Acronym | n/d-cuf |
Name |
n/d-grammal cupofastegium, 2n-gon - n/d-prismatic wedge, n/d-grammal orthobicupolaic ring, {2n/d} || n/d-prism, {n/d} || n/d-cupola |
Segmentochoron display | |
Circumradius | sqrt[(5+2 cos(π d/n)-4cos2(π d/n))/(8-12 cos2(π d/n))] |
Lace city in approx. ASCII-art |
x-n/d-x x-n/d-o x-n/d-o |
Face vector | 4n, 9n, 7n+3, 2n+3 |
Especially | {2n} || n-p (d=1) {4} || ortho {4} (n=2,d=1)* {6} || trip (n=3,d=1) {8} || cube (n=4,d=1) {10} || pip (n=5,d=1) |
* The case n=2,d=1 equally would be considerable here by concept, it just has a different incidence matrix as the n/d-grams become degenerate.
Incidence matrix according to Dynkin symbol
xx-n/d-ox xo&#x (2 < n/d < 5.104299) → height = sqrt[3 - 1/sin2(π d/n)]/2
(n/d-p || {2n/d})
o.-n/d-o. o. | 2n * | 2 1 2 0 0 | 1 2 2 1 2 0 | 1 1 2 1
.o-n/d-.o .o | * 2n | 0 0 2 1 1 | 0 0 2 2 1 1 | 0 2 1 1
----------------+-------+-------------+----------------+--------
x. .. .. | 2 0 | 2n * * * * | 1 1 1 0 0 0 | 1 1 1 0
.. .. x. | 2 0 | * n * * * | 0 2 0 0 2 0 | 1 0 2 1
oo-n/d-oo oo&#x | 1 1 | * * 4n * * | 0 0 1 1 1 0 | 0 1 1 1
.x .. .. | 0 2 | * * * n * | 0 0 2 0 0 1 | 0 2 1 0
.. .x .. | 0 2 | * * * * n | 0 0 0 2 0 1 | 0 2 0 1
----------------+-------+-------------+----------------+--------
x.-n/d-o. .. | n 0 | n 0 0 0 0 | 2 * * * * * | 1 1 0 0
x. .. x. | 4 0 | 2 2 0 0 0 | * n * * * * | 1 0 1 0
xx .. ..&#x | 2 2 | 1 0 2 1 0 | * * 2n * * * | 0 1 1 0
.. ox ..&#x | 1 2 | 0 0 2 0 1 | * * * 2n * * | 0 1 0 1
.. .. xo&#x | 2 1 | 0 1 2 0 0 | * * * * 2n * | 0 0 1 1
.x-n/d-.x .. | 0 2n | 0 0 0 n n | * * * * * 1 | 0 2 0 0
----------------+-------+-------------+----------------+--------
x.-n/d-o. x. ♦ 2n 0 | 2n n 0 0 0 | 2 n 0 0 0 0 | 1 * * *
xx-n/d-ox ..&#x ♦ n 2n | n 0 2n n n | 1 0 n n 0 1 | * 2 * *
xx .. xo&#x ♦ 4 2 | 2 2 4 1 0 | 0 1 2 0 2 0 | * * n *
.. ox xo&#x ♦ 2 2 | 0 1 4 0 1 | 0 0 0 2 2 0 | * * * n
xxx-n/d-oxo&#x → height(1,2) = height(2,3) = sqrt[4 - 1/sin2(π d/n)]/2 height(1,3) = 1 o..-n/d-o.. | n * * | 2 2 1 0 0 0 0 | 1 2 1 2 2 0 0 0 0 | 1 1 2 1 0 .o.-n/d-.o. | * 2n * | 0 1 0 1 1 1 0 | 0 1 1 0 1 1 1 1 0 | 1 0 1 1 1 ..o-n/d-..o | * * n | 0 0 1 0 0 2 2 | 0 0 0 2 2 0 2 1 1 | 0 1 2 1 1 ---------------+--------+-----------------+--------------------+---------- x.. ... | 2 0 0 | n * * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0 oo.-n/d-oo.