Acronym | n/d,2n/d-dipcup |
Name | n/d,2n/d-dip cupoliprism |
Circumradius | sqrt([7+4 cos(π d/n)-2 cos2(π d/n)]/[8-8 cos2(π d/n)]) |
Face vector | 4n2, 12n2, 12n2+6n, 4n2+10n, 4n+2 |
Especially | squatet (n=2,d=1)* pabexhix (n=3,d=1) |
Confer |
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These are all scaliform polytera. They even are convex if d=1.
* The case n/d = 2 (squatet) would be included here by concept. It just has a different incidence matrix, as the n/d-gons then become degenerate.
Incidence matrix according to Dynkin symbol
xx-n/d-xo xx-n/d-ox&#x (4/3 < n/d < 4) → height = sqrt[1 - 1/(2 sin2(π d/n))]
((2n/d,n/d)-dip || (n/d,2n/d)-dip)
o.-n/d-o. o.-n/d-o. | 2nn * | 1 1 2 2 0 0 0 | 1 2 2 1 2 2 2 1 0 0 0 0 | 2 1 1 2 2 1 2 1 1 0 0 0 | 1 2 1 1 1 0
.o-n/d-.o .o-n/d-.o | * 2nn | 0 0 0 2 2 1 1 | 0 0 0 0 2 1 2 2 1 2 2 1 | 0 0 0 1 2 2 1 1 2 1 1 2 | 0 1 1 2 1 1
-----------------------+---------+-------------------------+---------------------------------------+-------------------------------+------------
x. .. .. .. | 2 0 | nn * * * * * * | 1 2 0 0 2 0 0 0 0 0 0 0 | 2 1 0 2 2 1 0 0 0 0 0 0 | 1 2 1 1 0 0
.. x. .. .. | 2 0 | * nn * * * * * | 1 0 2 0 0 2 0 0 0 0 0 0 | 2 0 1 2 0 0 2 1 0 0 0 0 | 1 2 1 0 1 0
.. .. x. .. | 2 0 | * * 2nn * * * * | 0 1 1 1 0 0 1 0 0 0 0 0 | 1 1 1 0 1 0 1 0 1 0 0 0 | 1 1 0 1 1 0
oo-n/d-oo oo-n/d-oo&#x | 1 1 | * * * 4nn * * * | 0 0 0 0 1 1 1 1 0 0 0 0 | 0 0 0 1 1 1 1 1 1 0 0 0 | 0 1 1 1 1 0
.x .. .. .. | 0 2 | * * * * 2nn * * | 0 0 0 0 1 0 0 0 1 1 1 0 | 0 0 0 1 1 1 0 0 0 1 1 1 | 0 1 1 1 0 1
.. .. .x .. | 0 2 | * * * * * nn * | 0 0 0 0 0 0 2 0 0 2 0 1 | 0 0 0 0 2 0 1 0 2 1 0 2 | 0 1 0 2 1 1
.. .. .. .x | 0 2 | * * * * * * nn | 0 0 0 0 0 0 0 2 0 0 2 1 | 0 0 0 0 0 2 0 1 2 0 1 2 | 0 0 1 2 1 1
-----------------------+---------+-------------------------+---------------------------------------+-------------------------------+------------
x.-n/d-x. .. .. | 2n 0 | n n 0 0 0 0 0 | n * * * * * * * * * * * | 2 0 0 2 0 0 0 0 0 0 0 0 | 1 2 1 0 0 0
x. .. x. .. | 4 0 | 2 0 2 0 0 0 0 | * nn * * * * * * * * * * | 1 1 0 0 1 0 0 0 0 0 0 0 | 1 1 0 1 0 0
.. x. x. .. | 4 0 | 0 2 2 0 0 0 0 | * * nn * * * * * * * * * | 1 0 1 0 0 0 1 0 0 0 0 0 | 1 1 0 0 1 0
.. .. x.-n/d-o. | n 0 | 0 0 n 0 0 0 0 | * * * 2n * * * * * * * * | 0 1 1 0 0 0 0 0 1 0 0 0 | 1 0 0 1 1 0
xx .. .. ..&#x | 2 2 | 1 0 0 2 1 0 0 | * * * * 2nn * * * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 | 0 1 1 1 0 0
