Acronym bidhin (alt.: octahex, tetalrap, tetaltet octtric)
Name bidiminished hemipenteract,
oct || hex,
tet || inverted rap,
(tet,dual tet,oct)-trigonic
Lace city
in approx. ASCII-art
```O   t      T = x3o3o (tet)
O = o3x3o (oct)
T        t = o3o3x (dual tet)
```
Lace hyper city
in approx. ASCII-art
 ``` o o x x o o ``` ``` x o o x o x x o ```
(oct || alternate hex)
Coordinates
1. (1/sqrt(2), 0, 0, 0, 1/sqrt(8))   & all permutations and changes of sign in the first 3 coordinates only
(top oct)
2. (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), -1/sqrt(8))   & any even change of signs in the first 4 coordinates only
(bottom hex)
Confer
uniform relative:
hin
segmentotera:
dihin   tedhin   squasquasc   rappy
general polytopal classes:
segmentotera

The hemipenteract (hin) could be given as hex || gyro hex. Here the top-hex (itself being pt || pseudo oct || pt) has to be bidiminished at opposite vertices as hex - 2x octpy. Simultanuously the bottom-hex only gets marginally rasped at opposite tet (as hex equally is: tet || dual tet) according to hex - 2x tet. The sefa clearly is the vertex figure of hin, i.e. rap. In other words, one chops off 2 rappies.

Aside: There is also a polyteron which likewise uses O, T, and t (cf. Lace city), but has a not as sharp angle at O: the dot. (That relation kind of is like cube to trip.) As dot had edges connecting the opposite pair of octs, these clearly are missing in here. Conversely, the structure of hex, except of the pair of opposite bases (tets), were missing in dot but occurs here for sure.

Incidence matrix according to Dynkin symbol

```xoo3oxo3oox&#x   → heights(1,2) = height(2,3) = sqrt(5/8) = 0.790569
height(1,3) = 1/sqrt(2) = 0.707107
(oct || (tet || dual tet)

o..3o..3o..    | 4 * * ♦ 3  3  3  0  0 0 | 3 3  3  6  3  6 0 0  0 0 0 | 1 3 1 3 6 1  6  3  3 0 0 0 0 | 1 1 3 3 1 0
.o.3.o.3.o.    | * 6 * ♦ 0  2  0  4  2 0 | 0 1  4  0  0  4 2 2  4 1 0 | 0 2 2 0 0 0  2  4  2 1 2 2 0 | 1 0 2 1 2 1
..o3..o3..o    | * * 4 ♦ 0  0  3  0  3 3 | 0 0  0  3  6  6 0 0  3 3 3 | 0 0 0 1 6 3  3  3  6 0 1 3 1 | 0 1 1 3 3 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x.. ... ...    | 2 0 0 | 6  *  *  *  * * | 2 1  0  2  0  0 0 0  0 0 0 | 1 2 0 2 1 0  2  0  0 0 0 0 0 | 1 1 2 1 0 0
