    ♦  0 4 |  0  0  6 |  0  0  0  0  4 | * *  *  *  *  * 5 | 0 0  0 1 1
---------------+------+----------+----------------+-------------------+-----------
o.3x.3o.3o.&#x ♦ 10 0 | 30  0  0 | 10 20  0  0  0 | 5 5  0  0  0  0 0 | 1 *  * * *
oo3xo3oo ..&#x ♦  6 1 | 12  6  0 |  4  4 12  0  0 | 1 0  4  4  0  0 0 | * 5  * * *
oo3xo .. ox&#x ♦  3 2 |  3  6  1 |  1  0  6  3  0 | 0 0  2  0  3  0 0 | * * 10 * *
.. xo3oo3ox&#x ♦  4 4 |  6 12  6 |  0  4 12 12  4 | 0 1  0  4  6  4 1 | * *  * 5 *
.o3.o3.o3.x    ♦  0 5 |  0  0 10 |  0  0  0  0 10 | 0 0  0  0  0  0 5 | * *  * * 1
```

```ooxo3oxoo3ooox&#xr   → height(1,2) = height(3,4) = 1/sqrt(2) = 0.707107
height(1,4) = height(2,3) = height(2,4) sqrt(5/8) = 0.790569

o...3o...3o...     | 1 * * * ♦ 6 4  0  0  0 0  0 0 | 12 6 12 0 0 0  0  0 0  0 0  0  0 0 | 4 4 4 12 6 0 0 0 0  0  0  0 0 0 0 0 0 | 1 1 4 4 0 0 0 0
.o..3.o..3.o..     | * 6 * * ♦ 1 0  4  2  2 0  0 0 |  4 0  2 2 2 1  4  4 1  4 0  0  0 0 | 2 2 0  4 1 1 2 2 2  2  4  2 0 0 0 0 0 | 1 0 2 2 1 2 1 0
..o.3..o.3..o.     | * * 4 * ♦ 0 0  0  3  0 3  3 0 |  0 0  0 0 0 3  3  0 0  6 3  6  3 0 | 0 0 0  0 0 0 3 1 0  6  3  3 1 3 3 1 0 | 0 0 0 1 1 3 3 1
...o3...o3...o     | * * * 4 ♦ 0 1  0  0  3 0  3 3 |  0 3  3 0 0 0  0  3 3  6 0  3  6 3 | 0 0 3  3 3 0 0 0 1  3  3  6 0 1 3 3 1 | 0 1 1 3 0 1 3 1
-------------------+---------+---------------------+------------------------------------+---------------------------------------+----------------
oo..3oo..3oo..&#x  | 1 1 0 0 | 6 *  *  *  * *  * * ♦  4 0  2 0 0 0  0  0 0  0 0  0  0 0 | 2 2 0  4 1 0 0 0 0  0  0  0 0 0 0 0 0 | 1 0 2 2 0 0 0 0
o..o3o..o3o..o&#x  | 1 0 0 1 | * 4  *  *  * *  * * ♦  0 3  3 0 0 0  0  0 0  0 0  0  0 0 | 0 0 3  3 3 0 0 0 0  0  0  0 0 0 0 0 0 | 0 1 1 3 0 0 0 0
.... .x.. ....     | 0 2 0 0 | * * 12  *  * *  * * |  1 0  0 1 1 0  1  1 0  0 0  0  0 0 | 1 1 0  1 0 1 1 1 1  0  1  0 0 0 0 0 0 | 1 0 1 1 1 1 0 0
.oo.3.oo.3.oo.&#x  | 0 1 1 0 | * *  * 12  * *  * * |  0 0  0 0 0 1  2  0 0  2 0  0  0 0 | 0 0 0  0 0 0 2 1 0  2  2  1 0 0 0 0 0 | 0 0 0 1 1 2 1 0
.o.o3.o.o3.o.o&#x  | 0 1 0 1 | * *  *  * 12 *  * * ♦  0 0  1 0 0 0  0  2 1  2 0  0  0 0 | 0 0 0  2 1 0 0 0 1  1  2  2 0 0 0 0 0 | 0 0 1 2 0 1 1 0
