| 3/d | 4/d | 5/d | 6/d | 7/d | 8/d | 9/d | 10/d | {n/d}-appip |
|---|---|---|---|---|---|---|---|---|
| ope | squappip | pappip | happip | heappip | oappip | eappip | dappip | n/1 |
| * | stappip | * | shappip | * | steappip | * | n/2 | |
| starpip | * | gishappip | stoappip | * | stiddappip | n/3 | ||
| gisharpip | * | gisteappip | * | n/4 | ||||
| storpip | gisterpip | * | n/5 |
links
| Acronym | n/d-appip | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Name | n/d-antiprism prism | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Circumradius | sqrt[(5-4 cos(π d/n))/(8-8 cos(π d/n))] | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Face vector | 4n, 10n, 8n+4, 2n+4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Especially |
n-appip (d=1)
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External links |
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n=2, d=1 would result in the tepe. But for sure the top resp. bottom face prisms then become degenerate and hence that one needs a different incidence matrix.
The height-formula of the n/d-ap implies d<2n/3.
Note that for d odd the 2 n/d-p layers are aligned in a gyrated way, which then effects that top and bottom vertices are situated on gap. While for d even the 2 n/d-p layers are still aligned in a gyrated way, but this time it effects in a seeming ungyrated parallel alignment, just like in the mere 4,n/d-dip, but now with crossed lacings, so that the according height becomes lesser than 1 here.
Incidence matrix according to Dynkin symbol
x s-2-s-n/d-s (n>2) . demi( . . . ) | 4n | 1 1 1 2 | 1 1 1 2 3 | 1 1 3 ------------------------+----+-------------+-------------+------- . s-2-s . | 2 | 2n * * * | 1 0 0 0 2 | 0 1 2 . s . s2*b | 2 | * 2n * * | 0 1 0 0 2 | 0 1 2 x demi( . . . ) | 2 | * * 2n * | 1 1 0 2 0 | 1 0 3 . sefa( . s-n/d-s ) | 2 | * * * 4n | 0 0 1 1 1 | 1 1 1 ------------------------+----+-------------+-------------+------- x s-2-s . | 4 | 2 0 2 0 | n * * * * | 0 0 2 x s . s2*b | 4 | 0 2 2 0 | * n * * * | 0 0 2 . . s-n/d-s ♦ n | 0 0 0 n | * * 4 * * | 1 1 0 x sefa( . s-n/d-s ) | 4 | 0 0 2 2 | * * * 2n * | 1 0 1 . sefa( s-2-s-n/d-s ) | 3 | 1 1 0 1 | * * * * 4n | 0 1 1 ------------------------+----+-------------+-------------+------- x . s-n/d-s ♦ 2n | 0 0 n 2n | 0 0 2 n 0 | 2 * * . s-2-s-n/d-s ♦ 2n | n n 0 2n | 0 0 2 0 2n | * 2 * x sefa( s-2-s-n/d-s ) ♦ 6 | 2 2 3 2 | 1 1 0 1 2 | * * 2n
