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 > > > > - .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. > -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -.. > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC http://friam-comic.blogspot.com/
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