- general polytopal classes:
- segmentochora
| Acronym | n/d-pyp |
| Name | n/d-pyramidal prism, line || n/d-prism |
| Segmentochoron display |
|
| Circumradius | sqrt[1/4+1/(4-1/sin2[π d/n])] |
| Confer |
|
| Face vector | 2n+2, 5n+1, 4n+2, n+3 |
| Especially | tepe (n=3, d=1) squippyp (n=4, d=1) peppyp (n=5, d=1) stappyp (n=5, d=2) hippyp (n=6, d=1) |
The height formula given below shows that only 2 < n/d < 6 is possible. The maximal height would be obtained at n/d = 2 with upright latteral triangles but then degenerate base, the other extremal value n/d = 6 would generate a height of zero.
Incidence matrix according to Dynkin symbol
xx ox-n/d-oo&#x (2<n/d<6) → height = sqrt(1-[1/4 *sin^2(π d/n)])
(line || {n/d}-p)
o. o.-n/d-o. | 2 * | 1 n 0 0 | n n 0 0 | n 1 0
.o .o-n/d-.o | * 2n | 0 1 1 2 | 1 2 2 1 | 2 1 1
----------------+------+-----------+----------+------
x. .. .. | 2 0 | 1 * * * | n 0 0 0 | n 0 0
oo oo-n/d-oo&#x | 1 1 | * 2n * * | 1 2 0 0 | 2 1 0
.x .. .. | 0 2 | * * n * | 1 0 2 0 | 2 0 1
.. .x .. | 0 2 | * * * 2n | 0 1 1 1 | 1 1 1
----------------+------+-----------+----------+------
xx .. ..&#x | 2 2 | 1 2 1 0 | n * * * | 2 0 0
.. ox ..&#x | 1 2 | 0 2 0 1 | * 2n * * | 1 1 0
.x .x .. | 0 4 | 0 0 2 2 | * * n * | 1 0 1
.. .x-n/d-.o | 0 n | 0 0 0 n | * * * 2 | 0 1 1
----------------+------+-----------+----------+------
xx ox ..&#x ♦ 2 4 | 1 4 2 2 | 2 2 1 0 | n * *
.. ox-n/d-oo&#x ♦ 1 n | 0 n 0 n | 0 n 0 1 | * 2 *
.x .x-n/d-.o ♦ 0 2n | 0 0 n 2n | 0 0 n 2 | * * 1
{n/d}-py || {n/d}-py (2<n/d<6) → height = 1
1 * * * | n 1 0 0 0 0 | n n 0 0 0 0 | 1 n 0 0 top-tip
* n * * | 1 0 2 1 0 0 | 2 1 1 0 0 0 | 1 2 1 0 top-base
* * 1 * | 0 1 0 0 n 0 | 0 n 0 0 n 0 | 0 n 0 1 bottom-tip
* * * n | 0 0 0 1 1 2 | 0 1 0 2 2 1 | 0 2 1 1 bottom-base
----------+-------------+-------------+--------
1 1 0 0 | n * * * * * | 2 1 0 0 0 0 | 1 2 0 0
1 0 1 0 | * 1 * * * * | 0 n 0 0 0 0 | 0 n 0 0
0 2 0 0 | * * n * * * | 1 0 1 1 0 0 | 1 1 1 0
0 1 0 1 | * * * n * * | 0 1 0 2 0 0 | 0 2 1 0
0 0 1 1 | * * * * n * | 0 1 0 0 2 0 | 0 2 0 1
0 0 0 2 | * * * * * n | 0 0 0 1 1 1 | 0 1 1 1
----------+-------------+-------------+--------
1 2 0 0 | 2 0 1 0 0 0 | n * * * * * | 1 1 0 0
1 1 1 1 | 1 1 0 1 1 0 | * n * * * * | 0 2 0 0
0 n 0 0 | 0 0 n 0 0 0 | * * 1 * * * | 1 0 1 0
0 2 0 2 | 0 0 1 2 0 1 | * * * n * * | 0 1 1 0
0 0 1 2 | 0 0 0 0 2 1 | * * * * n * | 0 1 0 1
0 0 0 n | 0 0 0 0 0 n | * * * * * 1 | 0 0 1 1
----------+-------------+-------------+--------
♦ 1 n 0 0 | n 0 n 0 0 0 | n 0 1 0 0 0 | 1 * * *
♦ 1 2 1 2 | 2 1 1 2 2 1 | 1 2 0 1 1 0 | * n * *
♦ 0 n 0 n | 0 0 n n 0 n | 0 0 1 n 0 1 | * * 1 *
♦ 0 0 1 n | 0 0 0 0 n n | 0 0 0 0 n 1 | * * * 1
oxxo-n/d-oooo&#xr (2<n/d<6) → height(1,2) = height(3,4) = sqrt(1-[1/4 *sin^2(π/n)])
height(1,4) = height(2,3) = 1
( (pt || {n/d}) || (pt || {n/d}) )
o...-n/d-o... | 1 * * * | n 1 0 0 0 0 | n n 0 0 0 0 | 1 n 0 0
.o..-n/d-.o.. | * n * * | 1 0 2 1 0 0 | 2 1 1 2 0 0 | 1 2 1 0
..o.-n/d-..o. | * * n * | 0 0 0 1 2 1 | 0 1 0 2 1 2 | 0 2 1 1
