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

The lace city display shows that this polychoron can be dissected vertically into segmentochoric components: into 2 n-pufs; thereby adding one 2n-prism 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 orientation as stack of n-cupolae shows that those are all monostratic, except of the case n=2 those would not be segmentochora. This is because of the needed shift of these generally non-degenerate bases out of their circumcenter. Accordingly in general there will be no (full-dimensional) circumradius either. (But be aware, that the incidence matrices for n=2 would differ because of the degenerate n-gon.)

Incidence matrix according to Dynkin symbol

ox-n-xx&#x || xx-n-xo&#x   (1.243647<n<5.104299)   → height = ???
(n-cu || inv gyro n-cu)

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

xxo-n-oxx oxo&#xt   (1.243647<n<5.104299)   → both heights = sqrt[3 - 1/sin2(π/n)]/2
({n} || pseudo {2n}-p || dual {n})

o..-n-o.. o..    | n  * * | 2  4  0  0  0  0 0 | 1  4  2  2 0 0 0  0  0  0 0 | 2 2 1 0 0 0
.o.-n-.o. .o.    | * 4n * | 0  1  1  1  1  1 0 | 0  1  1  1 1 1 1  1  1  1 0 | 1 1 1 1 1 1
..o-n-..o ..o    | *  * n | 0  0  0  0  0  4 2 | 0  0  0  0 0 0 0  2  4  2 1 | 0 0 0 2 1 2
-----------------+--------+--------------------+-----------------------------+------------
x..   ... ...    | 2  0 0 | n  *  *  *  *  * * | 1  2  0  0 0 0 0  0  0  0 0 | 2 1 0 0 0 0
oo.-n-oo. oo.&#x | 1  1 0 | * 4n  *  *  *  * * | 0  1  1  1 0 0 0  0  0  0 0 | 1 1 1 0 0 0
.x.   ... ...    | 0  2 0 | *  * 2n  *  *  * * | 0  1  0  0 1 1 0  1  0  0 0 | 1 1 0 1 1 0
...   .x. ...    | 0  2 0 | *  *  * 2n  *  * * | 0  0  1  0 1 0 1  0  1  0 0 | 1 0 1 1 0 1
...   ... .x.    | 0  2 0 | *  *  *  * 2n  * * | 0  0  0  1 0 1 1  0  0  1 0 | 0 1 1 0 1 1
.oo-n-.oo .oo&#x | 0  1 1 | *  *  *  *  * 4n * | 0  0  0  0 0 0 0  1  1  1 0 | 0 0 0 1 1 1
...   ..x ...    | 0  0 2 | *  *  *  *  *  * n | 0  0  0  0 0 0 0  0  2  0 1 | 0 0 0 2 0 1
-----------------+--------+--------------------+-----------------------------+------------
x..-n-o.. ...    | n  0 0 | n  0  0  0  0  0 0 | 1  *  *  * * * *  *  *  * * | 2 0 0 0 0 0
xx.   ... ...&#x | 2  2 0 | 1  2  1  0  0  0 0 | * 2n  *  * * * *  *  *  * * | 1 1 0 0 0 0
...   ox. ...&#x | 1  2 0 | 0  2  0  1  0  0 0 | *  * 2n  * * * *  *  *  * * | 1 0 1 0 0 0
...   ... ox.&#x | 1  2 0 | 0  2  0  0  1  0 0 | *  *  * 2n * * *  *  *  * * | 0 1 1 0 0 0
.x.-n-.x. ...    | 0 2n 0 | 0  0  n  n  0  0 0 | *  *  *  * 2 * *  *  *  * * | 1 0 0 1 0 0
.x.   ... .x.    | 0  4 0 | 0  0  2  0  2  0 0 | *  *  *  * * n *  *  *  * * | 0 1 0 0 1 0
...   .x. .x.    | 0  4 0 | 0  0  0  2  2  0 0 | *  *  *  * * * n  *  *  * * | 0 0 1 0 0 1
.xo   ... ...&#x | 0  2 1 | 0  0  1  0  0  2 0 | *  *  *  * * * * 2n  *  * * | 0 0 0 1 1 0
...   .xx ...&#x | 0  2 2 | 0  0  0  1  0  2 1 | *  *  *  * * * *  * 2n  * * | 0 0 0 1 0 1
...   ... .xo&#x | 0  2 1 | 0  0  0  0  1  2 0 | *  *  *  * * * *  *  * 2n * | 0 0 0 0 1 1
..o-n-..x ...    | 0  0 n | 0  0  0  0  0  0 n | *  *  *  * * * *  *  *  * 1 | 0 0 0 2 0 0
-----------------+--------+--------------------+-----------------------------+------------
xx.-n-ox. ...&#x ♦ n 2n 0 | n 2n  n  n  0  0 0 | 1  n  n  0 1 0 0  0  0  0 0 | 2 * * * * *
xx.   ... ox.&#x ♦ 2  4 0 | 1  4  2  0  2  0 0 | 0  2  0  2 0 1 0  0  0  0 0 | * n * * * *
...   ox. ox.&#x ♦ 1  4 0 | 0  4  0  2  2  0 0 | 0  0  2  2 0 0 1  0  0  0 0 | * * n * * *
.xo-n-.xx ...&#x ♦ 0 2n n | 0  0  n  n  0 2n n | 0  0  0  0 1 0 0  n  n  0 1 | * * * 2 * *
.xo   ... .xo&#x ♦ 0  4 1 | 0  0  2  0  2  4 0 | 0  0  0  0 0 1 0  2  0  2 0 | * * * * n *
...   .xx .xo&#x ♦ 0  4 2 | 0  0  0  2  2  4 1 | 0  0  0  0 0 0 1  0  2  2 0 | * * * * * n