..x. .... ....     | 0 0 2 0 | * *  *  *  * 6  * * |  0 0  0 0 0 1  0  0 0  0 2  2  0 0 | 0 0 0  0 0 0 2 0 0  2  0  0 1 2 1 0 0 | 0 0 0 0 1 2 1 1
..oo3..oo3..oo&#x  | 0 0 1 1 | * *  *  *  * * 12 * ♦  0 0  0 0 0 0  0  0 0  2 0  2  2 0 | 0 0 0  0 0 0 0 0 0  2  1  2 0 1 2 1 0 | 0 0 0 1 0 1 2 1
.... .... ...x     | 0 0 0 2 | * *  *  *  * *  * 6 ♦  0 1  0 0 0 0  0  0 1  0 0  0  2 2 | 0 0 2  0 1 0 0 0 0  0  0  2 0 0 1 2 1 | 0 1 0 2 0 0 1 1
-------------------+---------+---------------------+------------------------------------+---------------------------------------+----------------
.... ox.. ....&#x  | 1 2 0 0 | 2 0  1  0  0 0  0 0 | 12 *  * * * *  *  * *  * *  *  * * | 1 1 0  1 0 0 0 0 0  0  0  0 0 0 0 0 0 | 1 0 1 1 0 0 0 0
.... .... o..x&#x  | 1 0 0 2 | 0 2  0  0  0 0  0 1 |  * 6  * * * *  *  * *  * *  *  * * | 0 0 2  0 1 0 0 0 0  0  0  0 0 0 0 0 0 | 0 1 0 2 0 0 0 0
oo.o3oo.o3oo.o&#x  | 1 1 0 1 | 1 1  0  0  1 0  0 0 |  * * 12 * * *  *  * *  * *  *  * * | 0 0 0  2 1 0 0 0 0  0  0  0 0 0 0 0 0 | 0 0 1 2 0 0 0 0
.o..3.x.. ....     | 0 3 0 0 | 0 0  3  0  0 0  0 0 |  * *  * 4 * *  *  * *  * *  *  * * | 1 0 0  0 0 1 1 0 1  0  0  0 0 0 0 0 0 | 1 0 1 0 1 1 0 0
.... .x..3.o..     | 0 3 0 0 | 0 0  3  0  0 0  0 0 |  * *  * * 4 *  *  * *  * *  *  * * | 0 1 0  0 0 1 0 1 0  0  0  0 0 0 0 0 0 | 1 0 0 1 1 0 0 0
.ox. .... ....&#x  | 0 1 2 0 | 0 0  0  2  0 1  0 0 |  * *  * * * 6  *  * *  * *  *  * * | 0 0 0  0 0 0 2 0 0  2  0  0 0 0 0 0 0 | 0 0 0 0 1 2 1 0
.... .xo. ....&#x  | 0 2 1 0 | 0 0  1  2  0 0  0 0 |  * *  * * * * 12  * *  * *  *  * * | 0 0 0  0 0 0 1 1 0  0  1  0 0 0 0 0 0 | 0 0 0 1 1 1 0 0
.... .x.o ....&#x  | 0 2 0 1 | 0 0  1  0  2 0  0 0 |  * *  * * * *  * 12 *  * *  *  * * | 0 0 0  1 0 0 0 0 1  0  1  0 0 0 0 0 0 | 0 0 1 1 0 1 0 0
.... .... .o.x&#x  | 0 1 0 2 | 0 0  0  0  2 0  0 1 |  * *  * * * *  *  * 6  * *  *  * * | 0 0 0  0 1 0 0 0 0  0  0  2 0 0 0 0 0 | 0 0 0 2 0 0 1 0
.ooo3.ooo3.ooo&#x  | 0 1 1 1 | 0 0  0  1  1 0  1 0 |  * *  * * * *  *  * * 24 *  *  * * | 0 0 0  0 0 0 0 0 0  1  1  1 0 0 0 0 0 | 0 0 0 1 0 1 1 0
..x.3..o. ....     | 0 0 3 0 | 0 0  0  0  0 3  0 0 |  * *  * * * *  *  * *  * 4  *  * * | 0 0 0  0 0 0 1 0 0  0  0  0 1 1 0 0 0 | 0 0 0 0 1 1 0 1
