Hi all,

Here is another recent rendering, of a zonohedral tiling, in this case, many
hundreds of zonohedra have been removed from the interior region and a red
light inserted. Also, the outer shells of zonohedra are missing. The tiling
is being built from the center out as it were. Eventually it would comprise
up a larger zonohedron (the Truncated Icosidodecahedron). The last set of
zonohedra added was colored transparent blue and given an irid finish.

I made the tiling using the software, Mathematica, in which I implemented
the "generalized dual" method to make a 3D tiling of zonohedra,
close-packing to comprise some larger zonohedron, from an "arrangement of
planes."

The thousands of zonohedra were organized into concentric sets, by distance
from the center. One hundred and eleven #include files were written, with
each zonohedron represented as a mesh object.

Each #include file had a number, as in "truico77.inc" etc.

In POV I set a WHILE loop from one file index to another, say, as in the
image, from 32 to 50.

Then all truico #includes from 32 to 50 are read in and assigned one name
and one texture.

Then (say) the 51st #include is read in and assigned a different texture.

I hope to develop many methods of visualizing these 3D tilings. Animation
seems crucial. If we were 4D beings, we could look down onto a nice flat
3-space and see all the zonohedra at once.

But we are not 4D beings.

Nevertheless, this zonohedral tiling has to do with the orthogonal
projection of a 15-dimension cube into a 3-space, or, more accurately, the
projection of an array of close-packing 15-cubes, fitting together to build
up a larger 15-cube, orthogonally and isometrically projected into a
3-space.

Now ordinarily such a projection would result in a horrible mishmash of
intersecting zonohedra.

The trick is to take this intersecting mish-mash and pick out some subset
which fit together face-to-face without intersecting.


RT

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