P3D Re: n-dimensional projections

[Bruce Springsteen <bsspringsteen@xxxxxxxxx>]

There are many web locations that provide both tutorials on hyperspace
(with figures of 4 and more dimensions) as well as mono and stereo, still
and animated and interactive depictions of regular n-space objects.  A
good place to start exploring links is the Geometry Junkyard at:

http://www.ics.uci.edu/~eppstein/junkyard/highdim.html

In the 1970's Thomas F. Banchoff at Brown University utilized computers to
create animated projections of various n-space objects.  His book "Beyond
the Third Dimension", published by the Scientific American Library
describes this work as well as providing a fascinating overview of
hyperspace in theory and art.  He became acquainted with Salvador Dali,
who made a number of stereographic paintings and incorporated
higher-dimensional objects into others. Banchoff's book serves as a prop
in my own Escher-inspired phantogram APEC stereocard "Knot Impossible",
which some of you have seen.

In the book, Banchoff mentions David Brisson, who founded an art group
called Hypergraphics, dedicated to making art depicting n-space ideas. 
Brisson invented what he called a "hyperstereogram" (unfortunately easy to
confuse with our notion of hyperstereo) in which a stereoscopic 3-d
projection of a 4-dimensional figure is viewed, but the parts of it only
can be fused one section at a time, by slightly tilting the head from side
to side.  This is supposed to convey some sense of the structure of
4-space, though it has never worked well for me.  Brisson describes his
work in an article entitled "Visual Comprehension of n-Dimensions", with
many stereo figures to illustrate.  I have an old photocopy of the
article, but unfortunately no idea what book it came from.

A. K. Dewdney in April 1986 wrote a "Computer Recreations" column in
Scientific American on creating hypercube rotation programs.  As I recall,
he mentions generating stereo pairs briefly as one option, with some
instruction.

One projection of the hypercube into 3-D space has an outer "shell" that
is a rhombic dodecahedron, which is a space-filling solid in 3-space.  An
artist friend I have mentioned here before works in glass and mirror, and
has produced a rhombic dodecahedron of half-silvered mirror which, when
lit from the inside, creates an infinite "froth" of similar figures.  It
sits in my living room.  This has implications of solid Esher-like
space-filling figures, as well as suggesting the possiblility of creating
of a true stereo-kaleidoscope as recently discussed on P-3D.  My friend
has made a mirrored central projection of a hypercube that was pictured in
the "Mathematical Reacretions" column of SciAm a while back, but as he
noted, a stereoscopic picture of his work would do it more justice.  I
recently mentioned a crude but enjoyable video that this friend made with
another man, showing stereoscopic projections of basic 4-space figures
rotating in all dimensions, to be viewed with the single-mirror stereo
method. 

Just a few leads and ideas to whet any appetities along these lines that
some of you may have.

Bruce 
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