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)  
External
links
wikipedia   mathworld  

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, even so 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)}

© 2004-2019
top of page