...o-n/d-...o | * * * 1 | 0 1 0 0 0 n | 0 n 0 0 0 n | 0 n 0 1
------------------+---------+-------------+-------------+--------
oo..-n/d-oo..&#x | 1 1 0 0 | n * * * * * | 2 1 0 0 0 0 | 1 2 0 0
o..o-n/d-o..o&#x | 1 0 0 1 | * 1 * * * * | 0 n 0 0 0 0 | 0 n 0 0
.x.. .... | 0 2 0 0 | * * n * * * | 1 0 1 1 0 0 | 1 1 1 0
.oo.-n/d-.oo.&#x | 0 1 1 0 | * * * n * * | 0 1 0 2 0 0 | 0 2 1 0
..x. .... | 0 0 2 0 | * * * * n * | 0 0 0 1 1 1 | 0 1 1 1
..oo-n/d-..oo&#x | 0 0 1 1 | * * * * * n | 0 1 0 0 0 2 | 0 2 0 1
------------------+---------+-------------+-------------+--------
ox.. ....&#x | 1 2 0 0 | 2 0 1 0 0 0 | n * * * * * | 1 1 0 0
oooo-n/d-oooo&#xr | 1 1 1 1 | 1 1 0 1 0 1 | * n * * * * | 0 2 0 0
.x..-n/d-.o.. | 0 n 0 0 | 0 0 n 0 0 0 | * * 1 * * * | 1 0 1 0
.xx. ....&#x | 0 2 2 0 | 0 0 1 2 1 0 | * * * n * * | 0 1 1 0
..x.-n/d-..o. | 0 0 n 0 | 0 0 0 0 n 0 | * * * * 1 * | 0 0 1 1
..xo ....&#x | 0 0 2 1 | 0 0 0 0 1 2 | * * * * * n | 0 1 0 1
------------------+---------+-------------+-------------+--------
ox..-n/d-oo..&#x ♦ 1 n 0 0 | n 0 n 0 0 0 | n 0 1 0 0 0 | 1 * * *
oxxo ....&#xr ♦ 1 2 2 1 | 2 1 1 2 1 2 | 1 2 0 1 0 1 | * n * *
.xx.-n/d-.oo.&#x ♦ 0 n n 0 | 0 0 n n n 0 | 0 0 1 n 1 0 | * * 1 *
..xo-n/d-..oo&#x ♦ 0 0 n 1 | 0 0 0 0 n n | 0 0 0 0 1 n | * * * 1
o(xo)x-n/d-o(oo)o&#xt (2<n/d<6) (pt || ({n/d} || pt) || para {n/d}) o(..).-n/d-o(..). | 1 * * * | n 1 0 0 0 0 | n n 0 0 0 0 | 1 n 0 0 .(o.).-n/d-.(o.). | * n * * | 1 0 2 1 0 0 | 2 1 1 2 0 0 | 1 2 1 0 .(.o).-n/d-.(.o). | * * 1 * | 0 1 0 0 n 0 | 0 n 0 0 n 0 | 0 n 0 1 .(..)o-n/d-.(..)o | * * * n | 0 0 0 1 1 2 | 0 1 0 2 2 1 | 0 2 1 1 ----------------------+---------+-------------+-------------+-------- o(o.).-n/d-o(o.).&#x | 1 1 0 0 | n * * * * * | 2 1 0 0 0 0 | 1 2 0 0 o(.o).-n/d-o(.o).&#x | 1 0 1 0 | * 1 * * * * | 0 n 0 0 0 0 | 0 n 0 0 .(x.). .(..). | 0 2 0 0 | * * n * * * | 1 0 1 1 0 0 | 1 1 1 0 .(o.)o-n/d-.(o.)o&#x | 0 1 0 1 | * * * n * * | 0 1 0 2 0 0 | 0 2 1 0 .(.o)o-n/d-.(.o)o&#x | 0 0 1 1 | * * * * n * | 0 1 0 0 2 0 | 0 2 0 1 .(..)x .(..). | 0 0 0 2 | * * * * * n | 0 0 0 1 1 1 | 0 1 1 1 ----------------------+---------+-------------+-------------+-------- o(x.). .(..).&#x | 1 2 0 0 | 2 0 1 0 0 0 | n * * * * * | 1 1 0 0 o(oo)o-n/d-o(oo)o&#xt | 1 1 1 1 | 1 1 0 1 1 0 | * n * * * * | 0 2 0 0 .(x.).-n/d-.(o.). | 0 n 0 0 | 0 0 n 0 0 0 | * * 1 * * * | 1 0 1 0 .(x.)x .(..).&#x | 0 2 0 2 | 0 0 1 2 0 1 | * * * n * * | 0 1 1 0 .(.o)x .(..).&#x | 0 0 1 2 | 0 0 0 0 2 1 | * * * * n * | 0 1 0 1 .(..)x-n/d-.(..)o | 0 0 0 n | 0 0 0 0 0 n | * * * * * 1 | 0 0 1 1 ----------------------+---------+-------------+-------------+-------- o(x.).-n/d-o(o.).&#x ♦ 1 n 0 0 | n 0 n 0 0 0 | n 0 1 0 0 0 | 1 * * * o(xo)x .(..).&#xt ♦ 1 2 1 2 | 2 1 1 2 2 1 | 1 2 0 1 1 0 | * n * * .(x.)x-n/d-.(o.)o&#x ♦ 0 n 0 n | 0 0 n n 0 n | 0 0 1 n 0 1 | * * 1 * .(.o)x-n/d-.(.o)o&#x ♦ 0 0 1 n | 0 0 0 0 n n | 0 0 0 0 n 1 | * * * 1
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