Acronym n-apt Name (line-)n-antiprismatic tegum,n-antiprismatic bipyramid,n-pyramidal antiprism ` ©` Lace cityin approx. ASCII-art ``` x-n-o o-n-o o-n-o o-n-x ``` Especially tete (n=2)   hex (n=3)   squapt (n=4)   papt (n=5) Confer related segmentochora: n-appy

The lace city display shows that this polychoron can be dissected vertically into segmentochoric components: into 2 n-appies; thereby adding one n-antiprism as further facet each, which here occurs as internal pseudo facet only. In fact, the other way round, this polychoron well can be considered as an external blend of those 2 components.

Although the antiprismatic orientation shows that those are all monostratic, but except of the case n=3 those would not be segmentochora. This is because of the needed shift of these non-degenerate bases out of their circumcenter. Accordingly there will be no (full-dimensional) circumradius either.

Note the conceptual difference between this pyramid-antiprism and the antiprism-pyramid n-appy. In fact it happens that the latter is half of the former.

Incidence matrix according to Dynkin symbol

```ox-n-oo&#x || oo-n-xo&#x (2<n<6)   → height = ???
n-py || inv gyro n-py

o.-n-o.      ..   ..    | 1 * * * ♦ n n 0  0 0 0 0 | n n 2n 0 0 0 0  0 0 0 | 1 1 n n 0 0 0 0
.o-n-.o      ..   ..    | * n * * | 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.-n-o.    | * * n * | 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
..   ..      .o-n-.o    | * * * 1 ♦ 0 0 0  0 n 0 n | 0 0  0 0 0 n 0 2n 0 n | 0 0 0 0 1 n n 1
------------------------+---------+----------------+-----------------------+----------------
oo-n-oo&#x   ..   ..    | 1 1 0 0 | n * *  * * * * | 2 0  2 0 0 0 0  0 0 0 | 1 0 2 1 0 0 0 0
o.-n-o.    || o.-n-o.    | 1 0 1 0 | * n *  * * * * | 0 2  2 0 0 0 0  0 0 0 | 0 1 1 2 0 0 0 0
.x   ..      ..   ..    | 0 2 0 0 | * * n  * * * * | 1 0  0 1 1 1 0  0 0 0 | 1 0 1 0 1 1 0 0
.o-n-.o    || o.-n-o.    | 0 1 1 0 | * * * 2n * * * | 0 0  1 0 1 0 1  1 0 0 | 0 0 1 1 0 1 1 0
.o-n-.o    || .o-n-.o    | 0 1 0 1 | * * *  * n * * | 0 0  0 0 0 2 0  2 0 0 | 0 0 0 0 1 2 1 0
..   ..      ..   x.    | 0 0 2 0 | * * *  * * n * | 0 1  0 0 0 0 1  0 0 2 | 0 1 0 1 0 0 1 1
..   ..      oo-n-oo&#x | 0 0 1 1 | * * *  * * * n | 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 | n *  * * * * *  * * * | 1 0 1 0 0 0 0 0
o.-n-o.    || ..   x.    | 1 0 2 0 | 0 2 0  0 0 1 0 | * n  * * * * *  * * * | 0 1 0 1 0 0 0 0
oo-n-oo&#x || o.-n-o.    | 1 1 1 0 | 1 1 0  1 0 0 0 | * * 2n * * * *  * * * | 0 0 1 1 0 0 0 0
.x-n-.o      ..   ..    | 0 n 0 0 | 0 0 n  0 0 0 0 | * *  * 1 * * *  * * * | 1 0 0 0 1 0 0 0
.x   ..    || o.-n-o.    | 0 2 1 0 | 0 0 1  2 0 0 0 | * *  * * n * *  * * * | 0 0 1 0 0 1 0 0
.x   ..    || .o-n-.o    | 0 2 0 1 | 0 0 1  0 2 0 0 | * *  * * * n *  * * * | 0 0 0 0 1 1 0 0
.o-n-.o    || ..   x.    | 0 1 2 0 | 0 0 0  2 0 1 0 | * *  * * * * n  * * * | 0 0 0 1 0 0 1 0
.o-n-.o    || oo-n-oo&#x | 0 1 1 1 | 0 0 0  1 1 0 1 | * *  * * * * * 2n * * | 0 0 0 0 0 1 1 0
..   ..      o.-n-x.    | 0 0 n 0 | 0 0 0  0 0 0 n | * *  * * * * *  * 1 * | 0 1 0 0 0 0 0 1
