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Acronym
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n/d-daf
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Name
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n/d-duoantifastegium,
(2,n)-duoantiwedge,
n/d-p atop bidual n/d-p
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Circumradius
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sqrt([3+1/(1-cos(pi d/n))]/8)
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Lace hyper city
in approx. ASCII-art
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x-n/d-o
x-n/d-o
o-n/d-x
o-n/d-x
(disphenoid in configuration space)
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Face vector
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4n, 14n, 18n+4, 10n+6, 2n+4
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Especially
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| triddaf |
squiddaf |
piddaf |
hiddaf |
hedaf |
odaf |
edaf |
deddaf |
| |
- |
steddaf
(stiddaf,
stardaf) |
- |
sithedaf |
- |
stendaf |
- |
| |
|
|
- |
gisthedaf |
stoddaf |
- |
stidadaf |
| |
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- |
gistendaf |
- |
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Confer
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- more general:
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(n,m)-dafup
- general polytopal classes:
-
segmentotera
scaliform
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External
links
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These all are scaliform polytera. They even are convex if d=1.
–
The boundary case n/d=2 not only would provide different incidences than the general matrix given below (degenerate {n/d}-gons), it even itself would become dimensionally
degenerate.
In fact it happens to be {4} || perp {4}, the hull of which is just hex.
Incidence matrix according to Dynkin symbol
xo ox xo-n/d-ox&#x (n > 2d) → height = sqrt([1-1/(1+cos(π d/n))]/2)
(n/d-p || bidual n/d-p)
o. o. o.-n/d-o. | 2n * | 1 2 4 0 0 | 2 1 4 2 4 2 0 0 | 1 2 4 2 2 1 2 0 | 2 1 2 1
.o .o .o-n/d-.o | * 2n | 0 0 4 1 2 | 0 0 2 4 2 4 2 1 | 0 2 1 2 2 4 2 1 | 1 2 1 2
-------------------+-------+--------------+---------------------+----------------------+--------
x. .. .. .. | 2 0 | n * * * * | 2 0 4 0 0 0 0 0 | 1 2 4 2 0 0 0 0 | 2 1 2 0
.. .. x. .. | 2 0 | * 2n * * * | 1 1 0 0 2 0 0 0 | 1 0 2 0 1 0 2 0 | 1 0 2 1
oo oo oo-n/d-oo&#x | 1 1 | * * 8n * * | 0 0 1 1 1 1 0 0 | 0 1 1 1 1 1 1 0 | 1 1 1 1
.. .x .. .. | 0 2 | * * * n * | 0 0 0 4 0 0 2 0 | 0 2 0 0 2 4 0 1 | 1 2 0 2
.. .. .. .x | 0 2 | * * * * 2n | 0 0 0 0 0 2 1 1 | 0 0 0 1 0 2 2 1 | 0 1 1 2
-------------------+-------+--------------+---------------------+----------------------+--------
x. .. x. .. | 4 0 | 2 2 0 0 0 | n * * * * * * * | 1 0 2 0 0 0 0 0 | 1 0 2 0
