After a further week blundering around his personal rabbit-warren of misdirected mathematics, incompetent programming, and malignantly inconsistant computer algebra systems (don't get him started!), our hero emerges clutching the prize to his emaciated breast --- behold the *** magic football *** !! https://www.dropbox.com/s/ajd0inoaag1wtbm/pseudo_icosidodeca_movie.gif https://www.dropbox.com/s/5yrmef1djjuatfc/pseudo_cubocta_movie.gif https://www.dropbox.com/s/dc337j827udfzd4/pseudo_tetratetra_movie.gif Current versions of video loops for all three pseudo-polyhedra have been uploaded to the links above. A cheap trick to clean up spherical polygon boundaries via tiny radial dilation succeeded tolerably; I have been resisting the temptation to try imitating Christian Lawson-Perfect's neatly incut boundaries, for fear of increasing already generous video file sizes. Each boundary pseudo-edge is an arc of a geodesic great circle, spinning around a radial axis joining opposite vertices of the fixed inscribed polyhedron; and each pseudo-corner lies at the intersection of two such arcs. Once this principle is grasped, it is quite easy to knock up coordinates of the (moving) pseudo polygonal faces for the pseudo-cuboctahedron, with only a little further fumbling required for the pseudo-octahedron. But as I have just painfully established, winging it doesn't work for the pseudo-icosidodecahedron. There are now 15 geodesics, and it is horribly easy to pick the wrong intersections for corners of trigons and pentagons, which then proceed to scramble indistinguishably. A path through this particular maze required some computational graft, involving the order-60 symmetry group of icosahedral rotations. Fred Lunnon On 7/19/19, Scott Kim <scott@scottkim.com> wrote:
Nice job!
On Thu, Jul 18, 2019 at 7:20 PM James Propp <jamespropp@gmail.com> wrote:
Someone should make soccer balls that look like that!
Jim
On Thu, Jul 18, 2019 at 8:37 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< Of course the same idea works for the pseudo-icosadodecahedron ... Want to animate that too? >>
I suspected that this case was not going to be nearly so straightforward as the previous two, and had been attempting to avoid getting interested --- and as usual, my intuition proves to have been more reliable than my progamming. [ A wise man said "The trouble with computers is that they do _exactly_ what you tell them". ]
So to cut an embarrassingly long story short, herewith the current fruit of my labours, an offering to the Singularity in earnest hope of things to come:
https://www.dropbox.com/s/gebaxjochsdkl7o/pseudo_icosidodeca_static.gif
WFL
On 7/13/19, Scott Kim <scott@scottkim.com> wrote:
Definitely you'll have to give up some symmetry in 4d. The challenge remains to design an interesting morph animation of the pseudo-16-cell. Giving up some symmetry is the most common case; the pseudo cuboctahedron has most but not all of the symmetry of the cuboctahedron because it is handed. Other pseudo polyhedra, like the pseudo cube, are less symmetrical; the most symmetrical spherical pseudo cube I know of has 4-bar symmetry, and the pseudo dodecahedron has 5-bar symmetry...but I haven't pushed hard on this.
I haven't animated it yet, but there's a beautiful animation of the rotating pseudo-cube I'll make to show the group. The logic is a bit different from the line Fred's been following...and also leads to a movie of a rotating pseudo-24-cell. I'll make the rotating pseudo-cube animation and perhaps someone else can tackle the higher-dimensional version, which should be spectacular.
On Sat, Jul 13, 2019 at 11:50 AM Fred Lunnon <fred.lunnon@gmail.com> wrote:
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 > > > >
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