n/d-cu

Acronym

n/d-cu

Name

n/d-gonal cupola

 © ©

Circumradius

sqrt[(7+4 cos(π d/n)-4 cos²(π d/n))/(12-16 cos²(π d/n))]

Vertex figures

[3,4,n/d,4], [3,4,2n/d]

General of army

if d=1: is itself convex
if gcd(n,d)=1: use d=1 for its general

Colonel of regiment

(is itself locally convex)

Face vector

3n, 5n, 2n+2

Especially

2/d	3/d	4/d	5/d	6/d	...	{n/d}-p
trip *	tricu	squacu	pecu	hicu **	-	n/1
-	ratricu		stacu			n/2 ‡
-	thah		stiscu			n/2 °
-	-	rasquacu	rastacu			n/3
-	-	-	rapescu			n/4 °

Confer

general polytopal classes:: cupola cuploid cupolaic blend

‡ / °: If d is even, the bottom face becomes some Grünbaum style polygon (case: ‡). Because the edges of that bottom face then coincide by pairs, that unusual face might be omitted without loss. This reduced figure (case: °) – not covered by the matrix below – is called a n/d-cuploid (a.k.a. semicupola).

*: The case n=2 fits here by concept too, it just has a different incidence matrix as the n-gons become degenerate.

**: The height formula given below shows that only 6/5 < n/d < 6 is possible. The maximal height is obtained at n/d = 2 with upright latteral triangles, the extremal values n/d = 6/5 or 6 would generate heights of zero.

The 3 non-degenerate cases in this range with d=1 are Johnson solids.

Incidence matrix according to Dynkin symbol

xx-n/d-ox&#x   (6/5<n/d<6 and n/d<>2)   → height = sqrt[1-1/(4 sin²(π d/n))]
({n/d} || {2n/d})

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

so-2n/d-ox&#x   (6/5<n/d<6 and n/d<>2)   → height = sqrt(1-[1/4 *sin^2(π d/n)])
({n/d} || {2n/d})

demi( o.-2n/d-o.    ) | n  * | 2  2 0 0 | 1 1 0 2
      .o-2n/d-.o      | * 2n | 0  1 1 1 | 0 1 1 1
----------------------+------+----------+--------
sefa( s.-2n/d-o.    ) | 2  0 | n  * * * | 1 0 0 1
demi( oo-2n/d-oo&#x ) | 1  1 | * 2n * * | 0 1 0 1
demi( ..      .x    ) | 0  2 | *  * n * | 0 0 1 1
demi( ..      .x    ) | 0  2 | *  * * n | 0 1 1 0
----------------------+------+----------+--------
      s.-2n/d-o.      | n  0 | n  0 0 0 | 1 * * *
demi( ..      ox&#x ) | 1  2 | 0  2 0 1 | * n * *
      .o-2n/d-.x      | 0 2n | 0  0 n n | * * 1 *
sefa( so-2n/d-ox&#x ) | 2  2 | 1  2 1 0 | * * * n

starting figure: xo-2n/d-ox&#x

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