Friday, March 21, 2014

TUVWXYZ Hexagons

TUVWXYZ Hexagons, by Meenakshi Mukerji

Some time back, I posted the WXYZ model, which consists of four intersecting planes.  Following the invention of the WXYZ, origami masters created so-called "planar models" with more and more intersecting planes.  This one is an intersection of 7 planes.  Each plane consists of 6 units, for a total of 42 units.

While you can make four planes intersect in a symmetric manner, you might not expect 7 planes to intersect very gracefully.  Here's a diagram showing how they go together (from here):

Each color represents a single plane.  The diagram on the left shows what's above the black plane, while the diagram on the right shows what's below the black plane.  It turns out that the symmetry group is that of a tetrahedron--note that there are exactly four hexagons.  This is actually better than you can do with, say, 8 or 9 planes (but worse than you can do with 10!).

Another thing to notice is that these are not, strictly speaking, planes.  Notice that the diagram on the left is smaller than the diagram on the right.  All the "planes" intersect at the center of the model, so you'd expect there to be an equal amount on either side of each "plane".  They are simply approximations of planes.  Isn't that funny?

There's got to be some interesting math behind which numbers of planes allow for symmetric planar models.  But it's completely beyond me.