Acronym n/d-appip
Name n/d-antiprism prism
Circumradius sqrt[(5-4 cos(π d/n))/(8-8 cos(π d/n))]
Especially n-appip (d=1)
3/d 4/d 5/d 6/d 7/d 8/d 9/d 10/d {n/d}-appip ope squappip pappip happip heappip oappip eappip dappip * stappip * shappip * steappip * starpip * gishappip stoappip * stiddappip gisharpip * gisteappip * storpip gisterpip *
*: Grünbaumian
External

n=2, d=1 would result in the tepe. But for sure the top resp. bottom face prisms then become degenerate and hence that one needs a different incidence matrix.

The height-formula of the n/d-ap implies d<2n/3.

Note that for d odd the 2 n/d-p layers are aligned in a gyrated way, which then effects that top and bottom vertices are situated on gap. While for d even the 2 n/d-p layers are still aligned in a gyrated way, but this time it effects in a seeming ungyrated parallel alignment, just like in the mere 4,n/d-dip, but now with crossed lacings, so that the according height becomes lesser than 1 here.

Incidence matrix according to Dynkin symbol

```x s-2-s-n/d-s   (n>2)

. demi( .   .     .   ) | 4n |  1  1  1  2 | 1 1 1  2  3 | 1 1  3
------------------------+----+-------------+-------------+-------
.       s-2-s     .     |  2 | 2n  *  *  * | 1 0 0  0  2 | 0 1  2
.       s   .     s2*b  |  2 |  * 2n  *  * | 0 1 0  0  2 | 0 1  2
x demi( .   .     .   ) |  2 |  *  * 2n  * | 1 1 0  2  0 | 1 0  3
. sefa( .   s-n/d-s   ) |  2 |  *  *  * 4n | 0 0 1  1  1 | 1 1  1
------------------------+----+-------------+-------------+-------
x       s-2-s     .     |  4 |  2  0  2  0 | n * *  *  * | 0 0  2
x       s   .     s2*b  |  4 |  0  2  2  0 | * n *  *  * | 0 0  2
.       .   s-n/d-s     ♦  n |  0  0  0  n | * * 4  *  * | 1 1  0
x sefa( .   s-n/d-s   ) |  4 |  0  0  2  2 | * * * 2n  * | 1 0  1
. sefa( s-2-s-n/d-s   ) |  3 |  1  1  0  1 | * * *  * 4n | 0 1  1
------------------------+----+-------------+-------------+-------
x       .   s-n/d-s     ♦ 2n |  0  0  n 2n | 0 0 2  n  0 | 2 *  *
.       s-2-s-n/d-s     ♦ 2n |  n  n  0 2n | 0 0 2  0 2n | * 2  *
x sefa( s-2-s-n/d-s   ) ♦  6 |  2  2  3  2 | 1 1 0  1  2 | * * 2n
```

```x s-2-s-2n/d-o   (n>2)

. demi( .   .      . ) | 4n |  2  1  2 |  2 1  2  3 | 1 1  3
-----------------------+----+----------+------------+-------
.       s-2-s      .   |  2 | 4n  *  * |  1 0  0  2 | 0 1  2
x demi( .   .      . ) |  2 |  * 2n  * |  2 0  2  0 | 1 0  3
. sefa( .   s-2n/d-o ) |  2 |  *  * 4n |  0 1  1  1 | 1 1  1
-----------------------+----+----------+------------+-------
x       s-2-s      .   |  4 |  2  2  0 | 2n *  *  * | 0 0  2
.       .   s-2n/d-o   ♦  n |  0  0  n |  * 4  *  * | 1 1  0
x sefa( .   s-2n/d-o ) |  4 |  0  2  2 |  * * 2n  * | 1 0  1
. sefa( s-2-s-2n/d-o ) |  3 |  2  0  1 |  * *  * 4n | 0 1  1
-----------------------+----+----------+------------+-------
x       .   s-2n/d-o   ♦ 2n |  0  n 2n |  0 2  n  0 | 2 *  *
.       s-2-s-2n/d-o   ♦ 2n | 2n  0 2n |  0 2  0 2n | * 2  *
x sefa( s-2-s-2n/d-o ) ♦  6 |  4  3  2 |  2 0  1  2 | * * 2n
```

```xx xo-n/d-ox&#x   (n>2)   → height = sqrt[(1+2*cos(πd/n))/(2+2*cos(πd/n))]
({n/d}-p || gyro {n/d}-p)

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

```s-2-s-n/d-s || s-2-s-n/d-s   (n>2)   → height = 1
({n/d}-ap || {n/d}-ap)

