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A couple years ago I made a codepen to generate arbitrarily long Genji-mon from a ruleset, and managed to write a rule that reduces the number of required special cases to one. However, these rules are very fragile, and "break" with lengths > 5 (edit: or maybe they don't! this deserves further exploration). Glad to see someone else discovering the structure behind this interesting system!

https://codepen.io/rezmason/pen/ejWogV

I tried setting `width` to 5 and found that your algorithm handles Yugiri correctly out-of-the-box. I tried commenting out various special case rules, such as lines 162-170, but it's the recursive algorithm itself that causes this. As you note on lines 204-214, your algorithm is not guaranteed to find the shortest solution; but the fact that it correctly handles Yugiri while mine does not suggests that it's closer to the intuitive, aesthetic judgement originally used to decide on the traditional layout.

That really surprised me (in a good way), because I had viewed Yugiri as an exception to an otherwise regular rule, while you showed it actually a natural consequence of a different rule! Of course, your algorithm also has special cases that need to be handled, so ultimately it may just be an aesthetic choice after all, but it proves that Yugiri may not be as irregular as I thought.

Great post, OWL! I love a good mathematical diversion with cultural context wrapped around it, it's like a beef wellington for the analytical mind.

Codepen is great for discourse like this but I want to put the conventional Genji-mon in a unit test and then go hog wild with rule construction, because I remember running out of steam exploring more robust rules that still satisfy `width` = 5.

I'm bearish on LLMs, for better or worse, but you seem to know your way around them. Do you suppose this problem could be attacked with some sort of trained rigmarole that writes rulesets?

This codepen's presentation strongly presents the "phantom squares crossroads" optical illusion.

This is a great article! Funny about Yugiri (39), because your version is a mirror image to Eawase (17), and the traditional Japanese Yugiri breaks that symmetry.

Every Genji-mon that is not self-symmetrical should have a mirror image, which I was able to quickly verify in the Genji-mon table you displayed.

Minori (40) is the only non symmetrical Genji-mon that breaks the rule, because it's the only one whose mirror image is also isomorphic to itself.

I can only imagine that there was something deliberate about the symmetry breaking for Yugiri, given the almost fanatical attention to detail in Japanese arts in general, and the equally strong penchant for deliberate imperfection in traditions like Kintsugi.

Loved this. Great writing and a bit of cultural arcana I knew nothing about but which resonates into the present day.

Obligatory entry in The On-Line Encyclopedia of Integer Sequences: https://oeis.org/A000110 which has 3 links mentioning Genji only one of which is not broken:

Xiaoling Dou, Hsien-Kuei Hwang and Chong-Yi Li, Bell numbers in Matsunaga's and Arima's Genjiko combinatorics [1]: Modern perspectives and local limit theorems, arXiv:2110.01156 [math.CO], 2021.

[1] https://arxiv.org/abs/2110.01156

I wonder if a constraint like “prefer the single lines to be shorter” would favor the traditional Yūgiri over the current optimal version.

No irony detected despite the tyranny of low expectations for the ability of bored Japanese nobles.

"Low expectations"? This article is about the specific language of literary allusion used to describe the solutions to competitions based on discerning subtly different incenses. Does that sound easy to you?

I don't think you understand what nobility was for. The point of aristocratic culture is to mediate alliances between powerful families. This:

> There has never been a group of people, in any time or place, who were so driven to display their sophistication and refinement. It wouldn’t do to merely put out a few sticks of incense; no, you would have to prove that your taste was more exquisite, your judgment more refined, your etiquette more oblique.

is the micro-politicking which undergirded macro-political moves like marriages, trade deals, and wartime alliances. Like a bird with colourful plumage, competence in this "frivolous" game represented having the wealth and power to spend all your time thinking about incense. Being good at this stuff was part of the job, and it mattered.

If your point is that aristocracy is a ridiculous system, sure, I don't disagree. But these people were not fooling around.