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

[Steve Smith]

PPS

Given  your preferences, you might want to check into Kepler's
Harmonices Mundi: https://en.wikipedia.org/wiki/Harmonices_Mundi though
doing it right would require becoming fluent in Latin.


>
> and why stop at 4 when you can go higher?
> https://www.youtube.com/watch?v=tfGf6gHQZQc
>>
>>>     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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