x s-2-s-2n/d-o (n>2) . demi( . . . ) | 4n | 2 1 2 | 2 1 2 3 | 1 1 3 -----------------------+----+----------+------------+------- . s-2-s . | 2 | 4n * * | 1 0 0 2 | 0 1 2 x demi( . . . ) | 2 | * 2n * | 2 0 2 0 | 1 0 3 . sefa( . s-2n/d-o ) | 2 | * * 4n | 0 1 1 1 | 1 1 1 -----------------------+----+----------+------------+------- x s-2-s . | 4 | 2 2 0 | 2n * * * | 0 0 2 . . s-2n/d-o ♦ n | 0 0 n | * 4 * * | 1 1 0 x sefa( . s-2n/d-o ) | 4 | 0 2 2 | * * 2n * | 1 0 1 . sefa( s-2-s-2n/d-o ) | 3 | 2 0 1 | * * * 4n | 0 1 1 -----------------------+----+----------+------------+------- x . s-2n/d-o ♦ 2n | 0 n 2n | 0 2 n 0 | 2 * * . s-2-s-2n/d-o ♦ 2n | 2n 0 2n | 0 2 0 2n | * 2 * x sefa( s-2-s-2n/d-o ) ♦ 6 | 4 3 2 | 2 0 1 2 | * * 2n
xx xo-n/d-ox&#x (n>2) → height = sqrt[(1+2*cos(πd/n))/(2+2*cos(πd/n))]
({n/d}-p || gyro {n/d}-p)
o. o.-n/d-o. | 2n * | 1 2 2 0 0 | 2 1 2 2 1 0 0 | 1 2 1 1 0
.o .o-n/d-.o | * 2n | 0 0 2 1 2 | 0 0 2 1 2 2 1 | 0 1 2 1 1
----------------+-------+--------------+------------------+----------
x. .. .. | 2 0 | n * * * * | 2 0 2 0 0 0 0 | 1 2 1 0 0
.. x. .. | 2 0 | * 2n * * * | 1 1 0 1 0 0 0 | 1 1 0 1 0
oo oo-n/d-oo&#x | 1 1 | * * 4n * * | 0 0 1 1 1 0 0 | 0 1 1 1 0
.x .. .. | 0 2 | * * * n * | 0 0 2 0 0 2 0 | 0 1 2 0 1
.. .. .x | 0 2 | * * * * 2n | 0 0 0 0 1 1 1 | 0 0 1 1 1
----------------+-------+--------------+------------------+----------
x. x. .. | 4 0 | 2 2 0 0 0 | n * * * * * * | 1 1 0 0 0
.. x.-n/d-o. | n 0 | 0 n 0 0 0 | * 2 * * * * * | 1 0 0 1 0
xx .. ..&#x | 2 2 | 1 0 2 1 0 | * * 2n * * * * | 0 1 1 0 0
.. xo ..&#x | 2 1 | 0 1 2 0 0 | * * * 2n * * * | 0 1 0 1 0
.. .. ox&#x | 1 2 | 0 0 2 0 1 | * * * * 2n * * | 0 0 1 1 0
.x .. .x | 0 4 | 0 0 0 2 2 | * * * * * n * | 0 0 1 0 1
.. .o-n/d-.x | 0 n | 0 0 0 0 n | * * * * * * 2 | 0 0 0 1 1
----------------+-------+--------------+------------------+----------
x. x.-n/d-o. ♦ 2n 0 | n 2n 0 0 0 | n 2 0 0 0 0 0 | 1 * * * *
xx xo ..&#x ♦ 4 2 | 2 2 4 1 0 | 1 0 2 2 0 0 0 | * n * * *
xx .. ox&#x ♦ 2 4 | 1 0 4 2 2 | 0 0 2 0 2 1 0 | * * n * *
.. xo-n/d-ox&#x ♦ n n | 0 n 2n 0 n | 0 1 0 n n 0 1 | * * * 2 *
.x .o-n/d-.x ♦ 0 2n | 0 0 0 n 2n | 0 0 0 0 0 n 2 | * * * * 1
s-2-s-n/d-s || s-2-s-n/d-s (n>2) → height = 1 ({n/d}-ap || {n/d}-ap) demi( . . . ) | 2n * | 1 1 2 1 0 0 0 | 1 3 1 1 2 0 0 | 1 1 3 0 demi( . . . ) | * 2n | 0 0 0 1 1 1 2 | 0 0 1 1 2 1 3 | 0 1 3 1 ------------------------------------------+-------+------------------+------------------+--------- s-2-s . | 2 0 | n * * * * * * | 0 2 1 0 0 0 0 | 1 0 1 0 s 2 s | 2 0 | * n * * * * * | 0 2 0 1 0 0 0 | 1 0 1 0 sefa( . s-n/d-s ) | 2 0 | * * 2n * * * * | 1 1 0 0 1 0 0 | 1 1 1 0 demi( . . . ) || demi( . . . ) | 1 1 | * * * 2n * * * | 0 0 1 1 2 0 0 | 0 1 3 0 s-2-s . | 0 2 | * * * * n * * | 0 0 1 0 0 0 2 | 