&#x | 1 1 0 | * 2n * * * * * | 0 1 1 0 1 0 0 0 0 | 1 0 1 1 0 o.o-n/d-o.o&#x | 1 0 1 | * * n * * * * | 0 0 0 2 2 0 0 0 0 | 0 1 2 1 0 .x. ... | 0 2 0 | * * * n * * * | 0 1 0 0 0 1 1 0 0 | 1 0 1 0 1 ... .x. | 0 2 0 | * * * * n * * | 0 0 1 0 0 1 0 1 0 | 1 0 0 1 1 .oo-n/d-.oo | 0 1 1 | * * * * * 2n * | 0 0 0 0 1 0 1 1 0 | 0 0 1 1 1 ..x ... | 0 0 2 | * * * * * * n | 0 0 0 1 0 0 1 0 1 | 0 1 1 0 1 ---------------+--------+-----------------+--------------------+---------- x..-n/d-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.x ...&#x | 2 0 2 | 1 0 2 0 0 0 1 | * * * n * * * * * | 0 1 1 0 0 ooo-n/d-ooo&#x | 1 1 1 | 0 1 1 0 0 1 0 | * * * * 2n * * * * | 0 0 1 1 0 .x.-n/d-.x. | 0 2n 0 | 0 0 0 n n 0 0 | * * * * * 1 * * * | 1 0 0 0 1 .xx ...&#x | 0 2 2 | 0 0 0 1 0 2 1 | * * * * * * n * * | 0 0 1 0 1 ... .xo&#x | 0 2 1 | 0 0 0 0 1 2 0 | * * * * * * * n * | 0 0 0 1 1 ..x-n/d-..o | 0 0 n | 0 0 0 0 0 0 n | * * * * * * * * 1 | 0 1 0 0 1 ---------------+--------+-----------------+--------------------+---------- xx.-n/d-ox.&#x ♦ n 2n 0 | n 2n 0 n n 0 0 | 1 n n 0 0 1 0 0 0 | 1 * * * * x.x-n/d-o.o&#x ♦ n 0 n | n 0 n 0 0 0 n | 1 0 0 n 0 0 0 0 1 | * 1 * * * xxx ...&#x ♦ 2 2 2 | 1 2 2 1 0 2 1 | 0 1 0 1 2 0 1 0 0 | * * n * * ... oxo&#x ♦ 1 2 1 | 0 2 1 0 1 2 0 | 0 0 1 0 2 0 0 1 0 | * * * n * .xx-n/d-.oo&#x ♦ 0 2n n | 0 0 0 n n 2n n | 0 0 0 0 0 1 n n 1 | * * * * 1
or o..-n/d-o.. & | 2n * | 2 2 1 0 0 | 1 2 1 2 2 0 | 1 1 2 1 .o.-n/d-.o. | * 2n | 0 2 0 1 1 | 0 2 2 0 1 1 | 2 0 1 1 ------------------+-------+-------------+----------------+-------- x.. ... & | 2 0 | 2n * * * * | 1 1 0 1 0 0 | 1 1 1 0 oo.-n/d-oo.&#x & | 1 1 | * 4n * * * | 0 1 1 0 1 0 | 1 0 1 1 o.o-n/d-o.o&#x | 2 0 | * * n * * | 0 0 0 2 2 0 | 0 1 2 1 .x. ... | 0 2 | * * * n * | 0 2 0 0 0 1 | 2 0 1 0 ... .x. | 0 2 | * * * * n | 0 0 2 0 0 1 | 2 0 0 1 ------------------+-------+-------------+----------------+-------- x..-n/d-o.. & | n 0 | n 0 0 0 0 | 2 * * * * * | 1 1 0 0 xx. ...&#x & | 2 2 | 1 2 0 1 0 | * 2n * * * * | 1 0 1 0 ... ox.&#x & | 1 2 | 0 2 0 0 1 | * * 2n * * * | 1 0 0 1 x.x ...&#x | 4 0 | 2 0 2 0 0 | * * * n * * | 0 1 1 0 ooo-n/d-ooo&#x | 2 1 | 0 2 1 0 0 | * * * * 2n * | 0 0 1 1 .x.-n/d-.x. | 0 2n | 0 0 0 n n | * * * * * 1 | 2 0 0 0 ------------------+-------+-------------+----------------+-------- xx.-n/d-ox.&#x & ♦ n 2n | n 2n 0 n n | 1 n n 0 0 1 | 2 * * * x.x-n/d-o.o&#x ♦ 2n 0 | 2n 0 n 0 0 | 2 0 0 n 0 0 | * 1 * * xxx ...&#x ♦ 4 2 | 2 4 2 1 0 | 0 2 0 1 2 0 | * * n * ... oxo&#x ♦ 2 2 | 0 4 1 0 1 | 0 0 2 0 2 0 | * * * n
{n/d} || n/d-cu → height = ???
