The icosohedron could be thought of as 12 overlapping pentagonal pyramids.
Let's focus on one of these pyramids. For convenience its base lies in the xy plane and the base center is the origin.

 
 
Within and without the pentagon base can be drawn an infinite series of pentagons. The pentagons are all bases of pyramids sharing the same apex.

 
 
| cos^n(pi/5)*cos(n*pi/5)   -cos^n(pi/5)*sin(n*pi/5)    0  |
|                                                          |
| cos^n(pi/5)*sin(n*pi/5)    cos^n(pi/5)*cos(n*pi/5)    0  |
|                                                          |
|          0                            0               1  |
If the pyramid of equilateral triangles were called 0, the rest of the pyramids could be assigned an interger n.
The nth pyramid would be pyramid 0 multipled by the above scaling, rotating matrix.
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Substitute the icosohedron's vertices with  pyramid # n,
cut off the pyramid past the lines of intersection and you have the nth solid.
When the pyramid faces are coplanar as with the icosohedron and rhombic triacontahedron,
draw the boundaries equidistant between closest vertices.
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These solids have 60 faces which are either strombi or isosceles triangles, with a few exceptions:
-
In the 0th polyhedra (icosohedron) , triplets of strombi are coplanar so there are 20 faces of equilateral triangles,
In the -1 polyhedra (rhombic triacontahedron) pairs of isosceles triangles are coplanar giving 30 rhombic faces.
At -infinity 5-tuples of strombi or isosceles triangles (take your pick) are coplanar giving 12 pentagon faces.
At infinity are 6 concentric line segments. The segments could be regarded as degenerate strombi or isosceles triangles.

 
 
 
Here are cardboard models of the polyhedra -2 through 2.

 
 
This illustration indicates the pentagon corners lie on continuous logarithmic spirals. Earlier each twistohedra was assigned an integer. But there are twistohedra for every real number. However, the non integer polyhedra are chiral. Also twistohedra that differ by less than 1 don't fit together as neatly.
I don't know how to construct the non integer twistohedra.

 

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