.. .. x.-n/d-o. | n 0 | 0 n 0 0 0 | * 2 * * * * * * | 1 0 0 0 0 0 2 0 | 0 0 2 1
xo .. .. ..&#x | 2 1 | 1 0 2 0 0 | * * 4n * * * * * | 0 1 1 1 0 0 0 0 | 1 1 1 0
.. ox .. ..&#x | 1 2 | 0 0 2 1 0 | * * * 4n * * * * | 0 1 0 0 1 1 0 0 | 1 1 0 1
.. .. xo ..&#x | 2 1 | 0 1 2 0 0 | * * * * 4n * * * | 0 0 1 0 1 0 1 0 | 1 0 1 1
.. .. .. ox&#x | 1 2 | 0 0 2 0 1 | * * * * * 4n * * | 0 0 0 1 0 1 1 0 | 0 1 1 1
.. .x .. .x | 0 4 | 0 0 0 2 2 | * * * * * * n * | 0 0 0 0 0 2 0 1 | 0 1 0 2
.. .. .o-n/d-.x | 0 n | 0 0 0 0 n | * * * * * * * 2 | 0 0 0 0 0 0 2 1 | 0 0 1 2
-------------------+-------+--------------+---------------------+----------------------+--------
x. .. x.-n/d-o. ♦ 2n 0 | n 2n 0 0 0 | n 2 0 0 0 0 0 0 | 1 * * * * * * * | 0 0 2 0
xo ox .. ..&#x ♦ 2 2 | 1 0 4 1 0 | 0 0 2 2 0 0 0 0 | * 2n * * * * * * | 1 1 0 0
xo .. xo ..&#x ♦ 4 1 | 2 2 4 0 0 | 1 0 2 0 2 0 0 0 | * * 2n * * * * * | 1 0 1 0
xo .. .. ox&#x ♦ 2 2 | 1 0 4 0 1 | 0 0 2 0 0 2 0 0 | * * * 2n * * * * | 0 1 1 0
.. ox xo ..&#x ♦ 2 2 | 0 1 4 1 0 | 0 0 0 2 2 0 0 0 | * * * * 2n * * * | 1 0 0 1
.. ox .. ox&#x ♦ 1 4 | 0 0 4 2 2 | 0 0 0 2 0 2 1 0 | * * * * * 2n * * | 0 1 0 1
.. .. xo-n/d-ox&#x ♦ n n | 0 n 2n 0 n | 0 1 0 0 n n 0 1 | * * * * * * 4 * | 0 0 1 1
.. .x .o-n/d-.x ♦ 0 2n | 0 0 0 n 2n | 0 0 0 0 0 0 n 2 | * * * * * * * 1 | 0 0 0 2
-------------------+-------+--------------+---------------------+----------------------+--------
xo ox xo ..&#x ♦ 4 2 | 2 2 8 1 0 | 1 0 4 4 4 0 0 0 | 0 2 2 0 2 0 0 0 | n * * *
xo ox .. ox&#x ♦ 2 4 | 1 0 8 2 2 | 0 0 4 4 0 4 1 0 | 0 2 0 2 0 2 0 0 | * n * *
xo .. xo-n/d-ox&#x ♦ 2n n | n 2n 4n 0 n | n 2 2n 0 2n 2n 0 1 | 1 0 n n 0 0 2 0 | * * 2 *
.. ox xo-n/d-ox&#x ♦ n 2n | 0 n 4n n 2n | 0 1 0 2n 2n 2n n 2 | 0 0 0 0 n n 2 1 | * * * 2
or
o. o. o.-n/d-o. & | 4n | 1 2 4 | 2 1 6 6 | 1 2 5 4 2 | 3 3
---------------------+----+----------+------------+--------------+-----
x. .. .. .. & | 2 | 2n * * | 2 0 4 0 | 1 2 4 2 0 | 3 2
.. .. x. .. & | 2 | * 4n * | 1 1 0 2 | 1 0 2 1 2 | 1 3
oo oo oo-n/d-oo&#x | 2 | * * 8n | 0 0 2 2 | 0 1 2 2 1 | 2 2
---------------------+----+----------+------------+--------------+-----
x. .. x. .. & | 4 | 2 2 0 | 2n * * * | 1 0 2 0 0 | 1 2
.. .. x.-n/d-o. & | n | 0 n 0 | * 4 * * | 1 0 0 0 2 | 0 3
xo .. .. ..&#x & | 3 | 1 0 2 | * * 8n * | 0 1 1 1 0 | 2 1
.. .. xo ..&#x & | 3 | 0 1 2 | * * * 8n | 0 0 1 1 1 | 1 2
---------------------+----+----------+------------+--------------+-----
x. .. x.-n/d-o. & ♦ 2n | n 2n 0 | n 2 0 0 | 2 * * * * | 0 2
xo ox .. ..&#x ♦ 4 | 2 0 4 | 0 0 4 0 | * 2n * * * | 2 0
xo .. xo ..&#x & ♦ 5 | 2 2 4 | 1 0 2 2 | * * 4n * * | 1 1
xo .. .. ox&#x & ♦ 4 | 1 1 4 | 0 0 2 2 | * * * 4n * | 1 1
.. .. xo-n/d-ox&#x ♦ 2n | 0 2n 2n | 0 2 0 2n | * * * * 4 | 0 2
---------------------+----+----------+------------+--------------+-----
xo ox xo ..&#x & ♦ 6 | 3 2 8 | 1 0 8 4 | 0 2 2 2 0 | 2n *
