Compounds of Catalan Solids

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Compounds of Catalan Solids

Postby ubersketch » Sat Jan 27, 2018 12:07 am

The compounds of Catalan solids here will be the duals of compound polyhedra made from Archimedean solids. I like to call these "combocatalans".

Trisso - Triakis stella octangula, triakis stellated octahedron
Compound of 2 tikit. Dual of tisso.

Trichi - Triakis chiricosahedron
Compound of 5 tikit. Dual of taki.

Tre - Triakis icosicosahedron
Compound of 10 tikit. Dual of te.

Trissi - Triakis small icosicosahedron
Compound of 5 tikko. Dual of tar.

Re - Rhombic icosicosahedron
Compound of 5 rad. Dual of arie.

Rile - Rhombilanceal icosicosahedron
Compound of 5 sladid. Dual of rasseri.

Pedisi - Pentagonal disicositetrahedron
Compound of 2 pedid. Dual of disco.

Pidis - Pentagonal dishexacontahedron
Compound of 2 sapedit. Dual of dissid.

Tell me if I should rename some and information I should add.
I'm going to try and make pictures of them.
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Re: Compounds of Catalan Solids

Postby Mercurial, the Spectre » Mon Jan 29, 2018 1:28 pm

Yes, these catalan compounds are, under its symmetry, face-transitive, meaning that the faces act transitively with each other; there is one type of face. The only restriction is that topology must dictate the symmetry of each individual polyhedron.

If you take a compound of two octahedra such that they form alternate vertices of a hexagonal prism (hence individual D3d symmetry), although they are vertex-transitive, it wouldn't be regular since by topology, there are two types of triangles (one always regular, and one isosceles). To see this, picture a compound of tall triangular antiprisms (which are topologically identical to octahedra but with D3d symmetry) such that they form the vertices of a hexagonal prism. Clearly the two regular hexagons are made from the equilateral triangles while the lateral rectangles are made from crisscrossing tall isosceles triangles.

Its dual, a compound of two trigonal trapezohedra (topologically identical to a compound of two cubes such that only 2 vertices coincide with the other in the direction of 3-fold symmetry which alternates to form a topological hexakis hexagonal prism), must have one face type (generally kites), but two types of vertices, one corresponding to the top and bottom ones, and one corresponding to the vertices of a hexagonal prism).

Wikipedia has a detailed collection of these archimedean compounds and their duals.
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Re: Compounds of Catalan Solids

Postby ubersketch » Thu Feb 01, 2018 12:40 am

Mercurial, the Spectre wrote:Yes, these catalan compounds are, under its symmetry, face-transitive, meaning that the faces act transitively with each other; there is one type of face. The only restriction is that topology must dictate the symmetry of each individual polyhedron.

If you take a compound of two octahedra such that they form alternate vertices of a hexagonal prism (hence individual D3d symmetry), although they are vertex-transitive, it wouldn't be regular since by topology, there are two types of triangles (one always regular, and one isosceles). To see this, picture a compound of tall triangular antiprisms (which are topologically identical to octahedra but with D3d symmetry) such that they form the vertices of a hexagonal prism. Clearly the two regular hexagons are made from the equilateral triangles while the lateral rectangles are made from crisscrossing tall isosceles triangles.

Its dual, a compound of two trigonal trapezohedra (topologically identical to a compound of two cubes such that only 2 vertices coincide with the other in the direction of 3-fold symmetry which alternates to form a topological hexakis hexagonal prism), must have one face type (generally kites), but two types of vertices, one corresponding to the top and bottom ones, and one corresponding to the vertices of a hexagonal prism).

Wikipedia has a detailed collection of these archimedean compounds and their duals.

Do you have a link?
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Re: Compounds of Catalan Solids

Postby Mercurial, the Spectre » Sun Feb 04, 2018 2:50 pm

ubersketch wrote:
Mercurial, the Spectre wrote:Yes, these catalan compounds are, under its symmetry, face-transitive, meaning that the faces act transitively with each other; there is one type of face. The only restriction is that topology must dictate the symmetry of each individual polyhedron.

If you take a compound of two octahedra such that they form alternate vertices of a hexagonal prism (hence individual D3d symmetry), although they are vertex-transitive, it wouldn't be regular since by topology, there are two types of triangles (one always regular, and one isosceles). To see this, picture a compound of tall triangular antiprisms (which are topologically identical to octahedra but with D3d symmetry) such that they form the vertices of a hexagonal prism. Clearly the two regular hexagons are made from the equilateral triangles while the lateral rectangles are made from crisscrossing tall isosceles triangles.

Its dual, a compound of two trigonal trapezohedra (topologically identical to a compound of two cubes such that only 2 vertices coincide with the other in the direction of 3-fold symmetry which alternates to form a topological hexakis hexagonal prism), must have one face type (generally kites), but two types of vertices, one corresponding to the top and bottom ones, and one corresponding to the vertices of a hexagonal prism).

Wikipedia has a detailed collection of these archimedean compounds and their duals.

Do you have a link?

Yeah, just search for "polyhedron compound".
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