Tuesday, July 7, 2009

Rhombic Hexecontahedron

If you ask Wolframalpha about its logo, it will tell you that it is a rhombic hexecontahedron. I had fun making one of these recently (here is a pdf of the pattern I used - you will need to print 3 copies onto card stock to make the model shown above).

One neat thing about the rhombic hexecontahedron is that it is a stellation of the rhombic triacontahedron (described a bit in this post), and both polyhedra have the same type of faces - golden rombs (how unusual is it for a stellation to have the same polygonal faces as the polyhedron that it is obtained from?).

A neat project would be to create a set of golden rhombohedra that could be used to build both the hexecontahedron and the tricontahedron. According to the mathworld entry, 20 acute rhombohedrons can be assembled to make up the hexacontahedron, and according to Coxeter (Regular Polytopes, page 27), you can build the tricontahedron with 10 acute and 10 obtuse rhombohedrons.

Another construction is suggested by the 'ball of whacks' toy - which shows that you can build up the tricontahedron with 30 rhombic pyramids.

2 comments:

  1. It is a very good net model, only it is absent which is hill and valley lines. There is a net for the Rhombic Hexecontahedron at Wolfram Math whithout any instruction of assembly, this way it is not useful

    ReplyDelete
    Replies
    1. Thanks for your note.

      The pdf isn't a net, since it isn't contiguous - it just groups the faces into separate sets of 5. You need 12 sets of these groups - they are assembled like you would put together a dodecahedron.

      In this model, all of the "internal edges" on each set of 5 is a valley fold, so each side caves inward. The edges on the "tabs" are all mountain/hill folds.

      Delete