oo.3oo.3oo.&#x | 1 1 0 | * 12  *  *  * * | 0 1  2  0  0  2 0 0  0 0 0 | 0 2 1 0 0 0  2  2  1 0 0 0 0 | 1 0 2 1 1 0
o.o3o.o3o.o&#x | 1 0 1 | *  * 12  *  * * ♦ 0 0  0  2  2  2 0 0  0 0 0 | 0 0 0 1 2 1  2  1  2 0 0 0 0 | 0 1 1 2 1 0
... .x. ...    | 0 2 0 | *  *  * 12  * * | 0 0  1  0  0  0 1 1  1 0 0 | 0 1 1 0 0 0  0  1  0 1 1 1 0 | 1 0 1 0 1 1
.oo3.oo3.oo&#x | 0 1 1 | *  *  *  * 12 * | 0 0  0  0  0  2 0 0  2 1 0 | 0 0 0 0 0 0  1  2  2 0 1 2 0 | 0 0 1 1 2 1
... ... ..x    | 0 0 2 | *  *  *  *  * 6 | 0 0  0  0  2  0 0 0  0 1 2 | 0 0 0 0 1 2  0  0  2 0 0 2 1 | 0 1 0 1 2 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x..3o.. ...    | 3 0 0 | 3  0  0  0  0 0 | 4 *  *  *  *  * * *  * * * | 1 1 0 1 0 0  0  0  0 0 0 0 0 | 1 1 1 0 0 0
xo. ... ...&#x | 2 1 0 | 1  2  0  0  0 0 | * 6  *  *  *  * * *  * * * | 0 2 0 0 0 0  2  0  0 0 0 0 0 | 1 0 2 1 0 0
... ox. ...&#x | 1 2 0 | 0  2  0  1  0 0 | * * 12  *  *  * * *  * * * | 0 1 1 0 0 0  0  1  0 0 0 0 0 | 1 0 1 0 1 0
x.o ... ...&#x | 2 0 1 | 1  0  2  0  0 0 | * *  * 12  *  * * *  * * * | 0 0 0 1 1 0  1  0  0 0 0 0 0 | 0 1 1 1 0 0
... ... o.x&#x | 1 0 2 | 0  0  2  0  0 1 | * *  *  * 12  * * *  * * * | 0 0 0 0 1 1  0  0  1 0 0 0 0 | 0 1 0 1 1 0
ooo3ooo3ooo&#x | 1 1 1 | 0  1  1  0  1 0 | * *  *  *  * 24 * *  * * * | 0 0 0 0 0 0  1  1  1 0 0 0 0 | 0 0 1 1 1 0
.o.3.x. ...    | 0 3 0 | 0  0  0  3  0 0 | * *  *  *  *  * 4 *  * * * | 0 1 0 0 0 0  0  0  0 1 1 0 0 | 1 0 1 0 0 1
... .x.3.o.    | 0 3 0 | 0  0  0  3  0 0 | * *  *  *  *  * * 4  * * * | 0 0 1 0 0 0  0  0  0 1 0 1 0 | 1 0 0 0 1 1
... .xo ...&#x | 0 2 1 | 0  0  0  1  2 0 | * *  *  *  *  * * * 12 * * | 0 0 0 0 0 0  0  1  0 0 1 1 0 | 0 0 1 0 1 1
... ... .ox&#x | 0 1 2 | 0  0  0  0  2 1 | * *  *  *  *  * * *  * 6 * | 0 0 0 0 0 0  0  0  2 0 0 2 0 | 0 0 0 1 2 1
... ..o3..x    | 0 0 3 | 0  0  0  0  0 3 | * *  *  *  *  * * *  * * 4 | 0 0 0 0 0 1  0  0  0 0 0 1 1 | 0 1 0 0 1 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x..3o..3o..    ♦ 4 0 0 | 6  0  0  0  0 0 | 4 0  0  0  0  0 0 0  0 0 0 | 1 * * * * *  *  *  * * * * * | 1 1 0 0 0 0
xo.3ox. ...&#x ♦ 3 3 0 | 3  6  0  3  0 0 | 1 3  3  0  0  0 1 0  0 0 0 | * 4 * * * *  *  *  * * * * * | 1 0 1 0 0 0
... ox.3oo.&#x ♦ 1 3 0 | 0  3  0  3  0 0 | 0 0  3  0  0  0 0 1  0 0 0 | * * 4 * * *  *  *  * * * * * | 1 0 0 0 1 0
