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I would like to find instances of the following dodecahedral wheel a geometric figure composed of equilateral triangles and squares

generalized to the situation where the red squares become rhombuses.

I tried this using RandomInstance[GeometricScene], but Mathematica could not complete it the way I was trying. I tried simplifying the Geometric Scence to some smaller subset of constraints (eliminating the outer 6 ring of equilateral triangles):

RandomInstance[
 GeometricScene[{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r,
    s},
  {
   t1 = Triangle[{a, b, c}],
   t2 = Triangle[{a, c, d}],
   t3 = Triangle[{a, d, e}],
   t4 = Triangle[{a, e, f}],
   t5 = Triangle[{a, f, g}],
   t6 = Triangle[{a, g, b}],
   
   s1 = Style[Polygon[{b, k, l, c}], Red],
   s2 = Style[Polygon[{c, m, n, d}], Red],
   s3 = Style[Polygon[{d, o, p, e}], Red],
   s4 = Style[Polygon[{e, q, r, f}], Red],
   s5 = Style[Polygon[{f, s, h, g}], Red],
   s6 = Style[Polygon[{g, i, j, b}], Red],
   
   GeometricAssertion[{t1, t2, t3, t4, t5, t6}, "Equilateral", 
    "Clockwise"],
   GeometricAssertion[{s1, s2, s3, s4, s5, s6}, "Equilateral", 
    "Clockwise"]
   }
  ]]

but this also couldn't complete.

This even smaller subfigure did complete:

RandomInstance[GeometricScene[{a, b, c, d, k, l, m, n},
  {
   t1 = Triangle[{a, b, c}],
   t2 = Triangle[{a, c, d}],
   t9 = Triangle[{c, l, m}],
   
   s1 = Style[Polygon[{b, k, l, c}], Red],
   s2 = Style[Polygon[{c, m, n, d}], Red],
   
   GeometricAssertion[{t1, t2, t9}, "Equilateral", "Clockwise"],
   GeometricAssertion[{s1, s2}, "Equilateral", "Clockwise"]
   }
  ]]

enter image description here

but was very slow.

This doesn't seem to be such a complicated constraints problem, and so I am wondering if there is a better way to do this with GeometricScene?

Thanks for any help you might provide.

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