xo .. xo-n/d-ox&#x & ♦ 3n | n 3n 4n | n n 2n 4n | 1 0 n n 2 | * 4
xxoo-n/d-ooxx&#x (n > 2d) → height(1,2) = height(3,4) = 1
height(1,3) = height(1,4) = height(2,3) = height(2,4) = sqrt[(1+2 cos(π d/n))/(2+2 cos(π d/n))]
o...-n/d-o... | n * * * | 2 1 2 2 0 0 0 0 0 0 | 1 2 2 1 2 1 2 2 2 0 0 0 0 0 0 0 0 0 | 1 1 1 2 1 2 1 2 1 2 0 0 0 0 0 | 1 1 1 2 1 0
.o..-n/d-.o.. | * n * * | 0 1 0 0 2 2 2 0 0 0 | 0 2 0 0 0 0 2 2 0 1 2 1 2 1 2 0 0 0 | 1 0 0 2 1 2 1 0 0 2 1 1 2 1 0 | 1 1 0 2 1 1
..o.-n/d-..o. | * * n * | 0 0 2 0 0 2 0 2 1 0 | 0 0 1 2 0 0 2 0 2 0 1 2 0 0 2 1 2 0 | 0 1 0 1 2 0 0 1 2 2 1 0 1 2 1 | 1 0 1 1 2 1
...o-n/d-...o | * * * n | 0 0 0 2 0 0 2 0 1 2 | 0 0 0 0 1 2 0 2 2 0 0 0 1 2 2 0 2 1 | 0 0 1 0 0 1 2 1 2 2 0 1 1 2 1 | 0 1 1 1 2 1
-----------------+---------+-------------------------+-----------------------------------------+--------------------------------+------------
x... .... | 2 0 0 0 | n * * * * * * * * * | 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 | 1 1 1 1 0 0
oo..-n/d-oo..&#x | 1 1 0 0 | * n * * * * * * * * | 0 2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 | 1 0 0 2 1 2 1 0 0 2 0 0 0 0 0 | 1 1 0 2 1 0
o.o.-n/d-o.o.&#x | 1 0 1 0 | * * 2n * * * * * * * | 0 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 | 0 1 0 1 1 0 0 1 1 1 0 0 0 0 0 | 1 0 1 1 1 0
o..o-n/d-o..o&#x | 1 0 0 1 | * * * 2n * * * * * * | 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 | 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 | 0 1 1 1 1 0
.x.. .... | 0 2 0 0 | * * * * n * * * * * | 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 | 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 | 1 1 0 1 0 1
.oo.-n/d-.oo.&#x | 0 1 1 0 | * * * * * 2n * * * * | 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 | 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 | 1 0 0 1 1 1
.o.o-n/d-.o.o&#x | 0 1 0 1 | * * * * * * 2n * * * | 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 | 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 | 0 1 0 1 1 1
.... ..x. | 0 0 2 0 | * * * * * * * n * * | 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 | 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 | 1 0 1 0 1 1
..oo-n/d-..oo&#x | 0 0 1 1 | * * * * * * * * n * | 0 0 0 0 0 0 0 0 2 0 0 0 0 0 2 0 2 0 | 0 0 0 0 0 0 0 1 2 2 0 0 1 2 1 | 0 0 1 1 2 1
.... ...x | 0 0 0 2 | * * * * * * * * * n | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 | 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1 | 0 1 1 0 1 1
-----------------+---------+-------------------------+-----------------------------------------+--------------------------------+------------
x...-n/d-o... | n 0 0 0 | n 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0
xx.. ....&#x | 2 2 0 0 | 1 2 0 0 1 0 0 0 0 0 | * n * * * * * * * * * * * * * * * * | 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 | 1 1 0 1 0 0
