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