What the heck is this thing?
(**) One thing it is: an MP4. See notes at end for semi-technical info.

In the Spring of 2014 I (susp?)ended my obsessive quest for super-tiles of the plane composed of the eleven hexominoes that form the nets of the cube. With 6x11 = 66 squares to be used, I found (or rather, confirmed by computer program) that there are no 6x11 or 3x22 super-tiles. But I discovered (*) three non-rectangular super-cube-net-tiles, as seen below, and showed them in the Friday Seminar at LSUS.

2

#1

#3

Note that these tile the plane using translations only. (Click on each to see how.) There are surely many more solutions. These three each fit in an 8x12 rectangle, and I wager there are more of those. Are there any that fit in a smaller rectangle? My own experiments suggest there are none that fit in a 7x11 or 6x12 rectangle. I'll bet some of the folks on the PUZZLE FUN Facebook group could find more. I shall inquire.

(Later that day...) Ha! I did inquire, and less than 24 hours later excellent answers arrived! Here is the "Netherlands" version by Edo Timmermans:

And a flapping version, one cube at a time.

The choices of base locations for the cubes within the flapping supertiles are variable but the decisions here were not entirely arbitrary. Mainly, I didn't want the unfoldings smacking into one another. And I tried for a wee bit of regularity. That was about the extent of any artistic consideration.

(*) It was around 25 April 2014, using a modification of Gerard's Universal Polyomino Solver.

(**) Technical note and reminder to myself: The original file was an animated GIF, created in Mathematica. The following was used to convert the GIF to MP4:
ffmpeg -i NAME.gif -movflags faststart -pix_fmt yuv420p -vf "scale=trunc(iw/2)*2:trunc(ih/2)*2" NAME.mp4
Without the -pix_fmt yuv420p option, the results were either poor quality or the inability to be loaded and played by browsers other than Chrome. I tried many variations and found the advice that worked on unix.stackexchange.com, which in turn came from the commercial website Rigor.com. I naively tried creating WEBM, OGV, and SWF versions using the same ffmpeg settings, but the WEBM was grainy, the OGV was nuts, and I didn't feel like downloading Flash to test the SWF. Different browsers did different things with the various formats. I tested the MP4s above in Chrome, Firefox, Opera, IE, and Edge (all the latest versions as of 23 Nov 2019, in Windows 10) and all worked well and the same. Thanks to Edo on PUZZLE FUN fo alerting me to the non-Chrome troubles.


If you have read this far, then maybe more of this sort will amuse.

And maybe this one to wrap it up.