o..-n/d-o.. | n * * | 2 2 1 0 0 0 0 | 1 2 1 2 2 0 0 0 0 | 1 1 2 1 0
.o.-n/d-.o. | * 2n * | 0 1 0 1 1 1 0 | 0 1 1 0 1 1 1 1 0 | 1 0 1 1 1
..o-n/d-..o | * * n | 0 0 1 0 0 2 2 | 0 0 0 2 2 0 2 1 1 | 0 1 2 1 1
---------------+--------+-----------------+--------------------+----------
x.. ... | 2 0 0 | n * * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0
oo.-n/d-oo.&#x | 1 1 0 | * 2n * * * * * | 0 1 1 0 1 0 0 0 0 | 1 0 1 1 0
o.o-n/d-o.o&#x | 1 0 1 | * * n * * * * | 0 0 0 2 2 0 0 0 0 | 0 1 2 1 0
.x. ... | 0 2 0 | * * * n * * * | 0 1 0 0 0 1 1 0 0 | 1 0 1 0 1
... .x. | 0 2 0 | * * * * n * * | 0 0 1 0 0 1 0 1 0 | 1 0 0 1 1
.oo-n/d-.oo | 0 1 1 | * * * * * 2n * | 0 0 0 0 1 0 1 1 0 | 0 0 1 1 1
..x ... | 0 0 2 | * * * * * * n | 0 0 0 1 0 0 1 0 1 | 0 1 1 0 1
---------------+--------+-----------------+--------------------+----------
x..-n/d-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.x ...&#x | 2 0 2 | 1 0 2 0 0 0 1 | * * * n * * * * * | 0 1 1 0 0
ooo-n/d-ooo&#x | 1 1 1 | 0 1 1 0 0 1 0 | * * * * 2n * * * * | 0 0 1 1 0
.x.-n/d-.x. | 0 2n 0 | 0 0 0 n n 0 0 | * * * * * 1 * * * | 1 0 0 0 1
.xx ...&#x | 0 2 2 | 0 0 0 1 0 2 1 | * * * * * * n * * | 0 0 1 0 1
... .xo&#x | 0 2 1 | 0 0 0 0 1 2 0 | * * * * * * * n * | 0 0 0 1 1
..x-n/d-..o | 0 0 n | 0 0 0 0 0 0 n | * * * * * * * * 1 | 0 1 0 0 1
---------------+--------+-----------------+--------------------+----------
xx.-n/d-ox.&#x ♦ n 2n 0 | n 2n 0 n n 0 0 | 1 n n 0 0 1 0 0 0 | 1 * * * *
x.x-n/d-o.o&#x ♦ n 0 n | n 0 n 0 0 0 n | 1 0 0 n 0 0 0 0 1 | * 1 * * *
xxx ...&#x ♦ 2 2 2 | 1 2 2 1 0 2 1 | 0 1 0 1 2 0 1 0 0 | * * n * *
... oxo&#x ♦ 1 2 1 | 0 2 1 0 1 2 0 | 0 0 1 0 2 0 0 1 0 | * * * n *
.xx-n/d-.oo&#x ♦ 0 2n n | 0 0 0 n n 2n n | 0 0 0 0 0 1 n n 1 | * * * * 1
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