Acronym squapt Name (line-)square-antiprismatic tegum,square-antiprismatic bipyramid,square-pyramidal antiprism Lace cityin approx. ASCII-art ```o4o x4o o4x o4o ``` Dihedral angles at {3} between tet and tet (within squappy):   arccos[(1-3 sqrt(2))/4] = 144.160482° at {3} between squippy and tet (within squappy):   arccos[(2-3 sqrt(2))/4] = 124.101465° at {4} between squippy and squippy (across rim):   arccos[-(2-sqrt(2))/2] = 107.031248° at {3} between tet and tet (across rim):   arccos[(3 sqrt(2)-4)/8] = 88.261948° Confer more general: n-apt   related segmentochora: squappy   general polytopal classes: bistratic lace towers

Incidence matrix according to Dynkin symbol

```pt || pseudo squap || pt   → both heights = sqrt[(4-sqrt(2))/8] = 0.568527

1 * * | 8 0 0 0 | 8 8 0 0 0 0 | 2 8 0 0  verf: squap
* 8 * | 1 2 2 1 | 2 2 1 3 2 2 | 1 3 1 3
* * 1 | 0 0 0 8 | 0 0 0 0 8 8 | 0 0 2 8  verf: squap
------+---------+-------------+--------
1 1 0 | 8 * * * | 2 2 0 0 0 0 | 1 3 0 0
0 2 0 | * 8 * * | 1 0 1 1 1 0 | 1 1 1 1
0 2 0 | * * 8 * | 0 1 0 2 0 1 | 0 2 0 2
0 1 1 | * * * 8 | 0 0 0 0 2 2 | 0 0 1 3
------+---------+-------------+--------
1 2 0 | 2 1 0 0 | 8 * * * * * | 1 1 0 0
1 2 0 | 2 0 1 0 | * 8 * * * * | 0 2 0 0
0 4 0 | 0 4 0 0 | * * 2 * * * | 1 0 1 0  {4}
0 3 0 | 0 1 2 0 | * * * 8 * * | 0 1 0 1
0 2 1 | 0 1 0 2 | * * * * 8 * | 0 0 1 1
0 2 1 | 0 0 1 2 | * * * * * 8 | 0 0 0 2
------+---------+-------------+--------
1 4 0 | 4 4 0 0 | 4 0 1 0 0 0 | 2 * * *  squippy
1 3 0 | 3 1 2 0 | 1 2 0 1 0 0 | * 8 * *  tet
0 4 1 | 0 4 0 4 | 0 0 1 0 4 0 | * * 2 *  squippy
0 3 1 | 0 1 2 3 | 0 0 0 1 1 2 | * * * 8  tet
```

```ox4oo&#x || oo4xo&#x   → height = sqrt[sqrt(8)-1]/2 = 0.676097
squippy || inv gyro squippy

o.4o.      .. ..    | 1 * * * ♦ 4 4 0 0 0 0 0 | 4 4 8 0 0 0 0 0 0 0 | 1 1 4 4 0 0 0 0
.o4.o      .. ..    | * 4 * * | 1 0 2 2 1 0 0 | 2 0 2 1 2 2 1 2 0 0 | 1 0 2 1 1 2 1 0
.. ..      o.4o.    | * * 4 * | 0 1 0 2 0 2 1 | 0 2 2 0 1 0 2 2 1 2 | 0 1 1 2 0 1 2 1
.. ..      .o4.o    | * * * 1 ♦ 0 0 0 0 4 0 4 | 0 0 0 0 0 4 0 8 0 4 | 0 0 0 0 1 4 4 1
--------------------+---------+---------------+---------------------+----------------
oo4oo&#x   .. ..    | 1 1 0 0 | 4 * * * * * * | 2 0 2 0 0 0 0 0 0 0 | 1 0 2 1 0 0 0 0
o.4o.    || o.4o.    | 1 0 1 0 | * 4 * * * * * | 0 2 2 0 0 0 0 0 0 0 | 0 1 1 2 0 0 0 0
.x ..      .. ..    | 0 2 0 0 | * * 4 * * * * | 1 0 0 1 1 1 0 0 0 0 | 1 0 1 0 1 1 0 0