x.o. ....&#x | 2 0 1 0 | 1 0 2 0 0 0 0 0 0 0 | * * n * * * * * * * * * * * * * * * | 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 | 1 0 1 1 0 0
.... o.x.&#x | 1 0 2 0 | 0 0 2 0 0 0 0 1 0 0 | * * * n * * * * * * * * * * * * * * | 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 | 1 0 1 0 1 0
x..o ....&#x | 2 0 0 1 | 1 0 0 2 0 0 0 0 0 0 | * * * * n * * * * * * * * * * * * * | 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 | 0 1 1 1 0 0
.... o..x&#x | 1 0 0 2 | 0 0 0 2 0 0 0 0 0 1 | * * * * * n * * * * * * * * * * * * | 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 | 0 1 1 0 1 0
ooo.-n/d-ooo.&#x | 1 1 1 0 | 0 1 1 0 0 1 0 0 0 0 | * * * * * * 2n * * * * * * * * * * * | 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 | 1 0 0 1 1 0
oo.o-n/d-oo.o&#x | 1 1 0 1 | 0 1 0 1 0 0 1 0 0 0 | * * * * * * * 2n * * * * * * * * * * | 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 | 0 1 0 1 1 0
o.oo-n/d-o.oo&#x | 1 0 1 1 | 0 0 1 1 0 0 0 0 1 0 | * * * * * * * * 2n * * * * * * * * * | 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 | 0 0 1 1 1 0
.x..-n/d-.o.. | 0 n 0 0 | 0 0 0 0 n 0 0 0 0 0 | * * * * * * * * * 1 * * * * * * * * | 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 | 1 1 0 0 0 1
.xo. ....&#x | 0 2 1 0 | 0 0 0 0 1 2 0 0 0 0 | * * * * * * * * * * n * * * * * * * | 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 | 1 0 0 1 0 1
.... .ox.&#x | 0 1 2 0 | 0 0 0 0 0 2 0 1 0 0 | * * * * * * * * * * * n * * * * * * | 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 | 1 0 0 0 1 1
.x.o ....&#x | 0 2 0 1 | 0 0 0 0 1 0 2 0 0 0 | * * * * * * * * * * * * n * * * * * | 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 | 0 1 0 1 0 1
.... .o.x&#x | 0 1 0 2 | 0 0 0 0 0 0 2 0 0 1 | * * * * * * * * * * * * * n * * * * | 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 | 0 1 0 0 1 1
.ooo-n/d-.ooo&#x | 0 1 1 1 | 0 0 0 0 0 1 1 0 1 0 | * * * * * * * * * * * * * * 2n * * * | 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 | 0 0 0 1 1 1
..o.-n/d-..x. | 0 0 n 0 | 0 0 0 0 0 0 0 n 0 0 | * * * * * * * * * * * * * * * 1 * * | 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 | 1 0 1 0 0 1
.... ..xx&#x | 0 0 2 2 | 0 0 0 0 0 0 0 1 2 1 | * * * * * * * * * * * * * * * * n * | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 | 0 0 1 0 1 1
...o-n/d-...x | 0 0 0 n | 0 0 0 0 0 0 0 0 0 n | * * * * * * * * * * * * * * * * * 1 | 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 | 0 1 1 0 0 1
-----------------+---------+-------------------------+-----------------------------------------+--------------------------------+------------
xx..-n/d-oo..&#x ♦ n n 0 0 | n n 0 0 n 0 0 0 0 0 | 1 n 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * * * | 1 1 0 0 0 0
x.o.-n/d-o.x.&#x ♦ n 0 n 0 | n 0 2n 0 0 0 0 n 0 0 | 1 0 n n 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | * 1 * * * * * * * * * * * * * | 1 0 1 0 0 0