..xo .... ....&#x  | 0 0 2 1 | 0 0  0  0  0 1  2 0 |  * *  * * * *  *  * *  * * 12  * * | 0 0 0  0 0 0 0 0 0  1  0  0 0 1 1 0 0 | 0 0 0 0 0 1 1 1
.... .... ..ox&#x  | 0 0 1 2 | 0 0  0  0  0 0  2 1 |  * *  * * * *  *  * *  * *  * 12 * | 0 0 0  0 0 0 0 0 0  0  0  1 0 0 1 1 0 | 0 0 0 1 0 0 1 1
.... ...o3...x     | 0 0 0 3 | 0 0  0  0  0 0  0 3 |  * *  * * * *  *  * *  * *  *  * 4 | 0 0 1  0 0 0 0 0 0  0  0  0 0 0 0 1 1 | 0 1 0 1 0 0 0 1
-------------------+---------+---------------------+------------------------------------+---------------------------------------+----------------
oo..3ox.. ....&#x  ♦ 1 3 0 0 | 3 0  3  0  0 0  0 0 |  3 0  0 1 0 0  0  0 0  0 0  0  0 0 | 4 * *  * * * * * *  *  *  * * * * * * | 1 0 1 0 0 0 0 0
.... ox..3oo..&#x  ♦ 1 3 0 0 | 3 0  3  0  0 0  0 0 |  3 0  0 0 1 0  0  0 0  0 0  0  0 0 | * 4 *  * * * * * *  *  *  * * * * * * | 1 0 0 1 0 0 0 0
.... o..o3o..x&#x  ♦ 1 0 0 3 | 0 3  0  0  0 0  0 3 |  0 3  0 0 0 0  0  0 0  0 0  0  0 1 | * * 4  * * * * * *  *  *  * * * * * * | 0 1 0 1 0 0 0 0
.... ox.o ....&#x  ♦ 1 2 0 1 | 2 1  1  0  2 0  0 0 |  1 0  2 0 0 0  0  1 0  0 0  0  0 0 | * * * 12 * * * * *  *  *  * * * * * * | 0 0 1 1 0 0 0 0
.... .... oo.x&#x  ♦ 1 1 0 2 | 1 2  0  0  2 0  0 1 |  0 1  2 0 0 0  0  0 1  0 0  0  0 0 | * * *  * 6 * * * *  *  *  * * * * * * | 0 0 0 2 0 0 0 0
.o..3.x..3.o..     ♦ 0 6 0 0 | 0 0 12  0  0 0  0 0 |  0 0  0 4 4 0  0  0 0  0 0  0  0 0 | * * *  * * 1 * * *  *  *  * * * * * * | 1 0 0 0 1 0 0 0
.ox.3.xo. ....&#x  ♦ 0 3 3 0 | 0 0  3  6  0 3  0 0 |  0 0  0 1 0 3  3  0 0  0 1  0  0 0 | * * *  * * * 4 * *  *  *  * * * * * * | 0 0 0 0 1 1 0 0
.... .xo.3.oo.&#x  ♦ 0 3 1 0 | 0 0  3  3  0 0  0 0 |  0 0  0 0 1 0  3  0 0  0 0  0  0 0 | * * *  * * * * 4 *  *  *  * * * * * * | 0 0 0 1 1 0 0 0
.o.o3.x.o ....&#x  ♦ 0 3 0 1 | 0 0  3  0  3 0  0 0 |  0 0  0 1 0 0  0  3 0  0 0  0  0 0 | * * *  * * * * * 4  *  *  * * * * * * | 0 0 1 0 0 1 0 0
.oxo .... ....&#x  ♦ 0 1 2 1 | 0 0  0  2  1 1  2 0 |  0 0  0 0 0 1  0  0 0  2 0  1  0 0 | * * *  * * * * * * 12  *  * * * * * * | 0 0 0 0 0 1 1 0
.... .xoo ....&#x  ♦ 0 2 1 1 | 0 0  1  2  2 0  1 0 |  0 0  0 0 0 0  1  1 0  2 0  0  0 0 | * * *  * * * * * *  * 12  * * * * * * | 0 0 0 1 0 1 0 0
