In 1976 John Skilling published "Uniform Compounds of Uniform Polyhedra [Ref]. Many, if not all, of these models may have appeared in Michael G. Harman, "Polyhedral Compounds" [Ref], an unpublished manuscript around 1974. However, the concept of Uniform Compounds with Rotational Freedom was first published in Skilling's paper. Historically, at least some of the Uniform Compounds were known. Peter Cromwell notes in his book Polyhedra [Ref] that the compound of two tetrahedra was first depicted in Pacioli's Divina Proportione [Ref] and the compounds of five and ten tetrahedra, of five cubes, and of five octahedra were first described by Edmund Hess [Ref] in 1876.
In 1996, George Hart generated many, but not all of them, as vrml models for the first time. In 2006 Piotr Pawlikowski noticed that there was still no one place where all of the 75 models could be found. With the help of Marcel Tunnissen and others, he completed the missing models for a collection of 75 Stella files. From those, the vrml models could be generated, completing the series. Those models served as the inspiration for this web page.
A Uniform Compound was described by Skilling as "a threedimensional combination of uniform polyhedra whose edgelengths are all equal and whose relative position is such that the symmetry group of the combination is transitive on the set of all vertices of the polyhedra. The polyhedra may intersect themselves and each other, but compounds in which some faces are either shared between constituents or totally hidden from exterior view are excluded".
Sometimes erroneously referred to as "vertex uniform", the property of the vertices of the Uniform Compounds is that they are isogonal (literally "samevertexed"). To be one of the Uniform Compounds, the vertices must be situated in a "kaleidoscopic" pattern such that if a section of it were displayed in 3 mirrors as in a sort of 3D kaleidoscope, and it was carefully rotated it would continue to look like the whole compound. The vertices of a Uniform Compound will also all be identical such that they all have the same valence. But just having vertex congruence within a given geometry is not enough. For instance, 20 cubes can be situated in an icosahedral pattern, but the vertices on their corners are not isogonal. While not instantly apparent, if the kaleidoscope test were done on the vertices, it could be seen as not quite right.
Isogonality and isohedrality are dual properties. Taking the dual of any of the Uniform Compounds, all which have one kind of vertex, will result in a compound with only one kind of face. For instance the dual of the Compound of 4 Hexagonal Prisms will be a compound of 4 Hexagonal Dipyramids and all faces will be triangles of the same shape and size (isohedral). Most of the time even when the dual has faces of regular polygons it is not a Uniform Compound. For instance, while the compound of 20 octahedra is isogonal, its dual is of all the square faces of 20 cubes but again it is not a Uniform Compound because it has two types of vertices. In fact, other than the selfdual tetrahedral compounds, there is only one case such that the dual of a Uniform Compound is also a Uniform Compound. In the case of the Compound of 5 Cubes, its dual, the Compound of 5 Octahedra is also a Uniform Compound.
Some interesting facts about the Uniform Compounds.
Links to other pages for further reading:
Question or comments about the web page should be directed to polyhedra@bigfoot.com
The compounds are presented in the order Skilling originally listed them. The Super Heading of each table is the group name from Skilling's paper for which those compounds belong. Each figure lists the symmetry (S) T  Tetrahedral, O  Octahedral, I  Icosahedral, number of faces of each type  Triangle, Square, Pentagon, Hexagon, Octagon and Decagon. Total Face, Edge and Vertex counts are given such that each compound constituent is a separate entity. A (C) denotes that solid is chiral. An (R) denotes rotational freedom. For Compounds with Rotation Freedom, animations of different aspects of the model are provided.
Miscellaneous 15  





 
Miscellaneous 610  




 
Miscellaneous 1115  




 
Miscellaneous 1619  



 
Prism Symmetry, Embedded in Prism Symmetry 2024 (Examples)  




 
Prism Symmetry, Embedded in Prism Symmetry 25 (Examples)  
 
Prism Symmetry, Embedded in Octahedral or Icosahedral Symmetry 2630  




 
Prism Symmetry, Embedded in Octahedral or Icosahedral Symmetry 3135  




 
Prism Symmetry, Embedded in Octahedral or Icosahedral Symmetry 3640  




 
Prism Symmetry, Embedded in Octahedral or Icosahedral Symmetry 4145  




 
Tetrahedral Symmetry, Embedded in Octahedral or Icosahedral Symmetry 4650  




 
Tetrahedral Symmetry, Embedded in Octahedral or Icosahedral Symmetry 5155  




 
Tetrahedral Symmetry, Embedded in Octahedral or Icosahedral Symmetry 5660  




 
Tetrahedral Symmetry, Embedded in Octahedral or Icosahedral Symmetry 6165  




 
Tetrahedral Symmetry, Embedded in Octahedral or Icosahedral Symmetry 6667  

 
Duplication of Enantiomorphs 6872  




 
Duplication of Enantiomorphs 7375  



Thanks to Guy Inchbald for a description of isogonality of Uniform Compound vertices
Generation of VRML models, OFF files, and Pictures was done with Antiprism. Stel were generated with Robert Webb's Stella application. The Hedron application by Jim McNeill was used to generate switch files.
I'd also like to thank Alex Doskey for his spreadsheet method which made the construction of this page much easier. I also use JovoToys in polyhedra contruction.
History:
20120229 Added bullet for regular compounds
20120222 Rebuilt page. (Old page is here)
20090130 Update site for Piotr Pawlikowski picture gallery
20080116 Edited part count for 5 Querco
20080107 Added Piotr Pawlikowski picture gallery. Added Wikipedia link. Listed isogonal cases. Clarified isogonality paragraph
20080106 Updated the beer statistics. Added Pictures of gidrissid paper models. Better explanation of "isogonality"
20080105 Added OFF files. Took spaces out of file names
20061221 Cylinder Files added. Big Pictures added
20061216 Jonathan Bowers's site added. #62 Stel file corrected
20061215 70 and 74 untangled
20061213 #67 Face count corrected
20061210 Initial Release
20061203 Beta
20061201 Alpha