Acronym | n/d-daf | |||||||||||||||||||||||||||||||||||||||||||||
Name |
n/d-duoantifastegium, (2,n)-duoantiwedge, n/d-p atop bidual n/d-p | |||||||||||||||||||||||||||||||||||||||||||||
Circumradius | sqrt([3+1/(1-cos(pi d/n))]/8) | |||||||||||||||||||||||||||||||||||||||||||||
Lace hyper city in approx. ASCII-art |
x-n/d-o x-n/d-o o-n/d-x o-n/d-x(disphenoid in configuration space) | |||||||||||||||||||||||||||||||||||||||||||||
Face vector | 4n, 14n, 18n+4, 10n+6, 2n+4 | |||||||||||||||||||||||||||||||||||||||||||||
Especially |
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Confer |
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External links |
These all are scaliform polytera. They even are convex if d=1. – The boundary case n/d=2 not only would provide different incidences than the general matrix given below (degenerate {n/d}-gons), it even itself would become dimensionally degenerate. In fact it happens to be {4} || perp {4}, the hull of which is just hex.
Incidence matrix according to Dynkin symbol
xo ox xo-n/d-ox&#x (n > 2d) → height = sqrt([1-1/(1+cos(pi d/n))]/2)
(n/d-p || bidual n/d-p)
o. o. o.-n/d-o. | 2n * | 1 2 4 0 0 | 2 1 4 2 4 2 0 0 | 1 2 4 2 2 1 2 0 | 2 1 2 1
.o .o .o-n/d-.o | * 2n | 0 0 4 1 2 | 0 0 2 4 2 4 2 1 | 0 2 1 2 2 4 2 1 | 1 2 1 2
-------------------+-------+--------------+---------------------+----------------------+--------
x. .. .. .. | 2 0 | n * * * * | 2 0 4 0 0 0 0 0 | 1 2 4 2 0 0 0 0 | 2 1 2 0
.. .. x. .. | 2 0 | * 2n * * * | 1 1 0 0 2 0 0 0 | 1 0 2 0 1 0 2 0 | 1 0 2 1
oo oo oo-n/d-oo&#x | 1 1 | * * 8n * * | 0 0 1 1 1 1 0 0 | 0 1 1 1 1 1 1 0 | 1 1 1 1
.. .x .. .. | 0 2 | * * * n * | 0 0 0 4 0 0 2 0 | 0 2 0 0 2 4 0 1 | 1 2 0 2
.. .. .. .x | 0 2 | * * * * 2n | 0 0 0 0 0 2 1 1 | 0 0 0 1 0 2 2 1 | 0 1 1 2
-------------------+-------+--------------+---------------------+----------------------+--------
x. .. x. .. | 4 0 | 2 2 0 0 0 | n * * * * * * * | 1 0 2 0 0 0 0 0 | 1 0 2 0
.. .. x.-n/d-o. | n 0 | 0 n 0 0 0 | * 2 * * * * * * | 1 0 0 0 0 0 2 0 | 0 0 2 1
xo .. .. ..&#x | 2 1 | 1 0 2 0 0 | * * 4n * * * * * | 0 1 1 1 0 0 0 0 | 1 1 1 0
.. ox .. ..&#x | 1 2 | 0 0 2 1 0 | * * * 4n * * * * | 0 1 0 0 1 1 0 0 | 1 1 0 1
.. .. xo ..&#x | 2 1 | 0 1 2 0 0 | * * * * 4n * * * | 0 0 1 0 1 0 1 0 | 1 0 1 1
.. .. .. ox&#x | 1 2 | 0 0 2 0 1 | * * * * * 4n * * | 0 0 0 1 0 1 1 0 | 0 1 1 1
.. .x .. .x | 0 4 | 0 0 0 2 2 | * * * * * * n * | 0 0 0 0 0 2 0 1 | 0 1 0 2
