hemiated 2n-gon - 2m-gon duoprism
- general polytopal classes:
- isogonal
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| Acronym | n,m-dap |
| Name |
n-gon - m-gon duoantiprism, hemiated 2n-gon - 2m-gon duoprism |
| Circumradius | sqrt[A2/(4 sin2(π/2n))+B2/(4 sin2(π/2m))] |
| Face vector | 2nm, 8nm, 8nm+2n+2m, 2nm+2n+2m |
| Especially | triddap (s3s2s3s) pedap (s3s2s5s) gudap (s5s2s5/3s, uniform) s2s2sns hex (s2s2s2s, uniform) |
| Confer |
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External links |
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These isogonal polychora are obtained by hemiation of the uniform 2n,2m-duoprism (with n>2, m>2). But because of lower degree of freedom the three edge sizes A*x(2n), B*x(2m), sqrt(A2+B2) cannot be made all alike.
When considering holosnubbing instead, the first matrix well can be applied to odd link marks too.
It also could be generalized to rationals, i.e. to hemiations of polygrammal duoprisms. Still the restriction to the link marks remains as to the degenerate hemiations of squares. The only uniform case here would be obtained with n=5, m=5/3 (gudap).
Those exceptional cases n=2 or m=2 could be considered similarily, but the incidence matrices below will be different because of degeneracy. There the special subcase of the hemiated tes (hex) even happens to be regular.
Incidence matrix according to Dynkin symbol
s-2n-o-2-s-2m-o (n>2, m>2)
demi( . . . . ) | 2nm | 2 4 2 | 1 6 6 1 | 2 4 2
------------------------+-----+-------------+---------------+----------
sefa( s-2n-o . . ) | 2 | 2nm * * | 1 2 0 0 | 2 1 0 A*x(2n)
s . s . | 2 | * 4nm * | 0 2 2 0 | 1 2 1 sqrt(A^2+B^2)
sefa( . . s-2m-o ) | 2 | * * 2nm | 0 0 2 1 | 0 1 2 B*x(2m)
------------------------+-----+-------------+---------------+----------
s-2n-o . . | n | n 0 0 | 2m * * * | 2 0 0
sefa( s-2n-o-2-s . ) | 3 | 1 2 0 | * 4nm * * | 1 1 0
sefa( s - 2 - s-2m-o ) | 3 | 0 2 1 | * * 4nm * | 0 1 1
. . s-2m-o | m | 0 0 m | * * * 2n | 0 0 2
------------------------+-----+-------------+---------------+----------
s-2n-o-2-s . ♦ 2n | 2n 2n 0 | 2 2n 0 0 | 2m * * n-ap
sefa( s-2n-o-2-s-2m-o ) ♦ 4 | 1 4 1 | 0 2 2 0 | * 2nm * tet
s - 2 - s-2m-o ♦ 2m | 0 2m 2m | 0 0 2m 2 | * * 2n m-ap
starting figure: x-2n-o x-2m-o
s-n-s-2-s-m-s (n>2,m>2)
demi( . . . . ) | 2nm | 2 4 2 | 1 6 6 1 | 2 4 2
------------------------+-----+-------------+---------------+----------
sefa( s-n-s . . ) | 2 | 2nm * * | 1 2 0 0 | 2 1 0 A*x(2n)
s - 2 - s . & | 2 | * 4nm * | 0 2 2 0 | 1 2 1 sqrt(A^2+B^2)
sefa( . . s-m-s ) | 2 | * * 2nm | 0 0 2 1 | 0 1 2 B*x(2m)
------------------------+-----+-------------+---------------+----------
s-n-s . . | n | n 0 0 | 2m * * * | 2 0 0
sefa( s-n-s-2-s . ) & | 3 | 1 2 0 | * 4nm * * | 1 1 0
sefa( . s-2-s-m-s ) & | 3 | 0 2 1 | * * 4nm * | 0 1 1
. . s-m-s | m | 0 0 m | * * * 2n | 0 0 2
------------------------+-----+-------------+---------------+----------
s-n-s-2-s . & ♦ 2n | 2n 2n 0 | 2 2n 0 0 | 2m * * n-ap
sefa( s-n-s-2-s-m-s ) ♦ 4 | 1 4 1 | 0 2 2 0 | * 2nm * disphenoids
. s-2-s-m-s & ♦ 2m | 0 2m 2m | 0 0 2m 2 | * * 2n m-ap
starting figure: x-n-x x-m-x
ao-n-oa bo-m-ob&#zc → height = 0,
a = A*x(2n),
b = B*x(2m),
c = sqrt(A^2+B^2)
(c-laced tegum sum of 2 bidual (a,b)-sized (n,m)-duoprisms)
o.-n-o. o.-m-o. | nm * | 2 2 4 0 0 | 1 1 4 2 4 2 0 0 | 2 2 2 2
.o-n-.o .o-m-.o | * nm | 0 0 4 2 2 | 0 0 2 4 2 4 1 1 | 2 2 2 2
--------------------+-------+-----------------+-------------------------+------------
a. .. .. .. | 2 0 | nm * * * * | 1 0 2 0 0 0 0 0 | 2 1 0 0
.. .. b. .. | 2 0 | * nm * * * | 0 1 0 0 2 0 0 0 | 0 0 1 2
oo-n-oo oo-m-oo&#c | 1 1 | * * 4nm * * | 0 0 1 1 1 1 0 0 | 1 1 1 1
.. .a .. .. | 0 2 | * * * nm * | 0 0 0 2 0 0 1 0 | 2 0 1 0
.. .. .. .b | 0 2 | * * * * nm | 0 0 0 0 0 2 0 1 | 0 1 0 2
--------------------+-------+-----------------+-------------------------+------------
a.-n-o. .. .. | n 0 | n 0 0 0 0 | m * * * * * * * | 2 0 0 0
.. .. b.-m-o. | m 0 | 0 m 0 0 0 | * n * * * * * * | 0 0 0 2
ao .. .. ..&#c | 2 1 | 1 0 2 0 0 | * * 2nm * * * * * | 1 1 0 0
.. oa .. ..&#c | 1 2 | 0 0 2 1 0 | * * * 2nm * * * * | 1 0 1 0
.. .. bo. ..&#c | 2 1 | 0 1 2 0 0 | * * * * 2nm * * * | 0 0 1 1
.. .. .. ob&#c | 1 2 | 0 0 2 0 1 | * * * * * 2nm * * | 0 1 0 1
.o-n-.a .. .. | 0 n | 0 0 0 n 0 | * * * * * * m * | 2 0 0 0
.. .. .o-m-.b | 0 m | 0 0 0 0 m | * * * * * * * n | 0 0 0 2
--------------------+-------+-----------------+-------------------------+------------
ao-n-oa .. ..&#c ♦ n n | n 0 2n n 0 | 1 0 n n 0 0 1 0 | m2 * * *
ao .. .. ob&#c ♦ 2 2 | 1 0 4 0 1 | 0 0 2 0 0 2 0 0 | * nm * *
.. oa bo ..&#c ♦ 2 2 | 0 1 4 1 0 | 0 0 0 2 2 0 0 0 | * * nm *
.. .. bo-m-ob&#c ♦ m m | 0 m 2m 0 m | 0 1 0 0 m m 0 1 | * * * 2n
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