Acronym | n-ap |
Name | n-gonal antiprism |
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Circumradius | sqrt[(3-2 cos(π/n))/(8-8 cos(π/n))] |
Height | sqrt[(1+2 cos(π/n))/(2+2 cos(π/n))] |
Coordinates |
(cos(k π/n)/[2 sin(π/n)], sin(k π/n)/[2 sin(π/n)], (-1)k h/2) all k integral where h is the height given above |
Vertex figure | [33,n] |
Snub derivation |
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General of army | (is itself convex) |
Colonel of regiment | (is itself locally convex) |
Dihedral angles |
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Face vector | 2n, 4n, 2n+2 |
Especially | tet (n=2)* oct (n=3) squap (n=4) pap (n=5) hap (n=6) oap (n=8) dap (n=10) azap (n=∞) |
Confer |
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External links |
* The case n=2 equally would be considerable here by concept, it just has a different incidence matrix as the n-gons become degenerate.
The compound of 2 mutually gyrated n-gonal antiprisms, one considered as xo-n-ox&#x, the other as ox-n-xo&#x, has for encasing convex hull a variant of the 2n-gonal prism, in fact the variant a-2n-o b, where a = 1/[2 cos(π/2n)] and b = sqrt[1 - 1/[2 cos(π/2n)]2] = sqrt[(1+2 cos(π/n))/(2+2 cos(π/n))]. This is how the below mentioned semiation as a vertex alternation (snubbing) truely works (when resizing would be considered first).
Incidence matrix according to Dynkin symbol
s2sns (n>2) demi( . . . ) | 2n | 1 1 2 | 1 3 ---------------+----+--------+----- s2s . | 2 | n * * | 0 2 s . s2*a | 2 | * n * | 0 2 sefa( . sns ) | 2 | * * 2n | 1 1 ---------------+----+--------+----- . sns ♦ n | 0 0 n | 2 * sefa( s2sns ) | 3 | 1 1 1 | * 2n starting figure: x xnx
s2s2no (n>2) demi( . . . ) | 2n | 2 2 | 1 3 ---------------+----+-------+----- s2s . | 2 | 2n * | 0 2 sefa( . s2no ) | 2 | * 2n | 1 1 ---------------+----+-------+----- . s2no ♦ n | 0 n | 2 * sefa( s2s2no ) | 3 | 2 1 | * 2n starting figure: x x2no
xonox&#x (n>2) → height = sqrt[(1+2 cos(π/n))/(2+2 cos(π/n))]
({n} || dual {n})
o.no. | n * | 2 2 0 | 1 2 1 0
.on.o | * n | 0 2 2 | 0 1 2 1
---------+-----+--------+--------
x. .. | 2 0 | n * * | 1 1 0 0
oonoo&#x | 1 1 | * 2n * | 0 1 1 0
.. .x | 0 2 | * * n | 0 0 1 1
---------+-----+--------+--------
x.no. | n 0 | n 0 0 | 1 * * *
xo ..&#x | 2 1 | 1 2 0 | * n * *
.. ox&#x | 1 2 | 0 2 1 | * * n *
.on.x | 0 n | 0 0 n | * * * 1
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