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
On 7/6/19, James Propp <jamespropp@gmail.com> wrote:
I've figured out one way to make a 3D realization of a
pseudo-octahedron,
by decorating the faces of a tetrahedron. (Maybe this was obvious to you all, but I didn't see it.) See
http://faculty.uml.edu/jpropp/mathenchant/semi-pseudo-octahedron.JPG
which shows how two of the four faces should be decorated, with three vertices per face. If you breathe into the tetrahedron to inflate it to become a sphere, the pattern of decorations on the tetrahedron should become a motley dissection of the sphere with 12 vertices, 6 pseudo-edges, and 8 three-sided pseudo-faces.
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.
Thanks,
Jim
On Thu, Jul 4, 2019 at 10:39 AM ed pegg <ed@mathpuzzle.com> wrote:
Related: Moritz W. Schmitt, On Space Groups and Dirichlet-Voronoi Stereohedra https://refubium.fu-berlin.de/handle/fub188/10176
I want to see pictures of all of these space filling plesiohedra. I've been wanting to see Engel's 38-sided spacefiller for years. --Ed Pegg Jr On Thursday, July 4, 2019, 07:58:05 AM CDT, Christian Lawson-Perfect <christianperfect@gmail.com> wrote:
Scott, I'm interested in giving it a go. I've got a couple of student working with me over the summer on 3d printing mathematical objects, so this'd be great.
On Thu, 4 Jul 2019 at 06:46, Scott Kim <scott@scottkim.com> wrote:
The pseudo octahedron as drawn has bar 3 symmetry, meaning 4-fold rotational symmetry plus top and lower hemispheres congruent but mirror images (I think...not sure). The pseudo-cube needs to be redrawn to work nicely on a sphere, as does the pseudo-dodecahedron. The 3d models I really want to have 3d printed are the pseudo 5-cell (easy) and pseudo 16-cell (hard), which have exteriors that are concave curivlinear tetrahedra. I can provide drawings if anyone's interested in giving them a try. — Scott
On Wed, Jul 3, 2019 at 9:07 PM James Propp <jamespropp@gmail.com> wrote:
And it would be good to have a picture of the pseudo-cube as an actual motley dissection of the sphere, but I have no idea how to realize it. I've put Scott's sketch of the combinatorial structure of the pseudo-cube at http://mathenchant.org/pseudo-cube.png; in both this picture and the pseudo-octahedron picture, it's important to keep in mind that the noncompact region in the picture is also a pseudo-face. In the symmetrical pseudo-cube, the six pseudo-faces would be centered at the vertices of an octahedron, but I don't see how the pseudo-faces fit together. Can any of you figure this out?
Jim
On Wed, Jul 3, 2019 at 9:28 PM James Propp <jamespropp@gmail.com> wrote:
> It would also be good to have a picture of the pseudo-octahedron. I've put > Scott Kim's picture of it at http://mathenchant.org/pseudo-octahedron.png; > there should be a realization of it as a motley dissection of the surface > of the sphere that has all the rotational symmetries of the octahedron (but > none of the reflection symmetries). Can Christian (or anyone else) > construct a virtual model? In some ways it might be harder to visually > parse the pseudo-octahedral motley dissection than the pseudo-cuboctahedral > motley dissection, even though the former is simpler, because the visible > hemisphere offers less information. But an interactive rotatable version > (or just a GIF that shows the object rotating) would solve that problem. > > Jim > > On Wed, Jul 3, 2019 at 2:49 PM James Propp < jamespropp@gmail.com> wrote: > >> Wow; this is great! >> >> Is there a way to rotate the view? >> >> Jim >> >> On Wed, Jul 3, 2019 at 3:38 AM Christian Lawson-Perfect < >> christianperfect@gmail.com> wrote: >> >>> I've made a 3d model in OpenSCAD and uploaded it to thingiverse: >>> https://www.thingiverse.com/thing:3726912. I've put a simple rendering >>> of >>> the shape on that page; I'm going to both 3d print it and do a nicer CGI >>> rendering now. >>> >>> On Tue, 2 Jul 2019 at 19:24, James Propp < jamespropp@gmail.com> wrote: >>> >>> > The link to the video apparently didn't survive the conversion from >>> HTML to >>> > PDF, so here's the URL for Scott Kim's video: >>> > >>> > https://youtu.be/xK1QA0Oi7iE >>> > >>> > Jim >>> > >>> > >>> > >>> > >>> > On Tue, Jul 2, 2019 at 12:03 PM James Propp < jamespropp@gmail.com> >>> wrote: >>> > >>> > > At http://faculty.uml.edu/jpropp/mathenchant/motley-draft3.pdf >>> you'll >>> > > find a draft of an essay that will be published on July 11th. >>> Comments >>> > are >>> > > welcome, but even more welcome would be a good picture of the >>> "spherical >>> > > pseudo-cuboctahedron"! In principle I know how to make such a >>> picture, >>> > sort >>> > > of: take the motley dissection of the surface of the cube (shown in >>> the >>> > > essay), push all the vertices out onto a sphere, and draw spherical >>> arcs >>> > > joining up those vertices. This seems like a job for Mathematica, >>> but my >>> > > mastery of 3D graphics isn't up to the job. I'll gratefully >>> acknowledge >>> > > anyone who can create a picture I can use! >>> > > >>> > > Thanks, >>> > > >>> > > Jim Propp >>> > > >>> > _______________________________________________