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<p>PPS</p>
<p>Given your preferences, you might want to check into Kepler's
Harmonices Mundi: <a
href="https://en.wikipedia.org/wiki/Harmonices_Mundi">https://en.wikipedia.org/wiki/Harmonices_Mundi</a>
though doing it right would require becoming fluent in Latin.<br>
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<div class="moz-cite-prefix">and why stop at 4 when you can go
higher?</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>
</p>
<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>
</p>
<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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