.. xo .. ..&#x | 2 1 | 0 1 0 2 0 0 0 | * * * * * 2nn * * * * * * | 0 0 0 1 0 0 1 1 0 0 0 0 | 0 1 1 0 1 0
.. .. xx ..&#x | 2 2 | 0 0 1 2 0 1 0 | * * * * * * 2nn * * * * * | 0 0 0 0 1 0 1 0 1 0 0 0 | 0 1 0 1 1 0
.. .. .. ox&#x | 1 2 | 0 0 0 2 0 0 1 | * * * * * * * 2nn * * * * | 0 0 0 0 0 1 0 1 1 0 0 0 | 0 0 1 1 1 0
.x-n/d-.o .. .. | 0 n | 0 0 0 0 n 0 0 | * * * * * * * * 2n * * * | 0 0 0 1 0 0 0 0 0 1 1 0 | 0 1 1 0 0 1
.x .. .x .. | 0 4 | 0 0 0 0 2 2 0 | * * * * * * * * * nn * * | 0 0 0 0 1 0 0 0 0 1 0 1 | 0 1 0 1 0 1
.x .. .. .x | 0 4 | 0 0 0 0 2 0 2 | * * * * * * * * * * nn * | 0 0 0 0 0 1 0 0 0 0 1 1 | 0 0 1 1 0 1
.. .. .x-n/d-.x | 0 2n | 0 0 0 0 0 n n | * * * * * * * * * * * n | 0 0 0 0 0 0 0 0 2 0 0 2 | 0 0 0 2 1 1
-----------------------+---------+-------------------------+---------------------------------------+-------------------------------+------------
x.-n/d-x. x. .. ♦ 4n 0 | 2n 2n 2n 0 0 0 0 | 2 n n 0 0 0 0 0 0 0 0 0 | n * * * * * * * * * * * | 1 1 0 0 0 0
x. .. x.-n/d-o. ♦ 2n 0 | n 0 2n 0 0 0 0 | 0 n 0 2 0 0 0 0 0 0 0 0 | * n * * * * * * * * * * | 1 0 0 1 0 0
.. x. x.-n/d-o. ♦ 2n 0 | 0 n 2n 0 0 0 0 | 0 0 n 2 0 0 0 0 0 0 0 0 | * * n * * * * * * * * * | 1 0 0 0 1 0
xx-n/d-xo .. ..&#x ♦ 2n n | n n 0 2n n 0 0 | 1 0 0 0 n n 0 0 1 0 0 0 | * * * 2n * * * * * * * * | 0 1 1 0 0 0
xx .. xx ..&#x ♦ 4 4 | 2 0 2 4 2 2 0 | 0 1 0 0 2 0 2 0 0 1 0 0 | * * * * nn * * * * * * * | 0 1 0 1 0 0
xx .. .. ox&#x ♦ 2 4 | 1 0 0 4 2 0 2 | 0 0 0 0 2 0 0 2 0 0 1 0 | * * * * * nn * * * * * * | 0 0 1 1 0 0
.. xo xx ..&#x ♦ 4 2 | 0 2 2 4 0 1 0 | 0 0 1 0 0 2 2 0 0 0 0 0 | * * * * * * nn * * * * * | 0 1 0 0 1 0
.. xo .. ox&#x ♦ 2 2 | 0 1 0 4 0 0 1 | 0 0 0 0 0 2 0 2 0 0 0 0 | * * * * * * * nn * * * * | 0 0 1 0 1 0
.. .. xx-n/d-ox&#x ♦ n 2n | 0 0 n 2n 0 n n | 0 0 0 1 0 0 n n 0 0 0 1 | * * * * * * * * 2n * * * | 0 0 0 1 1 0
.x-n/d-.o .x .. ♦ 0 2n | 0 0 0 0 2n n 0 | 0 0 0 0 0 0 0 0 2 n 0 0 | * * * * * * * * * n * * | 0 1 0 0 0 1
.x-n/d-.o .. .x ♦ 0 2n | 0 0 0 0 2n 0 n | 0 0 0 0 0 0 0 0 2 0 n 0 | * * * * * * * * * * n * | 0 0 1 0 0 1
.x .. .x-n/d-.x ♦ 0 4n | 0 0 0 0 2n 2n 2n | 0 0 0 0 0 0 0 0 0 n n 2 | * * * * * * * * * * * n | 0 0 0 1 0 1
-----------------------+---------+-------------------------+---------------------------------------+-------------------------------+------------
x.-n/d-x. x.-n/d-o. ♦ 2nn 0 | nn nn 2nn 0 0 0 0 | n nn nn 2n 0 0 0 0 0 0 0 0 | n n n 0 0 0 0 0 0 0 0 0 | 1 * * * * *
xx-n/d-xo xx ..&#x ♦ 4n 2n | 2n 2n 2n 4n 2n n 0 | 2 n n 0 2n 2n 2n 0 2 n 0 0 | 1 0 0 2 n 0 n 0 0 1 0 0 | * n * * * *