0 0 2 1 s 2 s | 0 2 | * * * * * n * | 0 0 0 1 0 0 2 | 0 0 2 1 sefa( . s-n/d-s ) | 0 2 | * * * * * * 2n | 0 0 0 0 1 1 1 | 0 1 1 1 ------------------------------------------+-------+------------------+------------------+--------- . s-n/d-s ♦ n 0 | 0 0 n 0 0 0 0 | 2 * * * * * * | 1 1 0 0 sefa( s-2-s-n/d-s ) | 3 0 | 1 1 1 0 0 0 0 | * 2n * * * * * | 1 0 1 0 s-2-s . || s-2-s . | 2 2 | 1 0 0 2 1 0 0 | * * n * * * * | 0 0 2 0 s 2 s || s 2 s | 2 2 | 0 1 0 2 0 1 0 | * * * n * * * | 0 0 2 0 sefa( . s-n/d-s ) || sefa( . s-n/d-s ) | 2 2 | 0 0 1 2 0 0 1 | * * * * 2n * * | 0 1 1 0 . s-n/d-s ♦ 0 n | 0 0 0 0 0 0 n | * * * * * 2 * | 0 1 0 1 sefa( s-2-s-n/d-s ) | 0 3 | 0 0 0 0 1 1 1 | * * * * * * 2n | 0 0 1 1 ------------------------------------------+-------+------------------+------------------+--------- s-2-s-n/d-s ♦ 2n 0 | n n 2n 0 0 0 0 | 2 2n 0 0 0 0 0 | 1 * * * . s-n/d-s || . s-n/d-s ♦ n n | 0 0 n n 0 0 n | 1 0 0 0 n 1 0 | * 2 * * sefa( s-2-s-n/d-s ) || sefa( s-2-s-n/d-s ) ♦ 3 3 | 1 1 1 3 1 1 1 | 0 1 1 1 1 0 1 | * * 2n * s-2-s-n/d-s ♦ 0 2n | 0 0 0 0 n n 2n | 0 0 0 0 0 2 2n | * * * 1
s-2-s-2n/d-o || s-2-s-2n/d-o (n>2) → height = 1 ({n/d}-ap || {n/d}-ap) demi( . . . ) | 2n * | 2 2 1 0 0 | 1 3 2 2 0 0 | 1 1 3 0 demi( . . . ) | * 2n | 0 0 1 2 2 | 0 0 2 2 1 3 | 0 1 3 1 --------------------------------------------+-------+----------------+-----------------+--------- s-2-s . | 2 0 | 2n * * * * | 0 2 1 0 0 0 | 1 0 2 0 sefa( . s-2n/d-o ) | 2 0 | * 2n * * * | 1 1 0 1 0 0 | 1 1 1 0 demi( . . . ) || demi( . . . ) | 1 1 | * * 2n * * | 0 0 2 2 0 0 | 0 1 3 0 s-2-s . ) | 0 2 | * * * 2n * | 0 0 1 0 0 2 | 0 0 2 1 sefa( . s-2n/d-o ) | 0 2 | * * * * 2n | 0 0 0 1 1 1 | 0 1 1 1 --------------------------------------------+-------+----------------+-----------------+--------- . s-2n/d-o ♦ n 0 | 0 n 0 0 0 | 2 * * * * * | 1 1 0 0 sefa( s-2-s-2n/d-o ) | 3 0 | 2 1 0 0 0 | * 2n * * * * | 1 0 1 0 s-2-s . ) || s-2-s . ) | 2 2 | 1 0 2 1 0 | * * 2n * * * | 0 0 2 0 sefa( . s-2n/d-o ) || sefa( . s-2n/d-o ) | 2 2 | 0 1 2 0 1 | * * * 2n * * | 0 1 1 0 . s-2n/d-o ♦ 0 n | 0 0 0 0 n | * * * * 2 * | 0 1 0 1 sefa( s-2-s-2n/d-o ) | 0 3 | 0 0 0 2 1 | * * * * * 2n | 0 0 1 1 --------------------------------------------+-------+----------------+-----------------+--------- s-2-s-2n/d-o ♦ 2n 0 | 2n 2n 0 0 0 | 2 2n 0 0 0 0 | 1 * * * . s-2n/d-o || . s-2n/d-o ♦ n n | 0 n n 0 n | 1 0 0 n 1 0 | * 2 * * sefa( s-2-s-2n/d-o ) || sefa( s-2-s-2n/d-o ) ♦ 3 3 | 2 1 3 2 1 | 0 1 2 1 0 1 | * * 2n * s-2-s-2n/d-o ♦ 0 2n | 0 0 0 2n 2n | 0 0 0 0 2 2n | * * * 1
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