Each group of isomorphic forms, characterized by variable indices, can be led to coincide with the corresponding archetypal Catalan polyhedron by the attribution of suitable values to the indices of the four forms:

• {τ+1/τ  1/τ² 0} Catalan triakis-icosahedron
• {τ+1/τ  τ  0} Catalan deltoid-hexecontahedron
• {1/τ  3 0} Catalan pentakis-dodecahedron
• {τ+2/τ  1/τ² 1/τ²} Catalan hexakis-icosahedron

All the 120 compound forms derived from the two Platonic and the five Catalan polyhedra are here represented, grouped in six sets; an equal distance from the centre of each solid has been attributed to all the faces of the compound forms.
Taking into account that the hexakis-icosahedron is the only icosahedral form belonging uniquely to the m 3 5 point group (whereas the other six forms are compatible also with 235, the other icosahedral point group, which is lacking in mirrors and centre of inversion), the first set includes the compound forms in which the hexakis-icosahedron is added to each combination of the other six single forms; it follows that the total number of such compound forms is 63 = 2⁶-1
The remaining sets include:

• thirteen compound forms (31 = 2⁵-1) in which the pentakis-dodecahedron is added to each combination of the remaining five forms
• fifteen compound forms (15 = 2⁴-1) in which the deltoid-hexecontahedron is added to each combination of the remaining four forms
• seven compound forms (7 = 2³-1) in which the triakis-icosahedron is added to each combination of the remaining three forms
• three compound forms (3 = 2²-1) in which the rhomb-triacontahedron is added to each combination of the remaining two forms
• one compound form (1 = 2¹-1) derived from the combination of the icosahedron with the dodecahedron

(clicking on each following image by the left button of the mouse, one can visualize the corresponding VRML dynamic image)

The seven single forms which characterize the m 3 5 icosahedral point group and the 120 derived compound forms

The seven single forms
1) dodecahedron	2) icosahedron	3) rhomb-triacontahedron	4) triakis-icosahedron	5) deltoid-hexecontahedron	6) pentakis-dodecahedron	7) hexakis-icosahedron

The sixty-three compound forms which include a Catalan hexakis-icosahedron
Six compound forms made of two forms
6+7	5+7	4+7	3+7	2+7	1+7
Fifteen compound forms made of three forms
5+6+7	4+6+7	3+6+7	2+6+7	1+6+7
4+5+7	3+5+7	2+5+7	1+5+7	3+4+7
2+4+7	1+4+7	2+3+7	1+3+7	1+2+7
Twenty compound forms made of four forms
4+5+6+7	3+5+6+7	2+5+6+7	1+5+6+7	3+4+6+7
2+4+6+7	1+4+6+7	2+3+6+7	1+3+6+7	1+2+6+7
3+4+5+7	2+4+5+7	1+4+5+7	2+3+5+7	1+3+5+7
1+2+5+7	2+3+4+7	1+3+4+7	1+2+4+7	1+2+3+7
Fifteeen compound forms made of five forms
3+4+5+6+7	2+4+5+6+7	1+4+5+6+7	2+3+5+6+7	1+3+5+6+7
1+2+5+6+7	2+3+4+6+7	1+3+4+6+7	1+2+4+6+7	1+2+3+6+7
2+3+4+5+7	1+3+4+5+7	1+2+4+5+7	1+2+3+5+7	1+2+3+4+7
Six compound forms made of six forms
2+3+4+5+6+7	1+3+4+5+6+7	1+2+4+5+6+7	1+2+3+5+6+7	1+2+3+4+6+7	1+2+3+4+5+7
Compound form made of seven forms
1+2+3+4+5+6+7

Thirty-one further compound forms including a Catalan pentakis-dodecahedron
Five compound forms made of two forms
5+6	4+6	3+6	2+6	1+6
Ten compound forms made of three forms
4+5+6	3+5+6	2+5+6	1+5+6	3+4+6
2+4+6	1+4+6	2+3+6	1+3+6	1+2+6
Ten compound forms made of four forms
3+4+5+6	2+4+5+6	1+4+5+6	2+3+5+6	1+3+5+6
1+2+5+6	2+3+4+6	1+3+4+6	1+2+4+6	1+2+3+6
Five compound forms made of five forms
2+3+4+5+6	1+3+4+5+6	1+2+4+5+6	1+2+3+5+6	1+2+3+4+6
Compound form made of six forms
1+2+3+4+5+6

Fifteen further compound forms including the Catalan deltoid-hexecontahedron
Four compound forms made of two forms
4+5	3+5	2+5	1+5
Six compound forms made of three forms
3+4+5	2+4+5	1+4+5	2+3+5	1+3+5	1+2+5
Four compound forms made of four forms
2+3+4+5	1+3+4+5	1+2+4+5	1+2+3+5
Compound form made of five forms
1+2+3+4+5

Seven further compound forms including the Catalan triakis-icosahedron
Three compound forms made of two forms
3+4	2+4	1+4
Three compound forms made of three forms
2+3+4	1+3+4	1+2+4
Compound form made of four forms
1+2+3+4

Three further compound forms including the rhomb-triacontahedron
Two compound forms made of two forms
2+3	1+3
Compound form made of three forms
1+2+3

Last compound form including an icosahedron and a dodecahedron
Compound form made of two forms
1+2

Afterwards one can see three examples of compound forms (not involving Catalan polyhedra), together with the single forms whose intersection can generate such particular compound forms, in case of suitable distances, from the centre of the solids, of the faces of each single form.

In detail:

• the compound form in the first row derives from the intersection of a triakis-icosahedron, an icosahedron and two different pentakis-dodecahedra
• the compound form in the second row derives from the intersection of a rhomb-triacontahedron, an icosahedron and two different deltoid-hexecontahedra
• the compound form in the third row derives from the intersection of a dodecahedron, a hexakis-icosahedron and two different triakis-icosahedra.

The compound form on the left derives from the intersection of the four single forms on the right: a triakis-icosahedron, an icosahedron and two different pentakis-dodecahedra.
The compound form on the left derives from the intersection of the four single forms on the right: a rhomb-triacontahedron, an icosahedron and two different deltoid-hexecontahedra
The compound form on the left derives from the intersection of the four single forms on the right: a dodecahedron, a hexakis-icosahedron and two different triakis-icosahedra.

Links

1. Zefiro L., Ardigo' M.R.
   Description of the Forms Belonging to the 235 and m35 Icosahedral Point Groups Starting from the Pairs of Dual Polyhedra: Icosahedron-Dodecahedron and Archimedean Polyhedra-Catalan Polyhedra
   VisMath, volume 9, No. 4, 2007


2. Zefiro L., Ardigo' M.R.
   Platonic and Catalan Polyhedra as Archetypes of Forms Belonging to the Cubic and Icosahedral Systems
   VisMath, volume 11, No. 2, 2009