Listener No 4858, “Lavish Border”, A Setter’s Blog by Hawk

The term “mathematical group” as I used it is not entirely accurate. I didn’t really want to get into formal group theory in this puzzle, and I’m not sure I’d be thanked if I had. A group is a very specific entity, consisting of a closed set of elements which transform into other elements under particular operations and rules.

The seven frieze patterns you discovered in this puzzle are not elements of a single group, but each is the basis of a distinct mathematical group in itself, with a closed set of patterns formed by reflecting and rotating the frieze in various ways. In the case of pattern “dddd….”, the other elements of this set would be “bbbb…”, “qqqq…” and “pppp…”. The other six frieze groups will have different sets of related elements under transformation.

Group theory is a cornerstone of modern mathematics, having practical significance in the understanding of symmetry and structure in disciplines such as molecular chemistry and particle physics, among others. My aim with Lavish Border was simply to introduce friezes as being one of seven types, each having a distinct set of symmetries, as I felt this was a reasonable concept for non-mathematicians to follow. I also wanted to have solvers categorise a known frieze pattern into one of those seven types. Realising that each type could be represented using lower-case letters, I knew that this theme was potential crossword fodder.

There were alternative letters available, but I tried as far as possible to stick to the b, d, p and q set. One of the frieze types required a minimum string length of 4 (pqbd…), so I expanded all the other strings to length-4 for neatness. I chose the simplest grid I could imagine, with 6-letter entries throughout, then wove the patterns into extra letters in wordplay, making sure that each cell was checked by a crossing clue answer. Finding that the word “frieze” fitted into the pattern was a bonus, as was the simple clue it spawned. The Greek key looked like a good candidate for the frieze to be categorised, but as some classical examples differ in their symmetries, I had to hide a description of what was required in the remaining clues.

Numbering the clues would have made the puzzle too easy, and having them as a jigsaw would have made it too difficult, so I compromised by having the two sets, each presented in normal order. Finally, I chose a shorter word list to generate the grid, as I thought simpler words would allow for shorter clues, to offset the long preamble, and by and large it did.

I must thank Flowerman, for testing the puzzle and providing helpful advice, and the Listener editors, who improved the preamble and several clues. Feedback I’ve seen from solvers suggests that it wasn’t easy to get a foothold in the grid, but once this was achieved, the rest of the puzzle fell fairly quickly.

So, what’s next? One thing I regret not being able to include in this puzzle is a reference to John H Conway’s work on popularizing frieze patterns, in which he describes them in the form of dance steps such as “hop”, “sidle”, “spinning jump”, etc. Conway died from Covid-19 in 2020, shortly before I started working on Lavish Border. There is certainly scope for using his descriptors in a future puzzle on this theme, but I’m not planning to write it, so any setters reading this might like to tackle that one in future.

Friezes are among the simpler forms of patterns, as they extend and repeat along a line. More complex ones repeating in two dimensions are commonly known as wallpaper patterns. There are seventeen groups of these, and I would love to write that puzzle one day. Unfortunately, I could not rely on a square-celled grid alone, as some wallpaper patterns require parallelograms, rhombi, rectangles or hexagons. The puzzle would probably take up an entire Magpie issue!

And there are three-dimensionally repeating “crystallographic” patterns too, with a total of 230 types. Tempting though it might be to render these in puzzle form, the 3-D lattices it would require might be just too difficult to present on paper to a solver. When future generations move to holographic Listener crosswords, however, watch this space…