# TikTok teen points to inside elbow, bites lip: "Heeeeeeeesch"May 26, 2021 7:55 AM   Subscribe

A shape's Heesch number is, roughly, the number of times it can be surrounded by copies of itself. 0 is boring: it just can't. Infinity is boring: it tiles the plane. Positive but finite? Now that's the good stuff. Craig S. Kaplan has been looking for specimens and has turned up a bunch of new examples recently.

This is his follow-up on a four part series about Heesch numbers from a few years ago, which does a good job of explaining both the idea itself and his and others' processes for searching for specimens that show that interesting positive-but-finite behavior: part 1, part 2, part 3, and part 4.
posted by cortex (19 comments total) 46 users marked this as a favorite

Ohhhh, this is so satisfying! Now I want to see what forms enable clustering - a heesch number for heesch numbers? Fascinating.
posted by freya_lamb at 8:25 AM on May 26

It was only a matter of time until cortex collided with Craig Kaplan. He's done some neat things, such as Interwoven Islamic geometric patterns, Woven book polyhedra, Escher-like Spiral Tilings and Spin Draw, an iOS app that's a generalized version of the prot-Spirograph "Hoot Nanny" toy.

A friend worked with Craig on some installations for Bridges. Sounded like fun.
posted by scruss at 9:08 AM on May 26 [3 favorites]

This is crazy and very very cool! It boggles my mind how some of these complex shapes actually fit. I wonder if any physical puzzles exist of these exist...reaaaallly wish I could access the laser cutter at work right now (lab shutdown for covid..... Maybe soon though!)
posted by lemonade at 9:09 AM on May 26

lemonade: "I wonder if any physical puzzles exist of these exist"

I had some "SHMUZZLE PUZZLES" as a kid.
posted by chavenet at 9:30 AM on May 26

Craig's a friend. He's got a phone game out called Good Fences where you get to figure out the first Heesch hull for a bunch of different shapes. Nice and relaxing.
posted by seanmpuckett at 9:49 AM on May 26 [1 favorite]

He offered a first version of the translation from geometry into Boolean logic

ok, what? He clearly indicated this is a very big deal but maybe he could go into a little more detail here? This is the piece that seems to make all this automated searching possible.
posted by GuyZero at 9:56 AM on May 26

ok, what? He clearly indicated this is a very big deal but maybe he could go into a little more detail here? This is the piece that seems to make all this automated searching possible.

Section 3 of his paper [pdf] (heading "Computing Heesch numbers with a SAT solver") is relatively approachable.
posted by jedicus at 10:22 AM on May 26

As a friend of Craig, it's a pretty small world how many friends of Craig are in this thread.

Figure 3 in the paper is technically everything you need to know to construct the formula. Actually constructing the massive formula is probably quite the coding exercise.
posted by allegedly at 11:40 AM on May 26

As a friend of Craig, it's a pretty small world how many friends of Craig are in this thread.

Yes, but can the friends of Craig tile the adjacent space around him without gaps? What about friends-of-friends?
posted by notoriety public at 11:56 AM on May 26 [13 favorites]

hey what say we get together and pop open some coronas and then calculate some coronas
posted by cortex at 12:07 PM on May 26 [2 favorites]

Not right now, because of corona
posted by Pronoiac at 1:28 PM on May 26 [11 favorites]

Positive but finite is interesting, but whatâ€™s really intriguing is when the Heesch number is finite and negative.
posted by nickmark at 5:19 PM on May 26

I'm a friend of Craig! He's an awesome guy. My wedding ring is an adaptation, made with his blessing, of his own 3d designs.
posted by jackbishop at 8:01 PM on May 26 [1 favorite]

He's got a phone game out called Good Fences

That sounds so much like my kind of game I went to the app store on my phone immediately. And I haven't bought a phone game since Threes.

Alas, nothing came up in the search for that. [sad face]
posted by ctmf at 8:29 PM on May 26

[turns out since I somehow ended up in Music (formerly iTunes) I thought I'd buy a song I heard the other day and loved, and I can't figure out how to fucking even do that. What a horrid interface, I'm out of give a shit. Thanks, Apple.]
posted by ctmf at 8:36 PM on May 26

This looks like his page for the Good Fences app.

Unfortunately it looks like it is not currently available in the US iOS App Store.
posted by unus sum at 12:04 PM on May 27 [1 favorite]

I too am sad the app is not in the US iOS store.
posted by nat at 2:51 AM on May 28

I couldn't find much information, but regarding the translation I wonder if it would've been possible to use some kind of SMT solver to represent some of the computational geometry rules at a higher level in the constraint formula, and maybe that would also dealt with the issue of representing and suppressing holes.
posted by polymodus at 5:18 AM on May 28

I guess Good Fences fell prey to the inevitable iOS code rot, where if you don't spend several days every couple of years updating your app to the latest OS rev that it will become essentially non-functional. Which is why I stopped selling shit on the iOS store also. Either that or the \$100/yr access fee to keep your apps in the store wasn't recouping its investment. Or maybe both. Apple is not kind to small developers who value their personal time.

Alas.
posted by seanmpuckett at 11:08 AM on May 28 [1 favorite]

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