Rectified {5,3,6}

Rectified {5,3,6}

The dodecahedron just might be the most beautiful platonic solid.  Even if you disagree, regular honeycombs of the form {5,3,r} are certainly engaging because they are built of dodecahedra.  When r is 6, the dodecahedral cells become ideal.  Their vertices have gone to infinity.

https://en.wikipedia.org/wiki/Order-6_dodecahedral_honeycomb

The rectified {5,3,6} has icosidodecahedral and triangular tiling cells.  Before peeking at the picture, I wondered if the icosidodecahedral cells might also be ideal.  But again rectification makes the infinite edge lengths of the regular honeycomb finite, and the cells are no longer ideal.

But I still wonder... do you think there exists a uniform honeycomb that does have ideal icosidodecahedral cells?  My guess is yes, and I bet wendy krieger knows!

Comments

  1. wendy kriegerApril 11, 2014 at 8:52 PM

    And of course Wendy knows it exists. But you will not find it among the wythoffs, which is where all of your piccies have been coming from. The vertex figure is a tiling of golden rectangles, though.

    There is a "compact" or finite extent tiling of alternating cells of icosadodecahedra and rhombododecahedra, it is a wythoff tiling, but comes at x3o5x3o5z, which is unrelated to the regular tilings.

    But the piccies are nice :)

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