x.o3o.o ...&#x ♦ 3 0 1 | 3  0  3  0  0 0 | 1 0  0  3  0  0 0 0  0 0 0 | * * * 4 * *  *  *  * * * * * | 0 1 1 0 0 0
x.o ... o.x&#x ♦ 2 0 2 | 1  0  4  0  0 1 | 0 0  0  2  2  0 0 0  0 0 0 | * * * * 6 *  *  *  * * * * * | 0 1 0 1 0 0
... o.o3o.x&#x ♦ 1 0 3 | 0  0  3  0  0 3 | 0 0  0  0  3  0 0 0  0 0 1 | * * * * * 4  *  *  * * * * * | 0 1 0 0 1 0
xoo ... ...&#x ♦ 2 1 1 | 1  2  2  0  1 0 | 0 1  0  1  0  2 0 0  0 0 0 | * * * * * * 12  *  * * * * * | 0 0 1 1 0 0
... oxo ...&#x ♦ 1 2 1 | 0  2  1  1  2 0 | 0 0  1  0  0  2 0 0  1 0 0 | * * * * * *  * 12  * * * * * | 0 0 1 0 1 0
... ... oox&#x ♦ 1 1 2 | 0  1  2  0  2 1 | 0 0  0  0  1  2 0 0  0 1 0 | * * * * * *  *  * 12 * * * * | 0 0 0 1 1 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 | * * * * * *  *  *  * 1 * * * | 1 0 0 0 0 1
.oo3.xo ...&#x ♦ 0 3 1 | 0  0  0  3  3 0 | 0 0  0  0  0  0 1 0  3 0 0 | * * * * * *  *  *  * * 4 * * | 0 0 1 0 0 1
... .xo3.ox&#x ♦ 0 3 3 | 0  0  0  3  6 3 | 0 0  0  0  0  0 0 1  3 3 1 | * * * * * *  *  *  * * * 4 * | 0 0 0 0 1 1
..o3..o3..x    ♦ 0 0 4 | 0  0  0  0  0 6 | 0 0  0  0  0  0 0 0  0 0 4 | * * * * * *  *  *  * * * * 1 | 0 1 0 0 0 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
xo.3ox.3oo.&#x ♦ 4 6 0 | 6 12  0 12  0 0 | 4 6 12  0  0  0 4 4  0 0 0 | 1 4 4 0 0 0  0  0  0 1 0 0 0 | 1 * * * * *
x.o3o.o3o.x&#x ♦ 4 0 4 | 6  0 12  0  0 6 | 4 0  0 12 12  0 0 0  0 0 4 | 1 0 0 4 6 4  0  0  0 0 0 0 1 | * 1 * * * *
xoo3oxo ...&#x ♦ 3 3 1 | 3  6  3  3  3 0 | 1 3  3  3  0  6 1 0  3 0 0 | 0 1 0 1 0 0  3  3  0 0 1 0 0 | * * 4 * * *
xoo ... oox&#x ♦ 2 1 2 | 1  2  4  0  2 1 | 0 1  0  2  2  4 0 0  0 1 0 | 0 0 0 0 1 0  2  0  2 0 0 0 0 | * * * 6 * *
... oxo3oox&#x ♦ 1 3 3 | 0  3  3  3  6 3 | 0 0  3  0  3  6 0 1  3 3 1 | 0 0 1 0 0 1  0  3  3 0 0 1 0 | * * * * 4 *
.oo3.xo3.ox&#x ♦ 0 6 4 | 0  0  0 12 12 6 | 0 0  0  0  0  0 4 4 12 6 4 | 0 0 0 0 0 0  0  0  0 1 4 4 1 | * * * * * 1
```

```oct || hex   → height = 1/sqrt(2) = 0.707107

o..3o..3o..     & | 8 * ♦  3  3  3  0 | 3  9  3  6  3 0 | 1 4 3 3  9  3 1 0 | 1 1 4 3
.o.3.o.3.o.       | * 6 ♦  0  0  4  4 | 0  0  2  4  8 4 | 0 0 0 4  4  4 4 1 | 0 2 4 1
------------------+-----+-------------+-----------------+-------------------+--------
x.. ... ...     & | 2 0 | 12  *  *  * | 2  2  1  0  0 0 | 1 2 1 2  2  0 0 0 | 1 1 2 1
