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? 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 > > > > > _______________________________________________ > > math-fun mailing list > > math-fun@mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
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