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     x-n-o     
o-n-o     o-n-o
     o-n-x     

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