o.o3o.o3o.o&#x    | 2 0 |  * 12  *  * ♦ 0  4  0  2  0 0 | 0 2 2 0  4  1 0 0 | 1 0 2 2
oo.3oo.3oo.&#x  & | 1 1 |  *  * 24  * | 0  0  1  2  2 0 | 0 0 0 2  3  2 1 0 | 0 1 3 1
... .x. ...       | 0 2 |  *  *  * 12 | 0  0  0  0  2 2 | 0 0 0 2  0  1 2 1 | 0 2 2 0
------------------+-----+-------------+-----------------+-------------------+--------
x..3o.. ...     & | 3 0 |  3  0  0  0 | 8  *  *  *  * * | 1 1 0 1  0  0 0 0 | 1 1 1 0
x.o ... ...&#x  & | 3 0 |  1  2  0  0 | * 24  *  *  * * | 0 1 1 0  1  0 0 0 | 1 0 1 1
xo. ... ...&#x  & | 2 1 |  1  0  2  0 | *  * 12  *  * * | 0 0 0 2  2  0 0 0 | 0 1 2 1
ooo3ooo3ooo&#x    | 2 1 |  0  1  2  0 | *  *  * 24  * * | 0 0 0 0  2  1 0 0 | 0 0 2 1
... ox. ...&#x  & | 1 2 |  0  0  2  1 | *  *  *  * 24 * | 0 0 0 1  0  1 1 0 | 0 1 2 0
.o.3.x. ...     & | 0 3 |  0  0  0  3 | *  *  *  *  * 8 | 0 0 0 1  0  0 1 1 | 0 2 1 0
------------------+-----+-------------+-----------------+-------------------+--------
x..3o..3o..     & ♦ 4 0 |  6  0  0  0 | 4  0  0  0  0 0 | 2 * * *  *  * * * | 1 1 0 0
x.o3o.o ...&#x  & ♦ 4 0 |  3  3  0  0 | 1  3  0  0  0 0 | * 8 * *  *  * * * | 1 0 1 0
x.o ... o.x&#x    ♦ 4 0 |  2  4  0  0 | 0  4  0  0  0 0 | * * 6 *  *  * * * | 1 0 0 1
xo.3ox. ...&#x  & ♦ 3 3 |  3  0  6  3 | 1  0  3  0  3 1 | * * * 8  *  * * * | 0 1 1 0
xoo ... ...&#x  & ♦ 3 1 |  1  2  3  0 | 0  1  1  2  0 0 | * * * * 24  * * * | 0 0 1 1
... oxo ...&#x    ♦ 2 2 |  0  1  4  1 | 0  0  0  2  2 0 | * * * *  * 12 * * | 0 0 2 0
... ox.3oo.&#x  & ♦ 1 3 |  0  0  3  3 | 0  0  0  0  3 1 | * * * *  *  * 8 * | 0 1 1 0
.o.3.x.3.o.       ♦ 0 6 |  0  0  0 12 | 0  0  0  0  0 8 | * * * *  *  * * 1 | 0 2 0 0
------------------+-----+-------------+-----------------+-------------------+--------
x.o3o.o3o.x&#x    ♦ 8 0 | 12 12  0  0 | 8 24  0  0  0 0 | 2 8 6 0  0  0 0 0 | 1 * * *
xo.3ox.3oo.&#x  & ♦ 4 6 |  6  0 12 12 | 4  0  6  0 12 8 | 1 0 0 4  0  0 4 1 | * 2 * *
xoo3oxo ...&#x  & ♦ 4 3 |  3  3  9  3 | 1  3  3  6  6 1 | 0 1 0 1  3  3 1 0 | * * 8 *
xoo ... oox&#x    ♦ 4 1 |  2  4  4  0 | 0  4  2  4  0 0 | 0 0 1 0  4  0 0 0 | * * * 6
```

```tet || inv rap   → height = sqrt(2/5) = 0.632456

o..3o..3o..    | 4 * * ♦ 3  3  3  0  0 0 | 3 3  3  6  3  6 0 0  0 0 0 | 1 3 1 3 6 1  6  3  3 0 0 0 0 | 1 1 3 3 1 0
