User:Padex/hypercupola

Hypercupola
I've found that, in 3 dimensions, cupolas are formed by an "expansion" of pyramids.

So, in 4D, I've found 4 hypercupolas:

They're composed of a {p,q} (all of the regular polyhedra, excepted the icosahedron) and a t0,2{p,q} (the cantellated polyhedron) linked by prisms and pyramids.

Cartesian coordinates
Tetrahedral cupola:

For the tetrahedral top:
 * (0, 0, √(6)/4, √(10)/4);
 * (±1/2, -1/(2√3), -√(2)/(4√3), √(5)/(2√2));
 * ( 0, 1/√(3), -√2/(4√3), √(5)/(2√2));

For the cuboctahedral base:

the hexagon:
 * (±1, 0, 0, 0)
 * (±1/2, ±√(3)/2, 0, 0)

the triangles:

n°1
 * (±1/2, 1/(2√3), √(2/3), 0)
 * (0, -1/√3, √(2/3), 0)

n°2
 * (±1/2, -1/(2√3), -√(2/3), 0)
 * (0, 1/√3, -√(2/3), 0)

Cubic cupola:


 * (±1/2, ±1/2, ±1/2, τ);
 * (±1/2, ±1/2, ± (1/2 + τ), 0);
 * (±1/2, ± (1/2 + τ), ±1/2, 0);
 * (±(1/2 + τ), ±1/2, ±1/2, 0);

where τ = √2/2

Octahedral cupola:


 * ( 0, 0, ±τ, 1/2);
 * (0, ±τ, 0, 1/2);
 * (±τ, 0, 0, 1/2);
 * (±1/2, ±1/2, ± (1/2 + τ), 0);
 * (±1/2, ± (1/2 + τ), ±1/2, 0);
 * (± (1/2 + τ), ±1/2, ±1/2, 0);

where τ = √2/2

fr:Utilisateur: Padex/Hypercoupole