xx-n/d-xo .. ox&#x ♦ 2n 2n | n n 0 4n 2n 0 n | 1 0 0 0 2n 2n 0 2n 2 0 n 0 | 0 0 0 2 0 n 0 n 0 0 1 0 | * * n * * *
xx .. xx-n/d-ox&#x ♦ 2n 4n | n 0 2n 4n 2n 2n 2n | 0 n 0 2 2n 0 2n 2n 0 n n 2 | 0 1 0 0 n n 0 0 2 0 0 1 | * * * n * *
.. xo xx-n/d-ox&#x ♦ 2n 2n | 0 n 2n 4n 0 n n | 0 0 n 2 0 2n 2n 2n 0 0 0 1 | 0 0 1 0 0 0 n n 2 0 0 0 | * * * * n *
.x-n/d-.o .x-n/d-.x ♦ 0 2nn | 0 0 0 0 2nn nn nn | 0 0 0 0 0 0 0 0 2n nn nn n | 0 0 0 0 0 0 0 0 0 n n n | * * * * * 1
or o.-n/d-o. o.-n/d-o. & | 4nn | 1 1 2 2 | 1 2 2 1 4 3 | 2 1 1 3 2 3 1 | 1 3 2 -------------------------+-----+-----------------+-----------------------+-----------------------+-------- x. .. .. .. & | 2 | 2nn * * * | 1 2 0 0 2 0 | 2 1 0 2 2 1 0 | 1 3 1 .. x. .. .. & | 2 | * 2nn * * | 1 0 2 0 0 1 | 2 0 1 2 0 2 1 | 1 2 2 .. .. x. .. & | 2 | * * 4nn * | 0 1 1 1 1 0 | 1 1 1 1 1 1 0 | 1 2 1 oo-n/d-oo oo-n/d-oo&#x | 2 | * * * 4nn | 0 0 0 0 2 2 | 0 0 0 2 1 2 1 | 0 2 2 -------------------------+-----+-----------------+-----------------------+-----------------------+-------- x.-n/d-x. .. .. & | 2n | n n 0 0 | 2n * * * * * | 2 0 0 2 0 0 0 | 1 2 1 x. .. x. .. & | 4 | 2 0 2 0 | * 2nn * * * * | 1 1 0 0 1 0 0 | 1 2 0 .. x. x. .. & | 4 | 0 2 2 0 | * * 2nn * * * | 1 0 1 0 0 1 0 | 1 1 1 .. .. x.-n/d-o. & | n | 0 0 n 0 | * * * 4n * * | 0 1 1 1 0 0 0 | 1 1 1 xx .. .. ..&#x & | 4 | 1 0 1 2 | * * * * 4nn * | 0 0 0 1 1 1 0 | 0 2 1 .. xo .. ..&#x & | 3 | 0 1 0 2 | * * * * * 4nn | 0 0 0 1 0 1 1 | 0 1 2 -------------------------+-----+-----------------+-----------------------+-----------------------+-------- x.-n/d-x. x. .. & ♦ 4n | 2n 2n 2n 0 | 2 n n 0 0 0 | 2n * * * * * * | 1 1 0 x. .. x.-n/d-o. & ♦ 2n | n 0 2n 0 | 0 n 0 2 0 0 | * 2n * * * * * | 1 1 0 .. x. x.-n/d-o. & ♦ 2n | 0 n 2n 0 | 0 0 n 2 0 0 | * * 2n * * * * | 1 0 1 xx-n/d-xo .. ..&#x & ♦ 3n | n n n 2n | 1 0 0 1 n n | * * * 4n * * * | 0 1 1 xx .. xx ..&#x ♦ 8 | 4 0 4 4 | 0 2 0 0 4 0 | * * * * nn * * | 0 2 0 xx .. .. ox&#x & ♦ 6 | 1 2 2 4 | 0 0 1 0 2 2 | * * * * * 2nn * | 0 1 1 .. xo .. ox&#x ♦ 4 | 0 2 0 4 | 0 0 0 0 0 4 | * * * * * * nn | 0 0 2 -------------------------+-----+-----------------+-----------------------+-----------------------+-------- x.-n/d-x. x.-n/d-o. & ♦ 2nn | nn nn 2nn 0 | n nn nn 2n 0 0 | n n n 0 0 0 0 | 2 * * xx-n/d-xo xx ..&#x & ♦ 6n | 3n 2n 4n 4n | 2 2n n 2 4n 2n | 1 1 0 2 n n 0 | * 2n * xx-n/d-xo .. ox&#x & ♦ 4n | n 2n 2n 4n | 1 0 n 2 2n 4n | 0 0 1 2 0 n n | * * 2n
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