.o.3.o.3.o.    | * 6 * ♦ 0  2  0  4  2 0 | 0 1  4  0  0  4 2 2  4 1 0 | 0 2 2 0 0 0  2  4  2 1 2 2 0 | 1 0 2 1 2 1
..o3..o3..o    | * * 4 ♦ 0  0  3  0  3 3 | 0 0  0  3  6  6 0 0  3 3 3 | 0 0 0 1 6 3  3  3  6 0 1 3 1 | 0 1 1 3 3 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x.. ... ...    | 2 0 0 | 6  *  *  *  * * | 2 1  0  2  0  0 0 0  0 0 0 | 1 2 0 2 1 0  2  0  0 0 0 0 0 | 1 1 2 1 0 0
oo.3oo.3oo.&#x | 1 1 0 | * 12  *  *  * * | 0 1  2  0  0  2 0 0  0 0 0 | 0 2 1 0 0 0  2  2  1 0 0 0 0 | 1 0 2 1 1 0
o.o3o.o3o.o&#x | 1 0 1 | *  * 12  *  * * ♦ 0 0  0  2  2  2 0 0  0 0 0 | 0 0 0 1 2 1  2  1  2 0 0 0 0 | 0 1 1 2 1 0
... .x. ...    | 0 2 0 | *  *  * 12  * * | 0 0  1  0  0  0 1 1  1 0 0 | 0 1 1 0 0 0  0  1  0 1 1 1 0 | 1 0 1 0 1 1
.oo3.oo3.oo&#x | 0 1 1 | *  *  *  * 12 * | 0 0  0  0  0  2 0 0  2 1 0 | 0 0 0 0 0 0  1  2  2 0 1 2 0 | 0 0 1 1 2 1
... ... ..x    | 0 0 2 | *  *  *  *  * 6 | 0 0  0  0  2  0 0 0  0 1 2 | 0 0 0 0 1 2  0  0  2 0 0 2 1 | 0 1 0 1 2 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x..3o.. ...    | 3 0 0 | 3  0  0  0  0 0 | 4 *  *  *  *  * * *  * * * | 1 1 0 1 0 0  0  0  0 0 0 0 0 | 1 1 1 0 0 0
xo. ... ...&#x | 2 1 0 | 1  2  0  0  0 0 | * 6  *  *  *  * * *  * * * | 0 2 0 0 0 0  2  0  0 0 0 0 0 | 1 0 2 1 0 0
... ox. ...&#x | 1 2 0 | 0  2  0  1  0 0 | * * 12  *  *  * * *  * * * | 0 1 1 0 0 0  0  1  0 0 0 0 0 | 1 0 1 0 1 0
x.o ... ...&#x | 2 0 1 | 1  0  2  0  0 0 | * *  * 12  *  * * *  * * * | 0 0 0 1 1 0  1  0  0 0 0 0 0 | 0 1 1 1 0 0
... ... o.x&#x | 1 0 2 | 0  0  2  0  0 1 | * *  *  * 12  * * *  * * * | 0 0 0 0 1 1  0  0  1 0 0 0 0 | 0 1 0 1 1 0
ooo3ooo3ooo&#x | 1 1 1 | 0  1  1  0  1 0 | * *  *  *  * 24 * *  * * * | 0 0 0 0 0 0  1  1  1 0 0 0 0 | 0 0 1 1 1 0
.o.3.x. ...    | 0 3 0 | 0  0  0  3  0 0 | * *  *  *  *  * 4 *  * * * | 0 1 0 0 0 0  0  0  0 1 1 0 0 | 1 0 1 0 0 1
... .x.3.o.    | 0 3 0 | 0  0  0  3  0 0 | * *  *  *  *  * * 4  * * * | 0 0 1 0 0 0  0  0  0 1 0 1 0 | 1 0 0 0 1 1
... .xo ...&#x | 0 2 1 | 0  0  0  1  2 0 | * *  *  *  *  * * * 12 * * | 0 0 0 0 0 0  0  1  0 0 1 1 0 | 0 0 1 0 1 1
... ... .ox&#x | 0 1 2 | 0  0  0  0  2 1 | * *  *  *  *  * * *  * 6 * | 0 0 0 0 0 0  0  0  2 0 0 2 0 | 0 0 0 1 2 1