.o4.o    || o.4o.    | 0 1 1 0 | * * * 8 * * * | 0 0 1 0 1 0 1 1 0 0 | 0 0 1 1 0 1 1 0
.o4.o    || .o4.o    | 0 1 0 1 | * * * * 4 * * | 0 0 0 0 0 2 0 2 0 0 | 0 0 0 0 1 2 1 0
.. ..      .. x.    | 0 0 2 0 | * * * * * 4 * | 0 1 0 0 0 0 1 0 0 2 | 0 1 0 1 0 0 1 1
.. ..      oo4oo&#x | 0 0 1 1 | * * * * * * 4 | 0 0 0 0 0 0 0 2 1 1 | 0 0 0 0 0 1 2 1
--------------------+---------+---------------+---------------------+----------------
ox ..&#x   .. ..    | 1 2 0 0 | 2 0 1 0 0 0 0 | 4 * * * * * * * * * | 1 0 1 0 0 0 0 0
o.4o.    || .. x.    | 1 0 2 0 | 0 2 0 0 0 1 0 | * 4 * * * * * * * * | 0 1 0 1 0 0 0 0
oo4oo&#x || o.4o.    | 1 1 1 0 | 1 1 0 1 0 0 0 | * * 8 * * * * * * * | 0 0 1 1 0 0 0 0
.x4.o      .. ..    | 0 4 0 0 | 0 0 4 0 0 0 0 | * * * 1 * * * * * * | 1 0 0 0 1 0 0 0
.x ..    || o.4o.    | 0 2 1 0 | 0 0 1 2 0 0 0 | * * * * 4 * * * * * | 0 0 1 0 0 1 0 0
.x ..    || .o4.o    | 0 2 0 1 | 0 0 1 0 2 0 0 | * * * * * 4 * * * * | 0 0 0 0 1 1 0 0
.o4.o    || .. x.    | 0 1 2 0 | 0 0 0 2 0 1 0 | * * * * * * 4 * * * | 0 0 0 1 0 0 1 0
.o4.o    || oo4oo&#x | 0 1 1 1 | 0 0 0 1 1 0 1 | * * * * * * * 8 * * | 0 0 0 0 0 1 1 0
.. ..      o.4x.    | 0 0 4 0 | 0 0 0 0 0 0 4 | * * * * * * * * 1 * | 0 1 0 0 0 0 0 1
.. ..      .. xo&#x | 0 0 2 1 | 0 0 0 0 0 2 1 | * * * * * * * * * 4 | 0 0 0 0 0 0 1 1
--------------------+---------+---------------+---------------------+----------------
ox4oo&#x   .. ..    ♦ 1 4 0 0 | 4 0 4 0 0 0 0 | 4 0 0 1 0 0 0 0 0 0 | 1 * * * * * * *
o.4o.    || o.4x.    ♦ 1 0 4 0 | 0 4 0 0 0 4 0 | 0 4 0 0 0 0 0 0 1 0 | * 1 * * * * * *
ox ..&#x || o.4o.    ♦ 1 2 1 0 | 2 1 1 2 0 0 0 | 1 0 2 0 1 0 0 0 0 0 | * * 4 * * * * *
oo4oo&#x || .. x.    ♦ 1 1 2 0 | 1 2 0 2 0 1 0 | 0 1 2 0 0 0 1 0 0 0 | * * * 4 * * * *
.x4.o    || .o4.o    ♦ 0 4 0 1 | 0 0 4 0 4 0 0 | 0 0 0 1 0 4 0 0 0 0 | * * * * 1 * * *
.x ..    || oo4oo&#x ♦ 0 2 1 1 | 0 0 1 2 2 0 1 | 0 0 0 0 1 1 0 2 0 0 | * * * * * 4 * *
. o4.o   || .. xo&#x ♦ 0 1 2 1 | 0 0 0 2 1 1 2 | 0 0 0 0 0 0 1 2 0 1 | * * * * * * 4 *
..  ..     oo4xo&#x ♦ 0 0 4 1 | 0 0 0 0 0 4 4 | 0 0 0 0 0 0 0 0 1 4 | * * * * * * * 1
```

```oxoo4ooox&#xr     → all heights = 1/sqrt(2) = 0.707107
(pt || pseudo ({4} || dual {4}) || pt)

o...4o...    | 1 * * * ♦ 4 4 0 0 0 0 0 | 4 8 4 0 0 0 0 0 0 0 | 1 1 4 4 0 0 0 0
.o..4.o..    | * 4 * * | 1 0 2 1 2 0 0 | 2 2 0 1 2 2 1 2 0 0 | 1 0 2 1 1 2 1 0
..o.4..o.    | * * 1 * ♦ 0 0 0 4 0 4 0 | 0 0 0 0 4 0 0 8 4 0 | 0 0 0 0 1 4 4 1