.. .. .o-n/d-.x | 0 n | 0 0 0 0 n | * * * * * * * 2 | 0 0 0 0 0 0 2 1 | 0 0 1 2
-------------------+-------+--------------+---------------------+----------------------+--------
x. .. x.-n/d-o. ♦ 2n 0 | n 2n 0 0 0 | n 2 0 0 0 0 0 0 | 1 * * * * * * * | 0 0 2 0
xo ox .. ..&#x ♦ 2 2 | 1 0 4 1 0 | 0 0 2 2 0 0 0 0 | * 2n * * * * * * | 1 1 0 0
xo .. xo ..&#x ♦ 4 1 | 2 2 4 0 0 | 1 0 2 0 2 0 0 0 | * * 2n * * * * * | 1 0 1 0
xo .. .. ox&#x ♦ 2 2 | 1 0 4 0 1 | 0 0 2 0 0 2 0 0 | * * * 2n * * * * | 0 1 1 0
.. ox xo ..&#x ♦ 2 2 | 0 1 4 1 0 | 0 0 0 2 2 0 0 0 | * * * * 2n * * * | 1 0 0 1
.. ox .. ox&#x ♦ 1 4 | 0 0 4 2 2 | 0 0 0 2 0 2 1 0 | * * * * * 2n * * | 0 1 0 1
.. .. xo-n/d-ox&#x ♦ n n | 0 n 2n 0 n | 0 1 0 0 n n 0 1 | * * * * * * 4 * | 0 0 1 1
.. .x .o-n/d-.x ♦ 0 2n | 0 0 0 n 2n | 0 0 0 0 0 0 n 2 | * * * * * * * 1 | 0 0 0 2
-------------------+-------+--------------+---------------------+----------------------+--------
xo ox xo ..&#x ♦ 4 2 | 2 2 8 1 0 | 1 0 4 4 4 0 0 0 | 0 2 2 0 2 0 0 0 | n * * *
xo ox .. ox&#x ♦ 2 4 | 1 0 8 2 2 | 0 0 4 4 0 4 1 0 | 0 2 0 2 0 2 0 0 | * n * *
xo .. xo-n/d-ox&#x ♦ 2n n | n 2n 4n 0 n | n 2 2n 0 2n 2n 0 1 | 1 0 n n 0 0 2 0 | * * 2 *
.. ox xo-n/d-ox&#x ♦ n 2n | 0 n 4n n 2n | 0 1 0 2n 2n 2n n 2 | 0 0 0 0 n n 2 1 | * * * 2
or o. o. o.-n/d-o. & | 4n | 1 2 4 | 2 1 6 6 | 1 2 5 4 2 | 3 3 ---------------------+----+----------+------------+--------------+----- x. .. .. .. & | 2 | 2n * * | 2 0 4 0 | 1 2 4 2 0 | 3 2 .. .. x. .. & | 2 | * 4n * | 1 1 0 2 | 1 0 2 1 2 | 1 3 oo oo oo-n/d-oo&#x | 2 | * * 8n | 0 0 2 2 | 0 1 2 2 1 | 2 2 ---------------------+----+----------+------------+--------------+----- x. .. x. .. & | 4 | 2 2 0 | 2n * * * | 1 0 2 0 0 | 1 2 .. .. x.-n/d-o. & | n | 0 n 0 | * 4 * * | 1 0 0 0 2 | 0 3 xo .. .. ..&#x & | 3 | 1 0 2 | * * 8n * | 0 1 1 1 0 | 2 1 .. .. xo ..&#x & | 3 | 0 1 2 | * * * 8n | 0 0 1 1 1 | 1 2 ---------------------+----+----------+------------+--------------+----- x. .. x.-n/d-o. & ♦ 2n | n 2n 0 | n 2 0 0 | 2 * * * * | 0 2 xo ox .. ..&#x ♦ 4 | 2 0 4 | 0 0 4 0 | * 2n * * * | 2 0 xo .. xo ..&#x & ♦ 5 | 2 2 4 | 1 0 2 2 | * * 4n * * | 1 1 xo .. .. ox&#x & ♦ 4 | 1 1 4 | 0 0 2 2 | * * * 4n * | 1 1 .. .. xo-n/d-ox&#x ♦ 2n | 0 2n 2n | 0 2 0 2n | * * * * 4 | 0 2 ---------------------+----+----------+------------+--------------+----- xo ox xo ..&#x & ♦ 6 | 3 2 8 | 1 0 8 4 | 0 2 2 2 0 | 2n * xo .. xo-n/d-ox&#x & ♦ 3n | n 3n 4n | n n 2n 4n | 1 0 n n 2 | * 4
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