<!DOCTYPE html><html><head><title></title><style type="text/css">p.MsoNormal,p.MsoNoSpacing{margin:0}</style></head><body><div style="font-family:Arial;">First,<br></div><div style="font-family:Arial;"><br></div><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></div><div style="font-family:Arial;"><br></div><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></div><div style="font-family:Arial;"><br></div><div style="font-family:Arial;">To the question/help request. Some roots of my problem:<br></div><div style="font-family:Arial;"><br></div><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></div><div style="font-family:Arial;"><br></div><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></div><div style="font-family:Arial;"><br></div><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></div><div style="font-family:Arial;"><br></div><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></div><div style="font-family:Arial;"><br></div><div style="font-family:Arial;">Please forgive the crude way of expressing/asking my question. I am both math and computer graphic ignorant.<br></div><div style="font-family:Arial;"><br></div><div style="font-family:Arial;">davew<br></div></body></html>