Uniform polyhedral compound

A uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.

The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering.

2 constituent polyhedra incident on each vertex

15{4}

60{4}

(n>0)

2np{4}

gcd(p,q)=1, p/q>2

(n>1)

np{4}

(n>0)

(q odd)

4np{3}

gcd(p,q)=1, p/q>3/2

D_nph (if n even)

(n>1)

(q odd)

2np{3}

D_nph (if n even)

(n>0)

(q even)

4np{3}

gcd(p,q)=1, p/q>3/2

(n>1)

(q even)

2np{3}

24{5}

12{5}

24{5/2}

12{5/2}

12{4}

24{4}

30{4}

60{4}

12{5}

24{5}

12{5/2}

24{5/2}

8{6}

20{6}

12{10}

12{10/3}

6{4}

12{4}

12{5/2}

24{5/2}

8{6}

20{6}

40{6}

30{8}

30{8/3}

30{4}

20{6}

(30+60){4}

30{8}

30{4}

30{8}

30{4}

30{8/3}

(30+60){4}

12{4}

24{5}

24{5/2}

24{5}

24{5/2}

24{5}

24{5/2}

24{5}

24{5/2}

...
Wikipedia