We may say that a polyomino is a connected set of squares in a grid; now I want to examine what we mean by “connected,” and what we can mean.

The usual definition requires cells to be connected along their edges. A common variation allows cells to connect either with neighbors either by edges or by corners. These have been variously called polykings, polyplets, and pseudo-polyominoes. My personal preference is polykings: I know what a chess king does, and it immediately suggests how a polyomino variation based on the concept would work. I don’t know what a plet is, and pseudo is both wrong, (there’s nothing fake or false about them) and unspecific, (if we are going to go around calling things false polyominoes, we have plenty of options.)

Wikipedia titles its page on them Pseudo-polyominoes, because Golomb called them that in his book, and printed books override all other forms of knowledge. Except, at the time I am writing this, everywhere on that page except for the title, where Wikipedia continues to calls them polykings.

Unfortunately, rather than leave well enough alone and stick with just “polykings”, I’m going to introduce my own confusing and awkward terminology. Because there are actually many more choices for ways to associate pairs of connected cells on a square grid, and we’re going to need a systematic method to keep track of them. As with “polykings”, chess piece moves more generally can correspond to allowed connections between cells in a type of polyomino. And we may turn to a standard notation for chess piece moves, which was developed by Ralph Betza and adopted more widely by chess variant designers.

The initials used to designate short piece moves mainly come from pieces in historical versions of chess. W, for Wazir, (Persian, minister or vizier) is the one orthogonal space move,. F, for Ferz, (Persian, counselor) is the one diagonal space move. D and A are for Dabbaba (Arabic, siege engine) and Alfil (Arabic, elephant) respectively. N is the standard initial for a chess knight, since the king takes the K. Instead of polykings, now we may speak of poly-WF-ominoes. Likewise, we can construct poly-WD-ominoes, and poly-WFD-ominoes, etc. Peter Esser has looked at the poly-WFD-ominoes, which he calls “Far-Bridged Polyominoes”, and also some symmetry variants based on them.

Not every possibility is equally interesting. For example, the poly-F-ominoes are equivalent in a way to the poly-W-ominoes: by rotating and dilating the positions of the cells, you can uniquely transform any of the former to one of the latter.

There is a practical problem with the poly-WF-ominoes and some of the other connection based variants we can come up with. The sets generally grow so fast with the size of polyomino that they skip right past the domain of practical manual solvability. Twenty-two 4-WF-ominoes is just too many, but five 3-WF-ominoes is trivial. But if we observe that just 12 of the 4-WF-ominoes aren’t W-ominoes or F-ominoes, we get a set that is manageable. We can call these the proper 4-WF-ominoes, and define them more formally as all of the 4-WF-ominoes for which every spanning tree of connections contains at least one W-connection and one one F-connection.

Unfortunately, the proper 4-WF-ominoes have odd checkerboard parity, so we can’t use them to tile a rectangle. I did however manually find a tiling of a 7×7 square torus with the proper 4-WF-ominoes plus a monomino.

Problem #51: There are some improvements upon that solution I’d like to see. How many of the following can be achieved, either individually or together?

1. Avoid any “Fault lines”, or runs of border segments that pass all the way through a tiling. With standard polyominoes, this is rarely difficult. These pieces have fewer orthogonal connections that can block a fault line. There are 14 rows and columns, and only 18 orthogonal connections, so they must be spent somewhat carefully.
2. “Crossroads”, or vertices that join four border segments are unavoidable due to the diagonal connections. A constraint that would be equivalent to “No crossroads” for standard polyominoes is “No four-piece corners”. Here, this constraint is weaker than the original, and should be achievable.
3. No crossed bridges. Physical sets of polykings require “bridges” joining cells across diagonal connections, and rounded corners that allow bridges to pass. Tilings where two bridges cross could not be made from such pieces. Bernd Karl Rennhak has explored bridged polyforms in depth. Kadon’s Roundominoes are another variation on the theme, where some “bridges” aren’t connected to cells on both sides.
4. Reduce the number of required colors. Since there are diagonal connections, same color pieces should also not meet at diagonals. (What I had previously called “strict” colorings.) But with rounded bridged pieces, that would not necessarily be visually confusing. Balanced colorings are nice.

Another way around the checkerboard parity issue is to use flexible rhombus based proper polykings. These can tile the following figure:

Problem #52: The same features that were desired in Problem #51 are also missing here. Can you improve it?

It seems that having spent so much time introducing Betza notation, I’m past my ideal post length, so I’ll have to close without using it enough to justify it for now. I hope to rectify this not too far in the future, when I intend to post about the proper poly-WFD-ominoes and other connection-based polyform variations.