n/d-dow

Acronym

n/d-dow

Name

n/d-grammal disphenoid,
n/d-grammal biwedge,
{n/d} || perp {n/d}

Circumradius

sqrt[1+1/(2 sin²(π d/n))]/2

Lace hyper city
in approx. ASCII-art

{n/d}

        o-n/d-o        
                       
                       
o-n/d-o         o-n/d-o
                       
                       
                       
                       
   o-n/d-o   o-n/d-o

perp {n/d} – example for n=5

Dual

(selfdual)

Face vector

2n, n²+2n, 2n²+2, n²+2n, 2n

Especially

hix (n=3,d=1) stadow (n=5,d=2) shadow (n=7,d=2) ogdow (n=8,d=3)

Confer

more general:: (n/d, m/b)-disphenoids
general polytopal classes:: scaliform

These self-dual polytera are obtained as the pyramid product of 2 n/d-grams.

In order to become at least scaliform, i.e. the lacing edges shall have the same size as those of the pair of offset perpendicular bases, there is the following restriction to the rational number n/d of the polygram: n < 4d < 2n. The first inequality here is derived from the below given height formula, the second is trivial.

Right this restriction then shows that the only convex case, i.e. using d = 1, would occur for n = 3, which then is even more symmetrical, in fact regular: hix.

Incidence matrix according to Dynkin symbol

xo-n/d-oo ox-n/d-oo&#x   → height = sqrt[1-1/(2 sin²(π d/n))]
                           n < 4d < 2n
(pyramid product of 2 {n/d}s)

o.-n/d-o. o.-n/d-o.    | n * ♦ 2  n 0 | 1 2n  n 0 | n 2n 1 | n 2
.o-n/d-.o .o-n/d-.o    | * n ♦ 0  n 2 | 0  n 2n 1 | 1 2n n | 2 n
-----------------------+-----+--------+-----------+--------+----
x.     .. ..     ..    | 2 0 | n  * * ♦ 1  n  0 0 | n  n 0 | n 1
oo-n/d-oo oo-n/d-oo&#x | 1 1 | * nn * ♦ 0  2  2 0 | 1  4 1 | 2 2
..     .. .x     ..    | 0 2 | *  * n ♦ 0  0  n 1 | 0  n n | 1 n
-----------------------+-----+--------+-----------+--------+----
x.-n/d-o. ..     ..    | n 0 | n  0 0 | 1  *  * * | n  0 0 | n 0
xo     .. ..     ..&#x | 2 1 | 1  2 0 | * nn  * * | 1  2 0 | 2 1
..     .. ox     ..&#x | 1 2 | 0  2 1 | *  * nn * | 0  2 1 | 1 2
..     .. .x-n/d-.o    | 0 n | 0  0 n | *  *  * 1 | 0  0 n | 0 n
-----------------------+-----+--------+-----------+--------+----
xo-n/d-oo ..     ..&#x ♦ n 1 | n  n 0 | 1  n  0 0 | n  * * | 2 0
xo     .. ox     ..&#x ♦ 2 2 | 1  4 1 | 0  2  2 0 | * nn * | 1 1
..     .. ox-n/d-oo&#x ♦ 1 n | 0  n n | 0  0  n 1 | *  * n | 0 2
-----------------------+-----+--------+-----------+--------+----
xo-n/d-oo ox     ..&#x ♦ n 2 | n 2n 1 | 1 2n  n 0 | 2  n 0 | n *
xo     .. ox-n/d-oo&#x ♦ 2 n | 1 2n n | 0  n 2n 1 | 0  n 2 | * n

or
o.-n/d-o. o.-n/d-o.    & |  2n ♦   2  n | 1  3n | n+1 2n | n+2
-------------------------+-----+--------+-------+--------+----
x.     .. ..     ..    & |   2 |  2n  * ♦ 1   n |   n  n | n+1
oo-n/d-oo oo-n/d-oo&#x   |   2 |   * nn ♦ 0   4 |   2  4 |   4
-------------------------+-----+--------+-------+--------+----
x.-n/d-o. ..     ..    & |   n |   n  0 | 2   * |   n  0 |   n
xo     .. ..     ..&#x & |   3 |   1  2 | * 2nn |   1  2 |   3
-------------------------+-----+--------+-------+--------+----
xo-n/d-oo ..     ..&#x & ♦ n+1 |   n  n | 1   n |  2n  * |   2
xo     .. ox     ..&#x   ♦   4 |   2  4 | 0   4 |   * nn |   2
-------------------------+-----+--------+-------+--------+----
xo-n/d-oo ox     ..&#x & ♦ n+2 | n+1 2n | 1  3n |   2  n |  2n

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