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<div class="moz-cite-prefix">and why stop at 4 when you can go
higher?</div>
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href="https://www.youtube.com/watch?v=tfGf6gHQZQc">https://www.youtube.com/watch?v=tfGf6gHQZQc</a></div>
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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>
</p>
<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"
moz-do-not-send="true">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"
moz-do-not-send="true">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"
moz-do-not-send="true">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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