..   ..      ..   xo&#x | 0 0 2 1 | 0 0 0  0 0 2 1 | * *  * * * * *  * * n | 0 0 0 0 0 0 1 1
------------------------+---------+----------------+-----------------------+----------------
ox-n-oo&#x   ..   ..    ♦ 1 n 0 0 | n 0 n  0 0 0 0 | n 0  0 1 0 0 0  0 0 0 | 1 * * * * * * *
o.-n-o.    || o.-n-x.    ♦ 1 0 n 0 | 0 n 0  0 0 n 0 | 0 n  0 0 0 0 0  0 1 0 | * 1 * * * * * *
ox   ..&#x || o.-n-o.    ♦ 1 2 1 0 | 2 1 1  2 0 0 0 | 1 0  2 0 1 0 0  0 0 0 | * * n * * * * *
oo-n-oo&#x || ..   x.    ♦ 1 1 2 0 | 1 2 0  2 0 1 0 | 0 1  2 0 0 0 1  0 0 0 | * * * n * * * *
.x-n-.o    || .o-n-.o    ♦ 0 n 0 1 | 0 0 n  0 n 0 0 | 0 0  0 1 0 n 0  0 0 0 | * * * * 1 * * *
.x   ..    || oo-n-oo&#x ♦ 0 2 1 1 | 0 0 1  2 2 0 1 | 0 0  0 0 1 1 0  2 0 0 | * * * * * n * *
. o-n-.o   || ..   xo&#x ♦ 0 1 2 1 | 0 0 0  2 1 1 2 | 0 0  0 0 0 0 1  2 0 1 | * * * * * * n *
..    ..     oo-n-xo&#x ♦ 0 0 n 1 | 0 0 0  0 0 n n | 0 0  0 0 0 0 0  0 1 n | * * * * * * * 1
```

```oxoo-n-ooox&#xr   (2<n<6)   → all heights = sqrt(1-1/[4 sin^2(π/n)])
(pt || pseudo ({n} || dual {n}) || pt)

o...-n-o...    | 1 * * * ♦ n n 0 0  0 0 0 | n 2n n 0 0 0 0  0 0 0 | 1 1 n n 0 0 0 0
.o..-n-.o..    | * n * * | 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.-n-..o.    | * * 1 * ♦ 0 0 0 n  0 n 0 | 0  0 0 0 n 0 0 2n n 0 | 0 0 0 0 1 n n 1
...o-n-...o    | * * * n | 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..-n-oo..&#x | 1 1 0 0 | n * * *  * * * | 2  2 0 0 0 0 0  0 0 0 | 1 0 2 1 0 0 0 0
o..o-n-o..o&#x | 1 0 0 1 | * n * *  * * * | 0  2 2 0 0 0 0  0 0 0 | 0 1 1 2 0 0 0 0
.x..   ....    | 0 2 0 0 | * * n *  * * * | 1  0 0 1 1 1 0  0 0 0 | 1 0 1 0 1 1 0 0
.oo.-n-.oo.&#x | 0 1 1 0 | * * * n  * * * | 0  0 0 0 2 0 0  2 0 0 | 0 0 0 0 1 2 1 0
.o.o-n-.o.o&#x | 0 1 0 1 | * * * * 2n * * | 0  1 0 0 0 1 1  1 0 0 | 0 0 1 1 0 1 1 0
..oo-n-..oo&#x | 0 0 1 1 | * * * *  * n * | 0  0 0 0 0 0 0  2 2 0 | 0 0 0 0 0 1 2 1
....   ...x    | 0 0 0 2 | * * * *  * * n | 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 | n  * * * * * *  * * * | 1 0 1 0 0 0 0 0
oo.o-n-oo.o&#x | 1 1 0 1 | 1 1 0 0  1 0 0 | * 2n * * * * *  * * * | 0 0 1 1 0 0 0 0
....   o..x&#x | 1 0 0 2 | 0 2 0 0  0 0 1 | *  * n * * * *  * * * | 0 1 0 1 0 0 0 0
.x..-n-.o..    | 0 n 0 0 | 0 0 n 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 | *  * * * n * *  * * * | 0 0 0 0 1 1 0 0
.x.o   ....&#x | 0 2 0 1 | 0 0 1 0  2 0 0 | *  * * * * n *  * * * | 0 0 1 0 0 1 0 0
....   .o.x&#x | 0 1 0 2 | 0 0 0 0  2 0 1 | *  * * * * * n  * * * | 0 0 0 1 0 0 1 0
.ooo-n-.ooo&#x | 0 1 1 1 | 0 0 0 1  1 1 0 | *  * * * * * * 2n * * | 0 0 0 0 0 1 1 0
....   ..ox&#x | 0 0 1 2 | 0 0 0 0  0 2 1 | *  * * * * * *  * n * | 0 0 0 0 0 0 1 1
...o-n-...x    | 0 0 0 n | 0 0 0 0  0 0 n | *  * * * * * *  * * 1 | 0 1 0 0 0 0 0 1
---------------+---------+----------------+-----------------------+----------------
ox..-n-oo..&#x ♦ 1 n 0 0 | n 0 n 0  0 0 0 | n  0 0 1 0 0 0  0 0 0 | 1 * * * * * * *
o..o-n-o..x&#x ♦ 1 0 0 n | 0 n 0 0  0 0 n | 0  0 n 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 | * * n * * * * *
....   oo.x&#x ♦ 1 1 0 2 | 1 2 0 0  2 0 1 | 0  2 1 0 0 0 1  0 0 0 | * * * n * * * *