demi( .   .     . )                       | 2n  * | 1 1  2  1 0 0  0 | 1  3 1 1  2 0  0 | 1 1  3 0
demi( .   .     . ) |  * 2n | 0 0  0  1 1 1  2 | 0  0 1 1  2 1  3 | 0 1  3 1
------------------------------------------+-------+------------------+------------------+---------
s-2-s     .                         |  2  0 | n *  *  * * *  * | 0  2 1 0  0 0  0 | 1 0  1 0
s   2     s                         |  2  0 | * n  *  * * *  * | 0  2 0 1  0 0  0 | 1 0  1 0
sefa( .   s-n/d-s )                       |  2  0 | * * 2n  * * *  * | 1  1 0 0  1 0  0 | 1 1  1 0
demi( .   .     . ) || demi( .   .     . ) |  1  1 | * *  * 2n * *  * | 0  0 1 1  2 0  0 | 0 1  3 0
s-2-s     .   |  0  2 | * *  *  * n *  * | 0  0 1 0  0 0  2 | 0 0  2 1
s   2     s   |  0  2 | * *  *  * * n  * | 0  0 0 1  0 0  2 | 0 0  2 1
sefa( .   s-n/d-s ) |  0  2 | * *  *  * * * 2n | 0  0 0 0  1 1  1 | 0 1  1 1
------------------------------------------+-------+------------------+------------------+---------
.   s-n/d-s                         ♦  n  0 | 0 0  n  0 0 0  0 | 2  * * *  * *  * | 1 1  0 0
sefa( s-2-s-n/d-s )                       |  3  0 | 1 1  1  0 0 0  0 | * 2n * *  * *  * | 1 0  1 0
s-2-s     .   ||       s-2-s     .   |  2  2 | 1 0  0  2 1 0  0 | *  * n *  * *  * | 0 0  2 0
s   2     s   ||       s   2     s   |  2  2 | 0 1  0  2 0 1  0 | *  * * n  * *  * | 0 0  2 0
sefa( .   s-n/d-s ) || sefa( .   s-n/d-s ) |  2  2 | 0 0  1  2 0 0  1 | *  * * * 2n *  * | 0 1  1 0
.   s-n/d-s   ♦  0  n | 0 0  0  0 0 0  n | *  * * *  * 2  * | 0 1  0 1
sefa( s-2-s-n/d-s ) |  0  3 | 0 0  0  0 1 1  1 | *  * * *  * * 2n | 0 0  1 1
------------------------------------------+-------+------------------+------------------+---------
s-2-s-n/d-s                         ♦ 2n  0 | n n 2n  0 0 0  0 | 2 2n 0 0  0 0  0 | 1 *  * *
.   s-n/d-s   ||       .   s-n/d-s   ♦  n  n | 0 0  n  n 0 0  n | 1  0 0 0  n 1  0 | * 2  * *
sefa( s-2-s-n/d-s ) || sefa( s-2-s-n/d-s ) ♦  3  3 | 1 1  1  3 1 1  1 | 0  1 1 1  1 0  1 | * * 2n *
s-2-s-n/d-s   ♦  0 2n | 0 0  0  0 n n 2n | 0  0 0 0  0 2 2n | * *  * 1
```

```s-2-s-2n/d-o || s-2-s-2n/d-o   (n>2)   → height = 1
({n/d}-ap || {n/d}-ap)

demi( .   .      . )                        | 2n  * |  2  2  1  0  0 | 1  3  2  2 0  0 | 1 1  3 0
demi( .   .      . ) |  * 2n |  0  0  1  2  2 | 0  0  2  2 1  3 | 0 1  3 1
--------------------------------------------+-------+----------------+-----------------+---------
s-2-s      .                          |  2  0 | 2n  *  *  *  * | 0  2  1  0 0  0 | 1 0  2 0
sefa( .   s-2n/d-o )                        |  2  0 |  * 2n  *  *  * | 1  1  0  1 0  0 | 1 1  1 0
demi( .   .      . ) || demi( .   .      . ) |  1  1 |  *  * 2n  *  * | 0  0  2  2 0  0 | 0 1  3 0
s-2-s      . ) |  0  2 |  *  *  * 2n  * | 0  0  1  0 0  2 | 0 0  2 1
sefa( .   s-2n/d-o ) |  0  2 |  *  *  *  * 2n | 0  0  0  1 1  1 | 0 1  1 1
--------------------------------------------+-------+----------------+-----------------+---------
.   s-2n/d-o                          ♦  n  0 |  0  n  0  0  0 | 2  *  *  * *  * | 1 1  0 0
sefa( s-2-s-2n/d-o )                        |  3  0 |  2  1  0  0  0 | * 2n  *  * *  * | 1 0  1 0
s-2-s      . ) ||       s-2-s      . ) |  2  2 |  1  0  2  1  0 | *  * 2n  * *  * | 0 0  2 0
sefa( .   s-2n/d-o ) || sefa( .   s-2n/d-o ) |  2  2 |  0  1  2  0  1 | *  *  * 2n *  * | 0 1  1 0
.   s-2n/d-o   ♦  0  n |  0  0  0  0  n | *  *  *  * 2  * | 0 1  0 1
sefa( s-2-s-2n/d-o ) |  0  3 |  0  0  0  2  1 | *  *  *  * * 2n | 0 0  1 1
--------------------------------------------+-------+----------------+-----------------+---------
s-2-s-2n/d-o                          ♦ 2n  0 | 2n 2n  0  0  0 | 2 2n  0  0 0  0 | 1 *  * *
.   s-2n/d-o   ||       .   s-2n/d-o   ♦  n  n |  0  n  n  0  n | 1  0  0  n 1  0 | * 2  * *
sefa( s-2-s-2n/d-o ) || sefa( s-2-s-2n/d-o ) ♦  3  3 |  2  1  3  2  1 | 0  1  2  1 0  1 | * * 2n *
s-2-s-2n/d-o   ♦  0 2n |  0  0  0 2n 2n | 0  0  0  0 2 2n | * *  * 1
```