The height formula according to its representation as line || perp n-gon implies: n ≤ 5.104299.

Incidence matrix according to Dynkin symbol

oox-n-ooo&#x   → height(1,2) = 1
                 height(1,3) = height(2,3) = sqrt(1-1/[4 sin^2(π/n)])

o..   o..    | 1 * * ♦ 1 n 0 0 | n n 0 0 | n 1 0
.o.   .o.    | * 1 * ♦ 1 0 n 0 | n 0 n 0 | n 0 1
..o   ..o    | * * n | 0 1 1 2 | 1 2 2 1 | 2 1 1
-------------+-------+---------+---------+------
oo.-n-oo.&#x | 1 1 0 | 1 * * * | n 0 0 0 | n 0 0
o.o-n-o.o&#x | 1 0 1 | * n * * | 1 2 0 0 | 2 1 0
.oo-n-.oo&#x | 0 1 1 | * * n * | 1 0 2 0 | 2 0 1
..x   ...    | 0 0 2 | * * * n | 0 1 1 1 | 1 1 1
-------------+-------+---------+---------+------
ooo-n-ooo&#x | 1 1 1 | 1 1 1 0 | n * * * | 2 0 0
o.x   ...&#x | 1 0 2 | 0 2 0 1 | * n * * | 1 1 0
.ox   ...&#x | 0 1 2 | 0 0 2 1 | * * n * | 1 0 1
..x-n-..o    | 0 0 n | 0 0 0 n | * * * 1 | 0 1 1
-------------+-------+---------+---------+------
oox   ...&#x ♦ 1 1 2 | 1 2 2 1 | 2 1 1 0 | n * *
o.x-n-o.o&#x ♦ 1 0 n | 0 n 0 n | 0 n 0 1 | * 1 *
.ox-n-.oo&#x ♦ 0 1 n | 0 0 n n | 0 n 5 1 | * * 1

pt || n-py   → height = ...

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

pyramid product of line with {n}   → height = sqrt[3 sin2(π/n)-1]/2 sin(π/n)

o..   o..     & | 2 * ♦ 1  n 0 | n  n 0 | n 1
..o   ..o       | * n | 0  2 2 | 1  4 1 | 2 2
----------------+-----+--------+--------+----
oo.-n-oo.&#x    | 2 0 | 1  * * | n  0 0 | n 0
o.o-n-o.o&#x  & | 1 1 | * 2n * | 1  2 0 | 2 1
..x   ...       | 0 2 | *  * n | 0  2 1 | 1 2
----------------+-----+--------+--------+----
ooo-n-ooo&#x    | 2 1 | 1  2 0 | n  * * | 2 0
o.x   ...&#x  & | 1 2 | 0  2 1 | * 2n * | 1 1
..x-n-..o       | 0 n | 0  0 n | *  * 1 | 0 2
----------------+-----+--------+--------+----
oox   ...&#x    ♦ 2 2 | 1  4 1 | 2  2 0 | n *
o.x-n-o.o&#x  & ♦ 1 n | 0  n n | 0  n 1 | * 2