RWG << How hard would it be to twist the whole scene so that the front face doesn't rotate? >> Face centres with tangent planes are fixed along axes, coordinates of vertices are known, so a suitable counter-rotation angle can be computed trivially and fed into each frame plot command of the animation. Simples! [ And on cue, from long ago, the ghost of Derek Morris taps me on the shoulder, reminding me with his customary patient resignation: "You should know that by now, Fred --- nothing is easy!" ] Speaking of which ... SK << The 3d models I really want to have 3d printed are the pseudo 5-cell (easy) and pseudo 16-cell (hard) ... >> Initially it looked plausible that analogous morphing spherical pseudo-octahedron & pseudo-cuboctahedron constructions should generalise to higher dimensions; thus near the end of Scott Kim's video https://youtu.be/xK1QA0Oi7iE he seems to discuss a pseudo-16-cell via which one 4-space simplex could morph into another. However, a day spent following many false trails convinced me that no such object exists: in particular, the obstruction to the latter example being the impossibility of arranging four large regular (spherical) tetrahedra around one small one, each small face coplanar with some large one, while maintaining full tetrahedral symmetry. [ Exercise: prove this on one line of text! ] Fred Lunnon [13/07/19] SK << The 3d models I really want to have 3d printed are the pseudo 5-cell (easy) and pseudo 16-cell (hard) ... >> Initially it looked vaguely plausible that analogous morphing spherical pseudo-octahedron & pseudo-cuboctahedron constructions should work in higher dimensions too; near the end of Scott Kim's video https://youtu.be/xK1QA0Oi7iE he seems to construct a pseudo-16-cell via which one 4-space simplex could morph into another. However, a day spent following many false trails convinced me that no such object exists: in particular, the obstruction to the latter example being the impossibility of arranging four large regular (spherical) tetrahedra around one small one, each small face coplanar with some large one, while maintaining full tetrahedral symmetry. (Exercise: prove this on one line of text!) Fred Lunnon [13/07/19] JP << Can Christian (or someone else) create an image of this for me? I'd love to use it in my next mini-essay (~ 1000 words) for the Big Internet Math-Off, as well as in the longer Mathematical Enchantments piece on the subject of motley dissections that I plan to write later this summer or in the Fall. >> Animations of pseudo-octahedron and pseudo-cuboctahedron (re-)posted at https://www.dropbox.com/s/jzc7z2clgeew1he/pseudo_octa_movie.gif https://www.dropbox.com/s/5yrmef1djjuatfc/pseudo_cubocta_movie.gif Static frames available to order on request. WFL [11/07/19] On 7/11/19, Bill Gosper <billgosper@gmail.com> wrote:
On Thu, Jul 11, 2019 at 3:14 AM rwg <rwg@ma.sdf.org> wrote:
-------- Original Message -------- Subject: Re: [math-fun] Draft of short essay on Scott Kim's motley dissections Date: 2019-07-10 12:59 From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com>
Beautiful!
Thanks, Fred!
Jim
I'll second that! How hard would it be to twist the whole scene so that the front face doesn't rotate?
We can change the problem so that the answer is an ancient Macsyma animation most of you should recall. Problem: Reshape the six "squares" so they can simply rotate, while four "triangles" alternately swell, then shrink away, while the other four complementarily follow suit in the opposite phase. The edges need no longer lie on great circles. Spoiler Lissajous arcs <http://gosper.org/lisspump.gif>
On Wed, Jul 10, 2019 at 3:41 PM Fred Lunnon <fred.lunnon@gmail.com>
wrote:
I have posted an animation at https://www.dropbox.com/s/5yrmef1djjuatfc/pseudo_movie.gif of a spherical pseudo-cuboctahedron (SPCO) morphing continuously via octahedron -- big triangles -- cuboctahedron -- big squares -- cube, then back again via the mirror-images; Maple program is available on request.
This should run continuously when the link is opened in a browser: please advise me of any problems!
Time is proportional to `twist' angle t between (plane containing) SPCO edge arc and associated edge of scaffolding cube, modulo pi . SPCO vertices are given by 24 cube symmetries (with even resigned permutations of components) of point with Cartesian coordinate P(t) = (a^2 + a*b, a^2 - a*b, a^2 + b^2)/d , where b = cos t , a = (1/sqrt2) sin t , d^2 = 1 - a^4 .
For octahedron t = 0 , for cube t = pi/2 ; for cuboctahedron t = arccos(-1/3)/2 . For Jim Propp's case, where edge plane meets 4 vertices of the scaffold cube, rather unexpectedly t = arccos(+1/3)/2 --- just pi/2 minus the cuboctahedron angle!
While I was busily upstaging his previous version, Christian has quietly replaced that static CGI view with a 3D-printed solid model: see https://www.thingiverse.com/thing:3726912
Hmmm ... a mechanically functional solid morphing model, anybody?
The only hope would be van Deventer. —rwg
Finally, note that a similar method should cope straightforwardly with morphing pseudo-octahedron and pseudo-icosidodecahedron.
Fred Lunnon