n-gonal anti-wedge,
2n-gon - n-antiprismatic wedge,
n-gonal gyrobicupolaic ring,
{2n} || n-antiprism,
{n} || gyrated n-cupola
in approx. ASCII-art
x-n-x
x-n-o o-n-x
- general polytopal classes:
- segmentochora
| Acronym | n-aw, {2n} || n-ap |
| Name |
anti-n-wedge, n-gonal anti-wedge, 2n-gon - n-antiprismatic wedge, n-gonal gyrobicupolaic ring, {2n} || n-antiprism, {n} || gyrated n-cupola |
| Segmentochoron display |
|
| Circumradius | sqrt[(3-2 cos(π/n))/(5-6 cos(π/n))] |
|
Lace city in approx. ASCII-art |
x-n-x
x-n-o o-n-x
|
| Face vector | 4n, 10n, 8n+3, 2n+3 |
| Confer |
|
| Especially | {4} || tet (n=2)* {6} || oct (n=3) {8} || squap (n=4) {10} || pap (n=5) {10/2} || stap (n=5/2) pseudo {10/2} || stap (n=5/2)* |
* The case n=2 equally would be considerable here by concept, it just has a different incidence matrix as the n-gons become degenerate. Similarily in the case of n=5/2 the 2n-gons could be either considered Grünbaumian or could as such be withdrawn, becoming then a pseudo face only, i.e. giving rise to a corresponding semi-anti-wedge instead.
Incidence matrix according to Dynkin symbol
xxo-n-oxx&#x → height(1,2) = height(2,3) = sqrt(1-1/[4 sin^2(π/n)])
height(1,3) = sqrt[(1+2*cos(π/n))/(2+2*cos(π/n))]
o..-n-o.. | n * * | 2 2 2 0 0 0 0 | 1 2 1 2 1 2 0 0 0 0 | 1 1 2 1 0
.o.-n-.o. | * 2n * | 0 1 0 1 1 1 0 | 0 1 1 0 0 1 1 1 1 0 | 1 0 1 1 1
..o-n-..o | * * n | 0 0 2 0 0 2 2 | 0 0 0 1 2 2 0 1 2 1 | 0 1 1 2 1
-------------+--------+------------------+----------------------+----------
x.. ... | 2 0 0 | n * * * * * * | 1 1 0 1 0 0 0 0 0 0 | 1 1 1 0 0
oo.-n-oo.&#x | 1 1 0 | * 2n * * * * * | 0 1 1 0 0 1 0 0 0 0 | 1 0 1 1 0
o.o-n-o.o&#x | 1 0 1 | * * 2n * * * * | 0 0 0 1 1 1 0 0 0 0 | 0 1 1 1 0
.x. ... | 0 2 0 | * * * n * * * | 0 1 0 0 0 0 1 1 0 0 | 1 0 1 0 1
... .x. | 0 2 0 | * * * * n * * | 0 0 1 0 0 0 1 0 1 0 | 1 0 0 1 1
.oo-n-.oo&#x | 0 1 1 | * * * * * 2n * | 0 0 0 0 0 1 0 1 1 0 | 0 0 1 1 1
... ..x | 0 0 2 | * * * * * * n | 0 0 0 0 1 0 0 0 1 1 | 0 1 0 1 1
-------------+--------+------------------+----------------------+----------
x..-n-o.. | n 0 0 | n 0 0 0 0 0 0 | 1 * * * * * * * * * | 1 1 0 0 0
xx. ...&#x | 2 2 0 | 1 2 0 1 0 0 0 | * n * * * * * * * * | 1 0 1 0 0
... ox.&#x | 1 2 0 | 0 2 0 0 1 0 0 | * * n * * * * * * * | 1 0 0 1 0
x.o ...&#x | 2 0 1 | 1 0 2 0 0 0 0 | * * * n * * * * * * | 0 1 1 0 0
... o.x&#x | 1 0 2 | 0 0 2 0 0 0 1 | * * * * n * * * * * | 0 1 0 1 0
ooo-n-ooo&#x | 1 1 1 | 0 1 1 0 0 1 0 | * * * * * 2n * * * * | 0 0 1 1 0
.x.-n-.x. | 0 2n 0 | 0 0 0 n n 0 0 | * * * * * * 1 * * * | 1 0 0 0 1
.xo ...&#x | 0 2 1 | 0 0 0 1 0 2 0 | * * * * * * * n * * | 0 0 1 0 1
... .xx&#x | 0 2 2 | 0 0 0 0 1 2 1 | * * * * * * * * n * | 0 0 0 1 1
..o-n-..x | 0 0 n | 0 0 0 0 0 0 n | * * * * * * * * * 1 | 0 1 0 0 1
-------------+--------+------------------+----------------------+----------
xx.-n-ox.&#x ♦ n 2n 0 | n 2n 0 n n 0 0 | 1 n n 0 0 0 1 0 0 0 | 1 * * * *
x.o-n-o.x&#x ♦ n 0 n | n 0 2n 0 0 0 n | 1 0 0 n n 0 0 0 0 1 | * 1 * * *
