Acronym n-appip, K-4.176 Name n-antiprism prism Segmentochoron display Circumradius sqrt[(5-4 cos(π/n))/(8-8 cos(π/n))] Lace cityin approx. ASCII-art ```xNo xNo -- x xNo (N-p) oNx oNx -- x oNx (gyro N-p) | | | +-- s2sNs (N-ap) +-------- s2sNs (N-ap) ``` Especially ope (n=3)   squappip (n=4)   pappip (n=5)   happip (n=6)   heappip (n=7)   oappip (n=8)   eappip (n=9)   dappip (n=10) Confer general antiprism-prisms: n/d-appip   general polytopal classes: segmentochora Externallinks

n=2 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.

Incidence matrix according to Dynkin symbol

```x s2sns   (n>2)

. demi( . . .   ) | 4n |  1  1  1  2 | 1 1 1  2  3 | 1 1  3
------------------+----+-------------+-------------+-------
.       s2s .     |  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( . sns   ) |  2 |  *  *  * 4n | 0 0 1  1  1 | 1 1  1
------------------+----+-------------+-------------+-------
x       s2s .     |  4 |  2  0  2  0 | n * *  *  * | 0 0  2
x       s . s2*b  |  4 |  0  2  2  0 | * n *  *  * | 0 0  2
.       . sns     ♦  n |  0  0  0  n | * * 4  *  * | 1 1  0
x sefa( . sns   ) |  4 |  0  0  2  2 | * * * 2n  * | 1 0  1
. sefa( s2sns   ) |  3 |  1  1  0  1 | * * *  * 4n | 0 1  1
------------------+----+-------------+-------------+-------
x       . sns     ♦ 2n |  0  0  n 2n | 0 0 2  n  0 | 2 *  *
.       s2sns     ♦ 2n |  n  n  0 2n | 0 0 2  0 2n | * 2  *
x sefa( s2sns   ) ♦  6 |  2  2  3  2 | 1 1 0  1  2 | * * 2n
```

```x s2s2no   (n>2)

. demi( . .  . ) | 4n |  2  1  2 |  2 1  2  3 | 1 1  3
-----------------+----+----------+------------+-------
.       s2s  .   |  2 | 4n  *  * |  1 0  0  2 | 0 1  2
x demi( . .  . ) |  2 |  * 2n  * |  2 0  2  0 | 1 0  3
. sefa( . s2no ) |  2 |  *  * 4n |  0 1  1  1 | 1 1  1
-----------------+----+----------+------------+-------
x       s2s  .   |  4 |  2  2  0 | 2n *  *  * | 0 0  2
.       . s2no   ♦  n |  0  0  n |  * 4  *  * | 1 1  0
x sefa( . s2no ) |  4 |  0  2  2 |  * * 2n  * | 1 0  1
. sefa( s2s2no ) |  3 |  2  0  1 |  * *  * 4n | 0 1  1
-----------------+----+----------+------------+-------
x       . s2no   ♦ 2n |  0  n 2n |  0 2  n  0 | 2 *  *
.       s2s2no   ♦ 2n | 2n  0 2n |  0 2  0 2n | * 2  *
x sefa( s2s2no ) ♦  6 |  4  3  2 |  2 0  1  2 | * * 2n
```

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

o. o.no.    | 2n  * | 1  2  2 0  0 | 2 1  2  2  1 0 0 | 1 2 1 1 0
.o .on.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 oonoo&#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.no.    |  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
.. .on.x    |  0  n | 0  0  0 0  n | * *  *  *  * * 2 | 0 0 0 1 1
------------+-------+--------------+------------------+----------
x. x.no.    ♦ 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 * *
.. xonox&#x ♦  n  n | 0  n 2n 0  n | 0 1  0  n  n 0 1 | * * * 2 *
.x .on.x    ♦  0 2n | 0  0  0 n 2n | 0 0  0  0  0 n 2 | * * * * 1
```

```s2sns || s2sns   (n>2)   → height = 1
({n}-ap || {n}-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
------------------------------+-------+------------------+------------------+---------
s2s .                   |  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( . sns )                 |  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
s2s .   |  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( . sns ) |  0  2 | * *  *  * * * 2n | 0  0 0 0  1 1  1 | 0 1  1 1
------------------------------+-------+------------------+------------------+---------
. sns                   ♦  n  0 | 0 0  n  0 0 0  0 | 2  * * *  * *  * | 1 1  0 0
sefa( s2sns )                 |  3  0 | 1 1  1  0 0 0  0 | * 2n * *  * *  * | 1 0  1 0
s2s .   ||       s2s .   |  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( . sns ) || sefa( . sns ) |  2  2 | 0 0  1  2 0 0  1 | *  * * * 2n *  * | 0 1  1 0
. sns   ♦  0  n | 0 0  0  0 0 0  n | *  * * *  * 2  * | 0 1  0 1
sefa( s2sns ) |  0  3 | 0 0  0  0 1 1  1 | *  * * *  * * 2n | 0 0  1 1
------------------------------+-------+------------------+------------------+---------
s2sns                   ♦ 2n  0 | n n 2n  0 0 0  0 | 2 2n 0 0  0 0  0 | 1 *  * *
. sns   ||       . sns   ♦  n  n | 0 0  n  n 0 0  n | 1  0 0 0  n 1  0 | * 2  * *
sefa( s2sns ) || sefa( s2sns ) ♦  3  3 | 1 1  1  3 1 1  1 | 0  1 1 1  1 0  1 | * * 2n *
s2sns   ♦  0 2n | 0 0  0  0 n n 2n | 0  0 0 0  0 2 2n | * *  * 1
```

```s2s2no || s2s2no   (n>2)   → height = 1
({n}-ap || {n}-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
--------------------------------+-------+----------------+-----------------+---------
s2s  .                    |  2  0 | 2n  *  *  *  * | 0  2  1  0 0  0 | 1 0  2 0
sefa( . s2no )                  |  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
s2s  . ) |  0  2 |  *  *  * 2n  * | 0  0  1  0 0  2 | 0 0  2 1
sefa( . s2no ) |  0  2 |  *  *  *  * 2n | 0  0  0  1 1  1 | 0 1  1 1
--------------------------------+-------+----------------+-----------------+---------
. s2no                    ♦  n  0 |  0  n  0  0  0 | 2  *  *  * *  * | 1 1  0 0
sefa( s2s2no )                  |  3  0 |  2  1  0  0  0 | * 2n  *  * *  * | 1 0  1 0
s2s  . ) ||       s2s  . ) |  2  2 |  1  0  2  1  0 | *  * 2n  * *  * | 0 0  2 0
sefa( . s2no ) || sefa( . s2no ) |  2  2 |  0  1  2  0  1 | *  *  * 2n *  * | 0 1  1 0
. s2no   ♦  0  n |  0  0  0  0  n | *  *  *  * 2  * | 0 1  0 1
sefa( s2s2no ) |  0  3 |  0  0  0  2  1 | *  *  *  * * 2n | 0 0  1 1
--------------------------------+-------+----------------+-----------------+---------
s2s2no                    ♦ 2n  0 | 2n 2n  0  0  0 | 2 2n  0  0 0  0 | 1 *  * *
. s2no   ||       . s2no   ♦  n  n |  0  n  n  0  n | 1  0  0  n 1  0 | * 2  * *
sefa( s2s2no ) || sefa( s2s2no ) ♦  3  3 |  2  1  3  2  1 | 0  1  2  1 0  1 | * * 2n *
s2s2no   ♦  0 2n |  0  0  0 2n 2n | 0  0  0  0 2 2n | * *  * 1
```