Acronym | ... |
Name |
dual of uniform n-gonal prism, n-gonal dipyramid |
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Vertex figure | [tn], [T4] |
Dihedral angles |
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Dual | n-gonal prism |
Especially | m m3o (n=3) oct (n=4) m m5o (n=5) m m6o (n=6) |
Face vector | n+2, 3n, 2n |
Confer |
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External links |
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 in general says nothing about the relative ratio of the edge sizes. 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 → both heights = 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)}
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