Acronym ...
Name complex polyhedron xp   xr || op   xr
Coordinates
  • (0, εrm/(2 sin(π/r)), sqrt[1-1/(4 sin2(π/p))])   for any 1≤m≤r
  • (εpn/(2 sin(π/p)), εrm/(2 sin(π/r)), 0)             for any 1≤n≤p and any 1≤m≤r
    where εp=exp(2πi/p) and εr=exp(2πi/r) respectively
Face vector r(p+1), pr+p+r+1, p+r+1
Confer
general polytopal classes:
complex polytopes  

This complex polyhedron is an extrapolation of square || line, which as such was just the extremal case x2   x2 || o2   x2, i.e. the case with p=r=2.

For r=2 it occurs (at least by structure, not by measurement) as vertex figure of x2-3-o2-3-x2-4-op, and for p=r=3 it occurs as such of x3-3-o3-3-x3-3-o3.


Incidence matrix according to Dynkin symbol

xp   xr || op   xr   → height = sqrt[1-1/(4 sin2(π/p))]

op   or   .    .  | pr * | 1 1  1 0 | 1 1 1
.    .    op   or |  * r | 0 0  p 1 | 0 1 p
-----------------+------+----------+------
xp   .    .    .  |  p 0 | r *  * * | 1 1 0
.    xr   .    .  |  r 0 | * p  * * | 1 0 1
op   or || op   or |  1 1 | * * pr * | 0 1 1
.    .    .    xr |  0 r | * *  * 1 | 0 0 p
-----------------+------+----------+------
xp   xr   .    .  ♦ pr 0 | r p  0 0 | 1 * *
xp   .  || op   .  |  p 1 | 1 0  p 0 | * r *
.    xr || .    xr ♦  r r | 0 1  r 1 | * * p

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