... ..o3..x    | 0 0 3 | 0  0  0  0  0 3 | * *  *  *  *  * * *  * * 4 | 0 0 0 0 0 1  0  0  0 0 0 1 1 | 0 1 0 0 1 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x..3o..3o..    ♦ 4 0 0 | 6  0  0  0  0 0 | 4 0  0  0  0  0 0 0  0 0 0 | 1 * * * * *  *  *  * * * * * | 1 1 0 0 0 0
xo.3ox. ...&#x ♦ 3 3 0 | 3  6  0  3  0 0 | 1 3  3  0  0  0 1 0  0 0 0 | * 4 * * * *  *  *  * * * * * | 1 0 1 0 0 0
... ox.3oo.&#x ♦ 1 3 0 | 0  3  0  3  0 0 | 0 0  3  0  0  0 0 1  0 0 0 | * * 4 * * *  *  *  * * * * * | 1 0 0 0 1 0
x.o3o.o ...&#x ♦ 3 0 1 | 3  0  3  0  0 0 | 1 0  0  3  0  0 0 0  0 0 0 | * * * 4 * *  *  *  * * * * * | 0 1 1 0 0 0
x.o ... o.x&#x ♦ 2 0 2 | 1  0  4  0  0 1 | 0 0  0  2  2  0 0 0  0 0 0 | * * * * 6 *  *  *  * * * * * | 0 1 0 1 0 0
... o.o3o.x&#x ♦ 1 0 3 | 0  0  3  0  0 3 | 0 0  0  0  3  0 0 0  0 0 1 | * * * * * 4  *  *  * * * * * | 0 1 0 0 1 0
xoo ... ...&#x ♦ 2 1 1 | 1  2  2  0  1 0 | 0 1  0  1  0  2 0 0  0 0 0 | * * * * * * 12  *  * * * * * | 0 0 1 1 0 0
... oxo ...&#x ♦ 1 2 1 | 0  2  1  1  2 0 | 0 0  1  0  0  2 0 0  1 0 0 | * * * * * *  * 12  * * * * * | 0 0 1 0 1 0
... ... oox&#x ♦ 1 1 2 | 0  1  2  0  2 1 | 0 0  0  0  1  2 0 0  0 1 0 | * * * * * *  *  * 12 * * * * | 0 0 0 1 1 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 | * * * * * *  *  *  * 1 * * * | 1 0 0 0 0 1
.oo3.xo ...&#x ♦ 0 3 1 | 0  0  0  3  3 0 | 0 0  0  0  0  0 1 0  3 0 0 | * * * * * *  *  *  * * 4 * * | 0 0 1 0 0 1
... .xo3.ox&#x ♦ 0 3 3 | 0  0  0  3  6 3 | 0 0  0  0  0  0 0 1  3 3 1 | * * * * * *  *  *  * * * 4 * | 0 0 0 0 1 1
..o3..o3..x    ♦ 0 0 4 | 0  0  0  0  0 6 | 0 0  0  0  0  0 0 0  0 0 4 | * * * * * *  *  *  * * * * 1 | 0 1 0 0 0 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
xo.3ox.3oo.&#x ♦ 4 6 0 | 6 12  0 12  0 0 | 4 6 12  0  0  0 4 4  0 0 0 | 1 4 4 0 0 0  0  0  0 1 0 0 0 | 1 * * * * *
x.o3o.o3o.x&#x ♦ 4 0 4 | 6  0 12  0  0 6 | 4 0  0 12 12  0 0 0  0 0 4 | 1 0 0 4 6 4  0  0  0 0 0 0 1 | * 1 * * * *
xoo3oxo ...&#x ♦ 3 3 1 | 3  6  3  3  3 0 | 1 3  3  3  0  6 1 0  3 0 0 | 0 1 0 1 0 0  3  3  0 0 1 0 0 | * * 4 * * *
xoo ... oox&#x ♦ 2 1 2 | 1  2  4  0  2 1 | 0 1  0  2  2  4 0 0  0 1 0 | 0 0 0 0 1 0  2  0  2 0 0 0 0 | * * * 6 * *
... oxo3oox&#x ♦ 1 3 3 | 0  3  3  3  6 3 | 0 0  3  0  3  6 0 1  3 3 1 | 0 0 1 0 0 1  0  3  3 0 0 1 0 | * * * * 4 *
.oo3.xo3.ox&#x ♦ 0 6 4 | 0  0  0 12 12 6 | 0 0  0  0  0  0 4 4 12 6 4 | 0 0 0 0 0 0  0  0  0 1 4 4 1 | * * * * * 1
```