Harman's Compounds
This
page presents the compound polyhedra which are described in Michael G.
Harman's interesting but unpublished 1974 paper "Polyhedral Compounds,"
cited in the references. Most of these polyhedra
had never been described before or since, and many have never been made
as paper models or otherwise illustrated. As background, one should be
familiar with the page about compounds.
Also, some of these fall under the category compounds
of cubes and/or uniform compounds.
Harman's general procedure can be paraphrased as follows:

Choose a polyhedron to make a compound from and an overall symmetry group
to attain. Think of their symmetries and pick some n and m
for which the component has an nfold axis
of symmetry and the overall compound has an mfold axis.

Imagine the symmetry axes of the compound as radiating from a point in
space and visualize one polyhedral component centered on that point, turned
so one of its nfold symmetry axes aligns with an mfold
symmetry axis of the compound.

Imagine rotating the component on the axis so that each of the m
planes of symmetry of the compound that pass through the axis has some
plane of symmetry of the component coincident with it. If n is not
divisible by m, this can not be achieved with a single copy of the
component along the axis, in which case m copies along the same
axis are first superimposed. Sometimes there is more than one angle which
covers the m symmetry planes, and then different variants of a compound
exist.

For every mfold axis of the desired compound, superimpose a copy
(or m copies) of the component with an nfold axis so oriented.

Check to see if the components so placed will overlap so that the number
of components needed is less than the number of mfold axes.
Illustrated above is an example in which a dodecahedron is embedded into
an octahedral compound and n=m=3. Four dodecahedra have their 3fold axes
aligned to the four 3fold axes of the cube. There is a second variant,
illustrated below, in which each component is rotated 60 degrees along
its axis relative to this version.
Harman uses the notation nX/mY to indicate that an nfold
axis of the X symmetry group in the component is placed along the
direction of the mfold axis of the Y symmetry group of the
compound. X and Y are i, c, or t, for
the icosahedral, cuboctahedral, or tetrahedral symmetries, respectively.
Icosahedral
Symmetry of Compound:
Icosahedral symmetry of component:
This first group of examples was detailed on the bottom of the page about
compounds:

6 icosahedra, 6
dodecahedra, 5i/5i.

10 icosahedra, 10
dodecahedra, 3i/3i.

5 icosahedra, 5
dodecahedra, 2i/2i, (although there are fifteen 2fold axes, groups
of three components coincide, leaving only five distinct components.)
Octahedral symmetry of component:
This group of examples was detailed on the page about compounds
of cubes:

10 cubes, 10
octahedra, 3c/3i, version A.

10 cubes, 10
octahedra, 3c/3i, version B.

5 cubes, 5
octahedra , 4c/2i, (these are the standard uniform compounds; again,
fifteen overlap to become five.)

15 cubes, 15
octahedra, 2c/2i. (This could be derived from 4c/2i as a second variant.)
Tetrahedral symmetry of component:
The following are equivalent to substituting stellae octangula into the
above compounds of cubes. Note that the familiar compounds of five and
ten tetrahedra do not show up by this technique of aligning planes of symmetry.

20 tetrahedra, 3t/3i,
version A.

20 tetrahedra, 3t/3i,
version B.

30 tetrahedra, 2t/2i.
Octahedral Symmetry of Compound:
Icosahedral symmetry of component:
Note
how the following four correspond to the above icosahedral compounds of
octahedral components. The number of nfold axes depends on the
overall symmetry, but the number of varieties agrees. This relation when
one "interchanges the role of component symmetry and overall symmetry"
is called outer duality by Harman. Shown at right is the 4 dodecahedra
version A, and at the top of the page is version B.

2 icosahedra, 2
dodecahedra, 2i/4c.

4 icosahedra, 4
dodecahedra, 3i/3c, version A.

4 icosahedra, 4
dodecahedra, 3i/3c, version B.

6 icosahedra,
6 dodecahedra,
2i/2c. (This can also be derived from 2i/4c as a second variant.)
Octahedral symmetry of component:
These are the three rigid octahedral compounds listed on the page about
compounds of cubes:

3 cubes, 3
octahedra, 4c/4c.

4 cubes, 4
octahedra, 3c/3c.

6 cubes, 6
octahedra, 2c/2c.
Tetrahedral symmetry of component:
The following can also be seen as following from the above compounds of
cubes (or, in the first case, a single cube) by substituting stellae octangula.

2 tetrahedra, 3t/3c.

8 tetrahedra, 3t/3c.

6 tetrahedra, 2t/4c.

12 tetrahedra, 2t/2c.
Tetrahedral
Symmetry of Compound:
Tetrahedral symmetry of component:
There is only one compound with tetrahedral symmetry on Harman's list.
A beautiful compound of four tetrahedra, I like to call it the "stella
sixteengula". If you combine it with its dual, you get the compound
of 8 tetrahedra listed above.

4 tetrahedra, 3t/3t.
Prism Symmetry of Compound:
in progress...
Harman's examples are all rigid compounds of Platonic solids, but the
idea is straightforward to apply to other symmetric polyhedra as components,
e.g., a compound
of five rhombic triacontahedra. Some examples with stellated components
were listed elsewhere. The idea of nonrigid
solids with rotational freedom which appears in Skilling's 1976 enumeration
of uniform compounds is excluded
by Harman's definitions. Most of the well
known compounds can be made by this technique but the compounds of
five tetrahedra and ten
tetrahedra are not, because they have no planes of symmetry of the
compound aligned with planes of symmetry of the components.
Note: Of course, under usual circumstances I would never consider
presenting the results of someone else's unpublished paper without their
expressed invitation. However, since Michael Harman's 1974 paper has remained
unpublished for over twenty years, and since these compounds are so beautiful
that they really deserve to be known, I feel it is appropriate for me to
present them here in this manner. The reader, of course, understands that
the credit for the general method and the first envisionment of these compounds
belongs to Harman, and his unpublished paper should be cited as their original
source.