The results for my Bridges 2019 submission “Animated Isohedral Tilings” are all animated GIFs. Click on each of the links below to see the animations (I didn't want to put them all on the same page—that would be visually overwhelming). When relevant, I include a few additional comments.
For more discussion, and a few additional results, please see my blog post about this paper.
The animation ih73_zoomout.gif is the one discussed in Figure 3 of the submission. See also the closely related animation ih73_noscale.gif. The latter moves between two different pinched configurations, meaning that no scaling is necessary.
Figure 3 also shows two animations based on IH90: these are ih90_1to3.gif (the large triangle splits into three smaller triangles) and ih90_1to4.gif (the large triangle splits into four).
Figure 5 talks about a topological transition from a hexagonal tiling to 60 degree rhombs. See the animation ih18_ih36.gif.
Figure 6 discusses a “grand tour” of the three regular tilings of the plane. That animation is provided in regular_grand_tour.gif. It would be good to create a longer director's cut that moves through more topological types. Stay tuned.
In response to a response on Twitter, I also created the animation hex_square.gif, which shows a different means of moving between squares and hexagons.
Also related is ih13_square_rhomb.gif, which moves from axis-aligned squares to squares oriented on the diagonal, by passing through hexagons as an intermediate step. Then we just rotate and scale separately to move back to squares.
Here's an animation not appearing in the paper, in which I use IH 18 to split a single hexagon into three smaller hexagons. It combines notions of point coincidence and edge coincidence: ih18_1to3.gif.
The animation in ih33_qbert.gif doesn't scale, but closes up on itself after a rotation of an edge of the tiling combined with a counterrotation of the tiling as a whole. The result plays with the tension between 2D and 3D.
Like IH73, IH61 provides a number of opportunities for simple, elegant animations. Here are a few that use different combinations of rotation, translation, and scaling in order to bring the final frame into alignment with the beginning: ih61_1.gif, ih61_2.gif, ih61_3.gif. I also created an animation ih61_zigzag.gif in which edges are brought into coincidence, but I'm not quite as happy with it.