x..o-n/d-o..x&#x ♦ n 0 0 n | n 0 0 2n 0 0 0 0 0 n | 1 0 0 0 n n 0 0 0 0 0 0 0 0 0 0 0 1 | * * 1 * * * * * * * * * * * * | 0 1 1 0 0 0
xxo. ....&#x ♦ 2 2 1 0 | 1 2 2 0 1 2 0 0 0 0 | 0 1 1 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 | * * * n * * * * * * * * * * * | 1 0 0 1 0 0
.... oox.&#x ♦ 1 1 2 0 | 0 1 2 0 0 2 0 1 0 0 | 0 0 0 1 0 0 2 0 0 0 0 1 0 0 0 0 0 0 | * * * * n * * * * * * * * * * | 1 0 0 0 1 0
xx.o ....&#x ♦ 2 2 0 1 | 1 2 0 2 1 0 2 0 0 0 | 0 1 0 0 1 0 0 2 0 0 0 0 1 0 0 0 0 0 | * * * * * n * * * * * * * * * | 0 1 0 1 0 0
.... oo.x&#x ♦ 1 1 0 2 | 0 1 0 2 0 0 2 0 0 1 | 0 0 0 0 0 1 0 2 0 0 0 0 0 1 0 0 0 0 | * * * * * * n * * * * * * * * | 0 1 0 0 1 0
x.oo ....&#x ♦ 2 0 1 1 | 1 0 2 2 0 0 0 0 1 0 | 0 0 1 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 | * * * * * * * n * * * * * * * | 0 0 1 1 0 0
.... o.xx&#x ♦ 1 0 2 2 | 0 0 2 2 0 0 0 1 2 1 | 0 0 0 1 0 1 0 0 2 0 0 0 0 0 0 0 1 0 | * * * * * * * * n * * * * * * | 0 0 1 0 1 0
oooo-n/d-oooo&#x ♦ 1 1 1 1 | 0 1 1 1 0 1 1 0 1 0 | 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 | * * * * * * * * * 2n * * * * * | 0 0 0 1 1 0
.xo.-n/d-.ox.&#x ♦ 0 n n 0 | 0 0 0 0 n 2n 0 n 0 0 | 0 0 0 0 0 0 0 0 0 1 n n 0 0 0 1 0 0 | * * * * * * * * * * 1 * * * * | 1 0 0 0 0 1
.x.o-n/d-.o.x&#x ♦ 0 n 0 n | 0 0 0 0 n 0 2n 0 0 n | 0 0 0 0 0 0 0 0 0 1 0 0 n n 0 0 0 1 | * * * * * * * * * * * 1 * * * | 0 1 0 0 0 1
.xoo ....&#x ♦ 0 2 1 1 | 0 0 0 0 1 2 2 0 1 0 | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 2 0 0 0 | * * * * * * * * * * * * n * * | 0 0 0 1 0 1
.... .oxx&#x ♦ 0 1 2 2 | 0 0 0 0 0 2 2 1 2 1 | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 0 1 0 | * * * * * * * * * * * * * n * | 0 0 0 0 1 1
..oo-n/d-..xx&#x ♦ 0 0 n n | 0 0 0 0 0 0 0 n n n | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n 1 | * * * * * * * * * * * * * * 1 | 0 0 1 0 0 1
-----------------+---------+-------------------------+-----------------------------------------+--------------------------------+------------
xxo.-n/d-oox.&#x ♦ n n n 0 | n n 2n 0 n 2n 0 n 0 0 | 1 n n n 0 0 2n 0 0 1 n n 0 0 0 1 0 0 | 1 1 0 n n 0 0 0 0 0 1 0 0 0 0 | 1 * * * * *
xx.o-n/d-oo.x&#x ♦ n n 0 n | n n 0 2n n 0 2n 0 0 n | 1 n 0 0 n n 0 2n 0 1 0 0 n n 0 0 0 1 | 1 0 1 0 0 n n 0 0 0 0 1 0 0 0 | * 1 * * * *
x.oo-n/d-o.xx&#x ♦ n 0 n n | n 0 2n 2n 0 0 0 n n n | 1 0 n n n n 0 0 2n 0 0 0 0 0 0 1 n 1 | 0 1 1 0 0 0 0 n n 0 0 0 0 0 1 | * * 1 * * *
xxoo ....&#x ♦ 2 2 1 1 | 1 2 2 2 1 2 2 0 1 0 | 0 1 1 0 1 0 2 2 2 0 1 0 1 0 2 0 0 0 | 0 0 0 1 0 1 0 1 0 2 0 0 1 0 0 | * * * n * *
.... ooxx&#x ♦ 1 1 2 2 | 0 1 2 2 0 2 2 1 2 1 | 0 0 0 1 0 1 2 2 2 0 0 1 0 1 2 0 1 0 | 0 0 0 0 1 0 1 0 1 2 0 0 0 1 0 | * * * * n *
.xoo-n/d-.oxx&#x ♦ 0 n n n | 0 0 0 0 n 2n 2n n n n | 0 0 0 0 0 0 0 0 0 1 n n n n 2n 1 n 1 | 0 0 0 0 0 0 0 0 0 0 1 1 n n 1 | * * * * * 1