Re: [tuning] Digest Number 1440

Greg: It's an Euler genus all right, just another projection, one that
causes some vertices to coincide in 2-D space. I was inspired to program
it because I came across a book on the Fourth Dimension in Art
("Foufield: Computers, Art & the 4th Dimension) by Tony Robbin, an
artist who tries to depict 4-D objects in his own work. Back in the
1970, I met him in Houston and had some correspondence with him, but had
lost contact.

In his book, he showed part of a diagram of a central projection of a
7-D hypercube and I immediately saw that it was based on the
centered-heptagon lattice that I had programmed earlier (based on Erv
Wilson's ideas). So, I merely had to insert a table of the notes
corresponding to the hypercube or Euler Genus.

As far as I can tell, hypercubes only in spaces with odd numbers of
dimensions can be projected meaningfully this way. In the cases of
even-dimensioned spaces, I have had to use some sort of rectilinear lattice.

Since I already had the programs written, I decided to compute (or write
routines to do so) the notes for all the hypercubes from 2 to 9
dimensions and plot them. Most look very comprehensible and attractive,
except, perhaps, the 9-D hc which is basically a black lace doily with a
nearly solid center due to the low resolution of TrueBASIC and my Mac.
However, the overly of red dots where the notes are looks quite good.

The hypercubes in even-numbered spaces are less attractive -- the 2, and
4 D cases are fine-- a square and a projected tesseract, but 6 and 8 are
blobby, though if I have time to adjust the angles and lengths of the
underlying lattices, they may improve somewhat.

--John