[FRIAM] mathematicians computer graphic-ians — a little? help please

[Steve Smith]

>     First,
>
>     Just finished reading, _the crest of the peacock_ (ibid
>     lowercase), by George Gheverghese Joseph. Subtitle is
>     "non-European roots of mathematics." Wonderful book, highest
>     recommendation and not just to mathematicians.
>
>     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.
>
>     To the question/help request. Some roots of my problem:
>
>     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?
>
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. 

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.

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.

>
>     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?
>
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: https://www.youtube.com/watch?v=fjwvMO-n2dY

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

>
>     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.
>
The mathematical objects you are talking about are called regular convex
4-polytopes,  Wikipedia has a good article on the topic:

https://en.wikipedia.org/wiki/Regular_4-polytope

>
>     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?
>
The video above tumbles you through some regular 4 polytopes... I'll
give everyone else the trigger-warning <trippy man!>

This guy: https://www.youtube.com/watch?v=2s4TqVAbfz4 has added 3D
printed models frozen in mid-4D tumble to give you (maybe) some added
intuition.

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.  

I do look forward to your "trip report" and will take you to task if *I*
start dreaming in hyperspace again!

- Steve


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