Acronym ... Name dual of uniform n-gonal prism,n-gonal dipyramid © © Vertex figure [tn], [T4] Dihedral angles at long edge:   arccos[(sin2(π/n)-1)/(sin2(π/n)+1)] at short edge:   arccos[(sin2(π/n)-1)/(sin2(π/n)+1)] Dual n-gonal prism Especially m m3o (n=3)   oct (n=4) Externallinks  The triangles {(t,T,T)} have vertex angles t = arccos[1-2 sin4(π/n)] resp. T = arccos[sin2(π/n)].

Note that the term n-gonal dipyramid, although stated above, in general says nothing about the relative ratio of the to be chosen lacing edge size. There are equilateral dipyramids as well, cf. tridpy (n=3), oct (n=4), and pedpy (n=5). – Whereas in here the edge ratio is chosen such as to match their duality to the n-gonal prisms!

Incidence matrix according to Dynkin symbol

m  m-n-o =
oxo-n-ooo&#yt   → height = cos(π/n)/(2 sin2(π/n))
y = 1/(2 sin2(π/n))

o..-n-o..    | 1 * * | n 0 0 | n 0  [tn]
.o.-n-.o.    | * n * | 1 2 1 | 2 2  [T4]
..o-n-..o    | * * 1 | 0 0 n | 0 n  [tn]
-------------+-------+-------+----
oo.-n-oo.&#y | 1 1 0 | n * * | 2 0  y
.x.   ...    | 0 2 0 | * n * | 1 1  x
.oo-n-.oo&#y | 0 1 1 | * * n | 0 2  y
-------------+-------+-------+----
ox.   ...&#y | 1 2 0 | 2 1 0 | n *  {(t,T,T)}
.xo   ...&#y | 0 2 1 | 0 1 2 | * n  {(t,T,T)}

m  m-n-o =
ao ox-n-oo&#zy   → height = 0
a = cos(π/n)/sin2(π/n)
y = 1/(2 sin2(π/n))

o. o.-n-o.    | 2 * |  n 0 |  n  [tn]
.o .o-n-.o    | * n |  2 2 |  4  [T4]
--------------+-----+------+---
oo oo-n-oo&#y | 1 1 | 2n * |  2  y
.. .x   ..    | 0 2 |  * n |  2  x
--------------+-----+------+---
.. ox   ..&#y | 1 2 |  2 1 | 2n  {(t,T,T)}