...o4...o    | * * * 4 | 0 1 0 0 2 1 2 | 0 2 2 0 0 1 2 2 2 1 | 0 1 1 2 0 1 2 1
-------------+---------+---------------+---------------------+----------------
oo..4oo..&#x | 1 1 0 0 | 4 * * * * * * | 2 2 0 0 0 0 0 0 0 0 | 1 0 2 1 0 0 0 0
o..o4o..o&#x | 1 0 0 1 | * 4 * * * * * | 0 2 2 0 0 0 0 0 0 0 | 0 1 1 2 0 0 0 0
.x.. ....    | 0 2 0 0 | * * 4 * * * * | 1 0 0 1 1 1 0 0 0 0 | 1 0 1 0 1 1 0 0
.oo.4.oo.&#x | 0 1 1 0 | * * * 4 * * * | 0 0 0 0 2 0 0 2 0 0 | 0 0 0 0 1 2 1 0
.o.o4.o.o&#x | 0 1 0 1 | * * * * 8 * * | 0 1 0 0 0 1 1 1 0 0 | 0 0 1 1 0 1 1 0
..oo4..oo&#x | 0 0 1 1 | * * * * * 4 * | 0 0 0 0 0 0 0 2 2 0 | 0 0 0 0 0 1 2 1
.... ...x    | 0 0 0 2 | * * * * * * 4 | 0 0 1 0 0 0 1 0 1 1 | 0 1 0 1 0 0 1 1
-------------+---------+---------------+---------------------+----------------
ox.. ....&#x | 1 2 0 0 | 2 0 1 0 0 0 0 | 4 * * * * * * * * * | 1 0 1 0 0 0 0 0
oo.o4oo.o&#x | 1 1 0 1 | 1 1 0 0 1 0 0 | * 8 * * * * * * * * | 0 0 1 1 0 0 0 0
.... o..x&#x | 1 0 0 2 | 0 2 0 0 0 0 1 | * * 4 * * * * * * * | 0 1 0 1 0 0 0 0
.x..4.o..    | 0 4 0 0 | 0 0 4 0 0 0 0 | * * * 1 * * * * * * | 1 0 0 0 1 0 0 0
.xo. ....&#x | 0 2 1 0 | 0 0 1 2 0 0 0 | * * * * 4 * * * * * | 0 0 0 0 1 1 0 0
.x.o ....&#x | 0 2 0 1 | 0 0 1 0 2 0 0 | * * * * * 4 * * * * | 0 0 1 0 0 1 0 0
.... .o.x&#x | 0 1 0 2 | 0 0 0 0 2 0 1 | * * * * * * 4 * * * | 0 0 0 1 0 0 1 0
.ooo4.ooo&#x | 0 1 1 1 | 0 0 0 1 1 1 0 | * * * * * * * 8 * * | 0 0 0 0 0 1 1 0
.... ..ox&#x | 0 0 1 2 | 0 0 0 0 0 2 1 | * * * * * * * * 4 * | 0 0 0 0 0 0 1 1
...o4...x    | 0 0 0 4 | 0 0 0 0 0 0 4 | * * * * * * * * * 1 | 0 1 0 0 0 0 0 1
-------------+---------+---------------+---------------------+----------------
ox..4oo..&#x ♦ 1 4 0 0 | 4 0 4 0 0 0 0 | 4 0 0 1 0 0 0 0 0 0 | 1 * * * * * * *
o..o4o..x&#x ♦ 1 0 0 4 | 0 4 0 0 0 0 4 | 0 0 4 0 0 0 0 0 0 1 | * 1 * * * * * *
ox.o ....&#x ♦ 1 2 0 1 | 2 1 1 0 2 0 0 | 1 2 0 0 0 1 0 0 0 0 | * * 4 * * * * *
.... oo.x&#x ♦ 1 1 0 2 | 1 2 0 0 2 0 1 | 0 2 1 0 0 0 1 0 0 0 | * * * 4 * * * *
.xo.4.oo.&#x ♦ 0 4 1 0 | 0 0 4 4 0 0 0 | 0 0 0 1 4 0 0 0 0 0 | * * * * 1 * * *
.xoo ....&#x ♦ 0 2 1 1 | 0 0 1 2 2 1 0 | 0 0 0 0 1 1 0 2 0 0 | * * * * * 4 * *
.... .oox&#x ♦ 0 1 1 2 | 0 0 0 1 2 2 1 | 0 0 0 0 0 0 1 2 1 0 | * * * * * * 4 *
..oo4..ox&#x ♦ 0 0 1 4 | 0 0 0 0 0 4 4 | 0 0 0 0 0 0 0 0 4 1 | * * * * * * * 1
```

```xoo4oox oyo&#xt   → both heights = 1/sqrt(sqrt(32)) = 0.420448