xxo ...&#x ♦ 2 2 1 | 1 2 2 1 0 2 0 | 0 1 0 1 0 2 0 1 0 0 | * * n * *
... oxx&#x ♦ 1 2 2 | 0 2 2 0 1 2 1 | 0 0 1 0 1 2 0 0 1 0 | * * * n *
.xo-n-.xx&#x ♦ 0 2n n | 0 0 0 n n 2n n | 0 0 0 0 0 0 1 n n 1 | * * * * 1
os2xo2nos&#x (2 < n < 5.363958) → height = sqrt[(5-6 cos(π/n))/(8-8 cos(π/n))]
({2n} || n-ap)
o.2o.2no. | 2n * | 2 2 0 0 | 1 1 2 0 2 0 | 2 0 2
demi( .o2.o2n.o ) | * 2n | 0 2 2 2 | 0 2 1 1 2 3 | 1 1 3
---------------------+-------+-------------+-----------------+-------
.. x. .. | 2 0 | 2n * * * | 1 0 1 0 1 0 | 2 0 1
demi( oo2oo2noo&#x ) | 1 1 | * 4n * * | 0 1 1 0 1 0 | 1 0 2
.s .2 .s | 0 2 | * * 2n * | 0 1 0 0 0 2 | 0 1 2
sefa( .. .o2n.s ) | 0 2 | * * * 2n | 0 0 0 1 1 1 | 1 1 1
---------------------+-------+-------------+-----------------+-------
.. x.2no. | 2n 0 | 2n 0 0 0 | 1 * * * * * | 2 0 0
os .2 os&#x | 1 2 | 0 2 1 0 | * 2n * * * * | 0 0 2
demi( .. xo ..&#x ) | 2 1 | 1 2 0 0 | * * 2n * * * | 1 0 1
.. .o2n.s | 0 n | 0 0 0 n | * * * 2 * * | 1 1 0
sefa( .. xo2nos&#x ) | 2 2 | 1 2 0 1 | * * * * 2n * | 1 0 1
sefa( .s2.o2n.s ) | 0 3 | 0 0 2 1 | * * * * * 2n | 0 1 1
---------------------+-------+-------------+-----------------+-------
.. xo2nos&#x ♦ 2n n | 2n 2n 0 n | 1 0 n 1 n 0 | 2 * *
.s2.o2n.s ♦ 0 2n | 0 0 2n 2n | 0 0 0 2 0 2n | * 1 *
sefa( os2xo2nos&#x ) ♦ 2 3 | 1 4 2 1 | 0 2 1 0 1 1 | * * 2n
starting figure: ox xo2nox&#x
{n} || gyro n-cu → height = ???
n * * | 2 2 2 0 0 0 0 | 1 2 2 1 2 1 0 0 0 0 | 1 1 2 1 0
* n * | 0 2 0 2 2 0 0 | 0 1 0 2 2 0 1 2 1 0 | 1 0 1 2 1
* * 2n | 0 0 1 0 1 1 1 | 0 0 1 0 1 1 0 1 1 1 | 0 1 1 1 1
---------+------------------+----------------------+----------
2 0 0 | n * * * * * * | 1 1 1 0 0 0 0 0 0 0 | 1 1 1 0 0
1 1 0 | * 2n * * * * * | 0 1 0 1 1 0 0 0 0 0 | 1 0 1 1 0
1 0 1 | * * 2n * * * * | 0 0 1 0 1 1 0 0 0 0 | 0 1 1 1 0
0 2 0 | * * * n * * * | 0 0 0 1 0 0 1 1 0 0 | 1 0 0 1 1
0 1 1 | * * * * 2n * * | 0 0 0 0 1 0 0 1 1 0 | 0 0 1 1 1
0 0 2 | * * * * * n * | 0 0 1 0 0 0 0 0 1 1 | 0 1 1 0 1
0 0 2 | * * * * * * n | 0 0 0 0 0 1 0 1 0 1 | 0 1 0 1 1
---------+------------------+----------------------+----------
n 0 0 | n 0 0 0 0 0 0 | 1 * * * * * * * * * | 1 1 0 0 0
2 1 0 | 1 2 0 0 0 0 0 | * n * * * * * * * * | 1 0 1 0 0
2 0 2 | 1 0 2 0 0 1 0 | * * n * * * * * * * | 0 1 1 0 0
1 2 0 | 0 2 0 1 0 0 0 | * * * n * * * * * * | 1 0 0 1 0
1 1 1 | 0 1 1 0 1 0 0 | * * * * 2n * * * * * | 0 0 1 1 0
1 0 2 | 0 0 2 0 0 0 1 | * * * * * n * * * * | 0 1 0 1 0
0 n 0 | 0 0 0 n 0 0 0 | * * * * * * 1 * * * | 1 0 0 0 1
0 2 2 | 0 0 0 1 2 0 1 | * * * * * * * n * * | 0 0 0 1 1
0 1 2 | 0 0 0 0 2 1 0 | * * * * * * * * n * | 0 0 1 0 1
0 0 2n | 0 0 0 0 0 n n | * * * * * * * * * 1 | 0 1 0 0 1
---------+------------------+----------------------+----------
♦ n n 0 | n 2n 0 n 0 0 0 | 1 n 0 n 0 0 1 0 0 0 | 1 * * * *
♦ n 0 2n | n 0 2n 0 0 n n | 1 0 n 0 0 n 0 0 0 1 | * 1 * * *
♦ 2 1 2 | 1 2 2 0 2 1 0 | 0 1 1 0 2 0 0 0 1 0 | * * n * *
♦ 1 2 2 | 0 2 2 1 2 0 1 | 0 0 0 1 2 1 0 1 0 0 | * * * n *
♦ 0 n 2n | 0 0 0 n 2n n n | 0 0 0 0 0 0 1 n n 1 | * * * * 1
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