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<div style="font-family:Arial">First,<br>
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<div style="font-family:Arial">Just finished reading, <u>the
crest of the peacock</u> (ibid lowercase), by George
Gheverghese Joseph. Subtitle is "non-European roots of
mathematics." Wonderful book, highest recommendation and
not just to mathematicians.<br>
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<div style="font-family:Arial">My three biggest shames in
life: losing my fluency in Japanese and Arabic; and
excepting one course in knot theory at UW-Madison,
stopping my math education at calculus in high school. I
still love reading about math and mathematicians but wish
I understood more.<br>
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<div style="font-family:Arial">To the question/help request.
Some roots of my problem:<br>
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<div style="font-family:Arial">One) I am studying origami
and specifically the way you can, in 2-dimensions, draw
the pattern of folds that will yield a specific 3-D
figure. And there are 'families' of 2-D patterns that an
origami expert can look at and tell you if the eventual
3-D figure will have 2, 3, or 4 legs. How it is possible
to 'see', in your mind, the 3-D in the 2-D?<br>
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<p>I've only dabbled with origami and share your implied questions
about the way people who work with it a lot seem to be not only
able to "guess" what a 2d pattern of folds will be in 3d but can
"design" in 2d to yield 3d shapes. I suspect a formalization of
how they do it is closer to group theory than geometry. As for
"how is it possible?" I think that is the fundamental question
for all forms of "fusing" sensory data of one type into higher
level abstractions. The only way I know to acquire such a skill
is to practice, practice, practice. <br>
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<p>For highD data, that means (for me) working in as
high-dimensional of a perception space as possible (e.g. stereo +
motion parallax with other depth cues like texture and saturation
and hue. Manipulating the object "directly" with a 3D pointer
(spaceball, etc.) or better "pinch gloves" or even better,
haptic-gloves (looking a bit edward scissorhandy). My best
experiences with all of this have been in a modestly good VR
environment (my preferred being Flatland from UNM, named after EA
Abbot's Victorian Romance in Many Dimensions (for the very reason
you are asking about this I'd say)) on an immersive workbench (8'
diagonal view surface tilted at 20+ degrees with active
stereography, head and hand tracking, and pinch gloves). You
literally "reach out and grab geometry and rotate/drag it
around". I'd also recommend "listening" to them, but that can be
a little trickier.<br>
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<p>Staring at clouds and other phenomena which are 3D ++ (the shape
of a cloud as observed is roughly an isosurface of temperature,
pressure, humidity over the three spatial dimensions) as they
evolve (facilitated by timelapse and best observed as they
"squeeze" over mountains or "form" over bodies of water.<br>
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<div style="font-family:Arial">Two) a quick look at several
animated hyper-cubes show the 'interior' cube remaining
cubical as the hypercube is manipulated. Must this always
be true, must the six facets of the 3-D cube remain
perfect squares? What degrees of freedom are allowed the
various vertices of the hyper-cube?<br>
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<p>The conventional projections of the Tesseract into 3D are only
rotated around the yz, xz, xz axes... the additional ones that
include the w axis do not present as "perfect cubes". See second
:40 and on in this video: <a
href="https://www.youtube.com/watch?v=fjwvMO-n2dY">https://www.youtube.com/watch?v=fjwvMO-n2dY</a></p>
<p>It might be easier to accept this if you notice that off-axis
rotations of a cube when projected into 2D yield non-square faces
in 2D<br>
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<div style="font-family:Arial">Three) can find static
hyper— for the five platonic solids, but not animations.
Is it possible to provide something analogous to the
hypercube animation for the other solids? I think this is
a problem in manifolds as many of you have talked about.<br>
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<p>The mathematical objects you are talking about are called regular
convex 4-polytopes, Wikipedia has a good article on the topic: <br>
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<p><a href="https://en.wikipedia.org/wiki/Regular_4-polytope">https://en.wikipedia.org/wiki/Regular_4-polytope</a></p>
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<div style="font-family:Arial">Question: If one had a series
of very vivid, very convincing, visions of animated
hyper-platonic solids with almost complete freedom of
movement of the various vertices (doesn't really apply to
hypersphere) — how would one go about finding
visualizations that would assist in
confirming/denying/making sense of the visions?<br>
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<p>The video above tumbles you through some regular 4 polytopes...
I'll give everyone else the trigger-warning <trippy man!></p>
<p>This guy: <a href="https://www.youtube.com/watch?v=2s4TqVAbfz4">https://www.youtube.com/watch?v=2s4TqVAbfz4</a>
has added 3D printed models frozen in mid-4D tumble to give you
(maybe) some added intuition.<br>
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<p>There are a plethora of commercial HMDs out now that would
facilitate a great deal more than just staring at your laptop
while geometry tumbles through 3, 4, nD. These days I bet you can
drop your phone into a google-cardboard device ($3 on amazon),
load up a copy of Mathematica or similar and find a program to let
you tumble yourself through these experiences. <br>
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<p>I do look forward to your "trip report" and will take you to task
if *I* start dreaming in hyperspace again!</p>
<p>- Steve<br>
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