.xo.-n-.oo.&#x ♦ 0 n 1 0 | 0 0 n n  0 0 0 | 0  0 0 1 n 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 | * * * * * n * *
....   .oox&#x ♦ 0 1 1 2 | 0 0 0 1  2 2 1 | 0  0 0 0 0 0 1  2 1 0 | * * * * * * n *
..oo-n-..ox&#x ♦ 0 0 1 n | 0 0 0 0  0 n n | 0  0 0 0 0 0 0  0 n 1 | * * * * * * * 1
```

```xoo-n-oox oyo&#xt   → both heights = sqrt([1+2 cos(π/n)]/[8+8 cos(π/n)])
({n} || perp y-line || dual {n})   y = sqrt([5-6 cos(π/n)]/[2-2 cos(π/n)])

o..-n-o.. o..     | n * * | 2  2  2  0 0 | 1  4 2 1  4  0 0 | 2  4  2 0
.o.-n-.o. .o.     | * 2 * ♦ 0  n  0  n 0 | 0  n 0 0 2n  n 0 | 1  n  n 1
..o-n-..o ..o     | * * n | 0  0  2  2 2 | 0  0 1 2  4  4 1 | 0  2  4 2
------------------+-------+--------------+------------------+----------
x..   ... ...     | 2 0 0 | n  *  *  * * | 1  2 1 0  0  0 0 | 2  2  0 0
oo.-n-oo. oo.&#x  | 1 1 0 | * 2n  *  * * | 0  2 0 0  2  0 0 | 1  2  1 0
o.o-n-o.o o.o&#x  | 1 0 1 | *  * 2n  * * | 0  0 1 1  2  0 0 | 0  2  2 0
.oo-n-.oo .oo&#x  | 0 1 1 | *  *  * 2n * | 0  0 0 0  2  2 0 | 0  1  2 1
...   ..x ...     | 0 0 2 | *  *  *  * n | 0  0 0 1  0  2 1 | 0  0  2 2
------------------+-------+--------------+------------------+----------
x..-n-o.. ...     | n 0 0 | n  0  0  0 0 | 1  * * *  *  * * | 2  0  0 0
xo.   ... ...&#x  | 2 1 0 | 1  2  0  0 0 | * 2n * *  *  * * | 1  1  0 0
x.o   ... ...&#x  | 2 0 1 | 1  0  2  0 0 | *  * n *  *  * * | 0  2  0 0
...   o.x ...&#x  | 1 0 2 | 0  0  2  0 1 | *  * * n  *  * * | 0  0  2 0
ooo-n-ooo ooo&#xt | 1 1 1 | 0  1  1  1 0 | *  * * * 4n  * * | 0  1  1 0
...   .ox ...&#x  | 0 1 2 | 0  0  0  2 1 | *  * * *  * 2n * | 0  0  1 1
..o-n-..x ...     | 0 0 n | 0  0  0  0 n | *  * * *  *  * 1 | 0  0  0 2
------------------+-------+--------------+------------------+----------
xo.-n-oo. ...&#x  ♦ n 1 0 | n  n  0  0 0 | 1  n 0 0  0  0 0 | 2  *  * *
xoo   ... ...&#xt ♦ 2 1 1 | 1  2  2  1 0 | 0  1 1 0  2  0 0 | * 2n  * *
...   oox ...&#xt ♦ 1 1 2 | 0  1  2  2 1 | 0  0 0 1  2  1 0 | *  * 2n *
.oo-n-.ox ...&#x  ♦ 0 1 n | 0  0  0  n n | 0  0 0 0  0  n 1 | *  *  * 2
```

```yo os2os-2n-os&#zx   → height = 0
(tegum sum of y-line and perp n-ap)   y = sqrt([5-6 cos(π/n)]/[2-2 cos(π/n)])
(tegum product of y-line with n-ap)

o. demi( o.2o.-2n-o. )    | 2  * ♦ 2n  0  0 | 2n 2n 0  0 | 2 2n
.o demi( .o2.o-2n-.o )    | * 2n |  2  2  2 |  4  4 1  3 | 2  6
--------------------------+------+----------+------------+-----
oo demi( oo2oo-2n-oo )&#x | 1  1 | 4n  *  * |  2  2 0  0 | 1  3
..       .s2.s    ..      | 0  2 |  * 2n  * |  2  0 0  2 | 0  4
.. sefa( .. .s-2n-.s )    | 0  2 |  *  * 2n |  0  2 1  1 | 2  2
--------------------------+------+----------+------------+-----
oo       os2os    ..  &#x | 1  2 |  2  1  0 | 4n  * *  * | 0  2
oo sefa( .. os-2n-os )&#x | 1  2 |  2  0  1 |  * 4n *  * | 1  1
..       .. .s-2n-.s      | 0  n |  0  0  n |  *  * 2  * | 2  0
.. sefa( .s2.s-2n-.s )    | 0  3 |  0  2  1 |  *  * * 2n | 0  2
--------------------------+------+----------+------------+-----
oo       .. os-2n-os  &#x ♦ 1  n |  n  0  n |  0  n 1  0 | 4  *
oo sefa( os2os-2n-os )&#x ♦ 1  3 |  3  2  1 |  2  1 0  1 | * 4n
```