.... .... .oox&#x  ♦ 0 1 1 2 | 0 0  0  1  2 0  2 1 |  0 0  0 0 0 0  0  0 1  2 0  0  1 0 | * * *  * * * * * *  *  * 12 * * * * * | 0 0 0 1 0 0 1 0
..x.3..o.3..o.     ♦ 0 0 4 0 | 0 0  0  0  0 6  0 0 |  0 0  0 0 0 0  0  0 0  0 4  0  0 0 | * * *  * * * * * *  *  *  * 1 * * * * | 0 0 0 0 1 0 0 1
..xo3..oo ....&#x  ♦ 0 0 3 1 | 0 0  0  0  0 3  3 0 |  0 0  0 0 0 0  0  0 0  0 1  3  0 0 | * * *  * * * * * *  *  *  * * 4 * * * | 0 0 0 0 0 1 0 1
..xo .... ..ox&#x  ♦ 0 0 2 2 | 0 0  0  0  0 1  4 1 |  0 0  0 0 0 0  0  0 0  0 0  2  2 0 | * * *  * * * * * *  *  *  * * * 6 * * | 0 0 0 0 0 0 1 1
.... ..oo3..ox&#x  ♦ 0 0 1 3 | 0 0  0  0  0 0  3 3 |  0 0  0 0 0 0  0  0 0  0 0  0  3 1 | * * *  * * * * * *  *  *  * * * * 4 * | 0 0 0 1 0 0 0 1
...o3...o3...x     ♦ 0 0 0 4 | 0 0  0  0  0 0  0 6 |  0 0  0 0 0 0  0  0 0  0 0  0  0 4 | * * *  * * * * * *  *  *  * * * * * 1 | 0 1 0 0 0 0 0 1
-------------------+---------+---------------------+------------------------------------+---------------------------------------+----------------
oo..3ox..3oo..&#x  ♦ 1 6 0 0 | 6 0 12  0  0 0  0 0 | 12 0  0 4 4 0  0  0 0  0 0  0  0 0 | 4 4 0  0 0 1 0 0 0  0  0  0 0 0 0 0 0 | 1 * * * * * * *
o..o3o..o3o..x&#x  ♦ 1 0 0 4 | 0 4  0  0  0 0  0 6 |  0 6  0 0 0 0  0  0 0  0 0  0  0 4 | 0 0 4  0 0 0 0 0 0  0  0  0 0 0 0 0 1 | * 1 * * * * * *
oo.o3ox.o ....&#x  ♦ 1 3 0 1 | 3 1  3  0  3 0  0 0 |  3 0  3 1 0 0  0  3 0  0 0  0  0 0 | 1 0 0  3 0 0 0 0 1  0  0  0 0 0 0 0 0 | * * 4 * * * * *
.... oxoo3ooox&#xr ♦ 1 3 1 3 | 3 3  3  3  6 0  3 3 |  3 3  6 0 1 0  3  3 3  6 0  0  3 1 | 0 1 1  3 3 0 0 1 0  0  3  3 0 0 0 1 0 | * * * 4 * * * *
.ox.3.xo.3.oo.&#x  ♦ 0 6 4 0 | 0 0 12 12  0 6  0 0 |  0 0  0 4 4 6 12  0 0  0 4  0  0 0 | 0 0 0  0 0 1 4 4 0  0  0  0 1 0 0 0 0 | * * * * 1 * * *
.oxo3.xoo ....&#x  ♦ 0 3 3 1 | 0 0  3  6  3 3  3 0 |  0 0  0 1 0 3  3  3 0  6 1  3  0 0 | 0 0 0  0 0 0 1 0 1  3  3  0 0 1 0 0 0 | * * * * * 4 * *
.oxo .... .oox&#x  ♦ 0 1 2 2 | 0 0  0  2  2 1  4 1 |  0 0  0 0 0 1  0  0 1  4 0  2  2 0 | 0 0 0  0 0 0 0 0 0  2  0  2 0 0 1 0 0 | * * * * * * 6 *
..xo3..oo3..ox&#x  ♦ 0 0 4 4 | 0 0  0  0  0 6 12 6 |  0 0  0 0 0 0  0  0 0  0 4 12 12 4 | 0 0 0  0 0 0 0 0 0  0  0  0 1 4 6 4 1 | * * * * * * * 1
```