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 >> >> >> >> - .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. >> -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -.. >> 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/ > > - .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. > -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -.. > 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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