In planning out blog posts, I typically use the rule of thumb that a post should contain between two and four images. If I have five or more images, I try to break the material up into multiple posts, and if I have only one I will usually wait until I have more material. A single image is a still life; when there are more, I can tell a small story. Unfortunately, this means that over time I build up a number of “orphans”, perfectly good results left to gather dust because I have nothing else that is closely related. I recently noticed that several of my orphans were square shaped. Perhaps I could use this to tie some disparate results that normally wouldn’t belong together into the same post.
1) I noticed last year that the tetrakis grid 4- and 5-tans had a combined area that could make a square. This seemed like a pretty basic result that had to have been known years before. I asked (and looked) around, but I couldn’t find any instances of it being found previously. Then in preparing this post, I discovered that I was almost right. It had, in fact, been found months before, by Thimo Rosenkranz. (That page isn’t visibly dated, but a timestamp in the html code indicates it was published in July of 2023.) For all I know, these pieces may have been studied many times previously, but having done the important step of preventing myself from falsely claiming primacy, I am content to leave off further historical research.
2) I claimed that I couldn’t do anything elegant with just the set of single-striped tetrominoes, but shortly after I posted that, I found the figure below. Here, I relaxed the rule that the stripes must form an unbroken line from one edge of a figure to another. Instead the stripes may have a singe-cell gap in order to allow a perpendicular stripe to pass through. A woven pattern, where each stripe crosses one other and allows another to cross it, seemed like the nicest possibility. (The faded dashed lines are artistic license on my part, not actual markings on the pieces.)
3) The pieces here are based on those in my L-Topia puzzle. That puzzle contained every way to mark two cells of an L tetromino with a circular and a square hole. When I had access to a laser cutter in 2009, I made a variation where, instead, the holes were double headed arrows that could point horizontally or vertically. I was recently reconsidering “pilings”, (tilings with overlap) and wondered what could be done with pieces with marking sites where the markings have two possible orientations by symmetry, and the overlaps must match one orientation with the other. My L-Topia variation came to mind, but I had better results with a similar variation, where the arrows would be diagonal instead. Here, instead of cutting out arrow-shaped holes, I cut out triangles on opposite corners, leaving a bar to overlap. There is a unique piling where the overlaps form a single loop joining all of the pieces.
4) At G4G15, I gave a talk on polykings and other polyomino variations where cells are connected in different ways from standard polyominoes. I based the talk on a couple of earlier blog posts, but I did throw in one additional result. One of my usual creative tricks is “mix ideas, even if they aren’t meant to be mixed.” In the context of tiling problems, that can work out to “tile with sets of pieces that aren’t meant to be mixed.” If we build the five tetrominoes with each of the five shortest cell connection types, we get 100 total cells. Why not throw them together into a 10×10 square? In the diagram below, each tetromino form is represented by a different color, while the crosses indicate the directions and magnitudes of the connection vectors.
Problem #64: Well, everything is always nicer if you can make things of the same kind not touch. Here that means both tetrominoes of the same form, and tetrominoes with the same connection type. But asking for both may be too much, so we may have to accept just one or the other. But wait! What exactly do we mean by not touching here? Simply using Wazirwise adjacency seems contrary to the spirit of the thing. Let us say that two pieces touch if the connection type of one of the pieces can reach, from one of its cells, a cell in the other piece.
We might like there to be some theme that we can tease out from these unrelated square tilings that can retroactively justify shoveling them into the same post. If there is, it might be parity of elegance. A square has high elegance, having maximal symmetry in a smooth, convex shell. But as a consequence of this elegance, we don’t have many sizes of squares available, just one for each integer, so we have to contrive somewhat inelegant sets of pieces to tile them. In the case of the striped tetrominoes, the fixed dimensions of the square forced me to fudge the stripe placement rules to get the stripes to fit.
Of course, there is no such thing as parity of elegance in the universe of possible tilings; many very inelegant tilings may be found. What we see is the effect of me selecting the most elegant tilings that I can find to share on my blog. In these cases, the elegance of the square justified less elegance in other aspects of the tilings.