({4} || perp y-line || dual {4})   y = sqrt[(4-sqrt(2))/2] = 1.137055

o..4o.. o..     | 4 * * | 2 2 2 0 0 | 1 4 2 1  4 0 0 | 2 4 2 0
.o.4.o. .o.     | * 2 * ♦ 0 4 0 4 0 | 0 4 0 0  8 4 0 | 1 4 4 1
..o4..o ..o     | * * 4 | 0 0 2 2 2 | 0 0 1 2  4 4 1 | 0 2 4 2
----------------+-------+-----------+----------------+--------
x.. ... ...     | 2 0 0 | 4 * * * * | 1 2 1 0  0 0 0 | 2 2 0 0
oo.4oo. oo.&#x  | 1 1 0 | * 8 * * * | 0 2 0 0  2 0 0 | 1 2 1 0
o.o4o.o o.o&#x  | 1 0 1 | * * 8 * * | 0 0 1 1  2 0 0 | 0 2 2 0
.oo4.oo .oo&#x  | 0 1 1 | * * * 8 * | 0 0 0 0  2 2 0 | 0 1 2 1
... ..x ...     | 0 0 2 | * * * * 4 | 0 0 0 1  0 2 1 | 0 0 2 2
----------------+-------+-----------+----------------+--------
x..4o.. ...     | 4 0 0 | 4 0 0 0 0 | 1 * * *  * * * | 2 0 0 0
xo. ... ...&#x  | 2 1 0 | 1 2 0 0 0 | * 8 * *  * * * | 1 1 0 0
x.o ... ...&#x  | 2 0 1 | 1 0 2 0 0 | * * 4 *  * * * | 0 2 0 0
... o.x ...&#x  | 1 0 2 | 0 0 2 0 1 | * * * 4  * * * | 0 0 2 0
ooo4ooo ooo&#xt | 1 1 1 | 0 1 1 1 0 | * * * * 16 * * | 0 1 1 0
... .ox ...&#x  | 0 1 2 | 0 0 0 2 1 | * * * *  * 8 * | 0 0 1 1
..o4..x ...     | 0 0 4 | 0 0 0 0 4 | * * * *  * * 1 | 0 0 0 2
----------------+-------+-----------+----------------+--------
xo.4oo. ...&#x  ♦ 4 1 0 | 4 4 0 0 0 | 1 4 0 0  0 0 0 | 2 * * *
xoo ... ...&#xt ♦ 2 1 1 | 1 2 2 1 0 | 0 1 1 0  2 0 0 | * 8 * *
... oox ...&#xt ♦ 1 1 2 | 0 1 2 2 1 | 0 0 0 1  2 1 0 | * * 8 *
.oo4.ox ...&#x  ♦ 0 1 4 | 0 0 0 4 4 | 0 0 0 0  0 4 1 | * * * 2
```

```yo os2os8os&#zx   → height = 0
(tegum sum of y-line and perp squap)   y = sqrt[(4-sqrt(2))/2] = 1.137055
(tegum product of y-line with squap)

o. demi( o.2o.8o. )    | 2 * ♦  8 0 0 |  8  8 0 0 | 2  8
.o demi( .o2.o8.o )    | * 8 |  2 2 2 |  4  4 1 3 | 2  6
-----------------------+-----+--------+-----------+-----
oo demi( oo2oo8oo )&#x | 1 1 | 16 * * |  2  2 0 0 | 1  3
..       .s2.s ..      | 0 2 |  * 8 * |  2  0 0 2 | 0  4
.. sefa( .. .s8.s )    | 0 2 |  * * 8 |  0  2 1 1 | 2  2
-----------------------+-----+--------+-----------+-----
oo       os2os ..  &#x | 1 2 |  2 1 0 | 16  * * * | 0  2
oo sefa( .. os8os )&#x | 1 2 |  2 0 1 |  * 16 * * | 1  1
..       .. .s8.s      | 0 4 |  0 0 4 |  *  * 2 * | 2  0
.. sefa( .s2.s8.s )    | 0 3 |  0 2 1 |  *  * * 8 | 0  2
-----------------------+-----+--------+-----------+-----
oo       .. os8os  &#x ♦ 1 4 |  4 0 4 |  0  4 1 0 | 4  *
oo sefa( os2os8os )&#x ♦ 1 3 |  3 2 1 |  2  1 0 1 | * 16
```