Re: [math-fun] Draft of short essay on Scott Kim's motley dissections
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 >>> > > >>> > _______________________________________________
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
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
<< 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
_______________________________________________ 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
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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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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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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Very pretty! Jim On Fri, Jul 26, 2019 at 1:23 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
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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Scott, Fred, Christian, and others, It occurs to me that the sort of animation that works so beautifully for morphing motley dissections of the sphere is equally suited to morphing motley dissections of the torus presented as periodic motley dissection of the plane. (Picture an orderly field of rotating lines, which are truncated at places where they cross other lines in accordance with some precedence rule so as to create three-way junctions.) I’m planning to rewrite my piece on motley dissections for use as a Mathematical Enchantments essay sometime around the end of the year, and the more cool animations I have, the better! Jim On Fri, Jul 26, 2019 at 2:03 PM James Propp <jamespropp@gmail.com> wrote:
Very pretty!
Jim
On Fri, Jul 26, 2019 at 1:23 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
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 > >> > > >> > > > _______________________________________________ > 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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Videos of planar morphing pseudo-tilings: Square-square chessboard at https://www.dropbox.com/s/6ut8jbc1rz4s497/pseudo_squasqua_movie.gif Triangular-hexagonal at https://www.dropbox.com/s/il2hl1r2izfvbg6/pseudo_trihex_movie.gif --- completing the family of 5, unless somebody can dream up another (groan)! The chessboard is very reminiscent of a video on the web a few years ago, showing alternating sets of square tiles morphing in similar fashion, but with random sizes and colours. Anybody have a reference? The hexagonal tri-hex region can be `wrapped' into a homotopy torus, via identifying (hexagonal tiles across) 3 pairs of opposite sides, and corners in alternate triplets. This preserves the full symmetry; while opening a didactic can of worms: it is not at all apparent how --- or even whether --- the resulting immersive `bottle' surface is deformable into a standard annular torus! Tilings of the torus wrapped from a rectangular region are topologically distinct from those above, suggesting further knotty questions: eg. counting distinct tilings of the torus with given number of hexagonal tiles at trivalent vertices. Nobody will be surprised that my initial assumption that the earlier polyhedral program could be quickly be tweaked to cope with planar tilings proved optimistic. However, retaining a 3-D framework will at least make it straightforward to clip such regions into nice clean rectangles --- at least, I'm almost sure it will ... WFL On Thu, Aug 29, 2019 at 12:52 PM James Propp <jamespropp@gmail.com> wrote:
Scott, Fred, Christian, and others,
It occurs to me that the sort of animation that works so beautifully for morphing motley dissections of the sphere is equally suited to morphing motley dissections of the torus presented as periodic motley dissection of the plane. (Picture an orderly field of rotating lines, which are truncated at places where they cross other lines in accordance with some precedence rule so as to create three-way junctions.)
I’m planning to rewrite my piece on motley dissections for use as a Mathematical Enchantments essay sometime around the end of the year, and the more cool animations I have, the better!
Jim
On Fri, Jul 26, 2019 at 2:03 PM James Propp <jamespropp@gmail.com> wrote:
Very pretty!
Jim
On Fri, Jul 26, 2019 at 1:23 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
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
Nice! Thanks. Jim On Wed, Sep 4, 2019 at 3:37 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Videos of planar morphing pseudo-tilings: Square-square chessboard at https://www.dropbox.com/s/6ut8jbc1rz4s497/pseudo_squasqua_movie.gif Triangular-hexagonal at https://www.dropbox.com/s/il2hl1r2izfvbg6/pseudo_trihex_movie.gif --- completing the family of 5, unless somebody can dream up another (groan)!
The chessboard is very reminiscent of a video on the web a few years ago, showing alternating sets of square tiles morphing in similar fashion, but with random sizes and colours. Anybody have a reference?
The hexagonal tri-hex region can be `wrapped' into a homotopy torus, via identifying (hexagonal tiles across) 3 pairs of opposite sides, and corners in alternate triplets. This preserves the full symmetry; while opening a didactic can of worms: it is not at all apparent how --- or even whether --- the resulting immersive `bottle' surface is deformable into a standard annular torus!
Tilings of the torus wrapped from a rectangular region are topologically distinct from those above, suggesting further knotty questions: eg. counting distinct tilings of the torus with given number of hexagonal tiles at trivalent vertices.
Nobody will be surprised that my initial assumption that the earlier polyhedral program could be quickly be tweaked to cope with planar tilings proved optimistic. However, retaining a 3-D framework will at least make it straightforward to clip such regions into nice clean rectangles --- at least, I'm almost sure it will ...
WFL
On Thu, Aug 29, 2019 at 12:52 PM James Propp <jamespropp@gmail.com> wrote:
Scott, Fred, Christian, and others,
It occurs to me that the sort of animation that works so beautifully for morphing motley dissections of the sphere is equally suited to morphing motley dissections of the torus presented as periodic motley dissection of the plane. (Picture an orderly field of rotating lines, which are truncated at places where they cross other lines in accordance with some precedence rule so as to create three-way junctions.)
I’m planning to rewrite my piece on motley dissections for use as a Mathematical Enchantments essay sometime around the end of the year, and the more cool animations I have, the better!
Jim
On Fri, Jul 26, 2019 at 2:03 PM James Propp <jamespropp@gmail.com> wrote:
Very pretty!
Jim
On Fri, Jul 26, 2019 at 1:23 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
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
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participants (4)
-
Bill Gosper -
Fred Lunnon -
James Propp -
Scott Kim