I'm sorry to have to correct myself, but the first method with rotations α,β,γ,δ is incomplete and lacks a rotation ε. Indeed, after rotation γ (3:41) OYV (red and orange axes) are still collinear, and after rotation δ (3:48) OYVZ (with green axis) are still coplanar! We need a rotation with X (blue axis) to leave coplanarity and achieve a truly 'arbitrary' 4-axes orientation, hence an extra rotation ε(X,V). The Desmos files and their links in my webpage are [being] adapted.
wait I don't understand what makes this a "true" 4-D projection _(vs._ e.g. the famous rotating hypercube animation) like, I mean, if this is true, what's the false one? just the previous attempt with elongation, or do you mean something bolder (like _"other videos aren't doing true projections!")_ I guess is what I'm asking also if you'd explain in baby steps & narrate, I think you'd get way more traction. (I have an excellent voice & good mic if you need a narrator in exchange for explaining to dumb ol' Kvel how this works-)
Hello Kvel, many thanks for your reply and, yes you were right: I meant my "corrected 4D projection" with "true 4D", in contrast with my earlier "bad, elongated 4D". But then I used to mean also with "true 4D" (even when with bad projections as before;-) those graphs that show an object in full 4D expansion, admittedly in projection of course, but with different results from the usual 3D graphs which cut off some 4th dimension part, say only Real or Imaginary parts or that lot. The popular views of rotating hypercubes you mentioned belong also to the family of "true 4D" views. I've been considering voicing my videos but I don't like doing it and hearing myself. Yet I'm dreaming of getting this fascinating stuff wider known, and at last adopted in professional software. However these guys keep, I don't understand why, stubbornly ignoring 4D possibilities (except for, you guessed it, the hypercube): Oh, I've just uploaded a video with a demo for doing Clifford tori, Dupin cyclides and Hopf fibration with my "Desmos4D".
I'm sorry to have to correct myself, but the first method with rotations α,β,γ,δ is incomplete and lacks a rotation ε. Indeed, after rotation γ (3:41) OYV (red and orange axes) are still collinear, and after rotation δ (3:48) OYVZ (with green axis) are still coplanar! We need a rotation with X (blue axis) to leave coplanarity and achieve a truly 'arbitrary' 4-axes orientation, hence an extra rotation ε(X,V). The Desmos files and their links in my webpage are [being] adapted.
wait I don't understand what makes this a "true" 4-D projection _(vs._ e.g. the famous rotating hypercube animation)
like, I mean, if this is true, what's the false one? just the previous attempt with elongation, or do you mean something bolder (like _"other videos aren't doing true projections!")_ I guess is what I'm asking
also if you'd explain in baby steps & narrate, I think you'd get way more traction.
(I have an excellent voice & good mic if you need a narrator in exchange for explaining to dumb ol' Kvel how this works-)
Hello Kvel, many thanks for your reply and, yes you were right: I meant my "corrected 4D projection" with "true 4D", in contrast with my earlier "bad, elongated 4D".
But then I used to mean also with "true 4D" (even when with bad projections as before;-) those graphs that show an object in full 4D expansion, admittedly in projection of course, but with different results from the usual 3D graphs which cut off some 4th dimension part, say only Real or Imaginary parts or that lot.
The popular views of rotating hypercubes you mentioned belong also to the family of "true 4D" views.
I've been considering voicing my videos but I don't like doing it and hearing myself.
Yet I'm dreaming of getting this fascinating stuff wider known, and at last adopted in professional software. However these guys keep, I don't understand why, stubbornly ignoring 4D possibilities (except for, you guessed it, the hypercube):
Oh, I've just uploaded a video with a demo for doing Clifford tori, Dupin cyclides and Hopf fibration with my "Desmos4D".