Rectified {3,6,3}

Rectified {3,6,3}

The regular {3,6,3} honeycomb has triangular tiling cells. The vertex figure is a hexagonal tiling.  

http://en.wikipedia.org/wiki/Triangular_tiling_honeycomb

It is one of three especially exotic regular honeycombs because infinities exist in both its cells and vertex figure.  

- Each cell has an infinite number of facets.
- Every cell facet is infinite in extent.  
- All the vertices live at infinity.  
- An infinite number of cells meet at every vertex. 
- All edges are infinitely long.  
- The in-radius, mid-radius, and circum-radius are all infinite.  
- The volume of each cell is infinite.  
- Each cell of the honeycomb is inscribed by the entirety of hyperbolic space, which makes the cells somehow feel "bigger" than those in many of the other honeycombs.

In short, just about every property of this thing has flown the coop.

But check out the rectified {3,6,3} in these images.  Rectification produces finite edge lengths again!  How'd that happen?

Can you figure out the two rectified {3,6,3} cell types?  (hint: one of them has shown up multiple times the past few days.)

Comments

  1. Math SolverSeptember 17, 2015 at 9:56 AM

    These all honeycomb (presented above (and below)) are really very very beautiful.

    ReplyDelete

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