fogleworm permutatations

Fogleworm I

(Store links: Fogleworms prints broken into four sets 1-24, 25-48, 49-72, and 73-96; Fogleworm Ghost prints 1-4.

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Over the course of two days in early February of 2021 I created 96 modular linocut prints (and four associated "ghost" prints) based on a mathematical proposal by Michael Fogleman the prior month. Michael (who posts as @FogleBird on twitter) had, on January 13th, said:

"How many ways can you pack N worms of length N in an NxN square grid?"

In other words, if you have e.g. a 4x4 grid, and a set of "worm" lines of various somewhat tetris-y shapes that each cover four consecutive dots of that grid, how many distinct ways are there to arrange 4 worms to exactly fill that grid? (Or likewise for 3 worms on a 3x3 grid, 5 worms on a 5x5 grid, etc.) Which is an interesting puzzle to me, for a few reasons: 1. It's very simply stated. 2. The actual problem is harder to solve than it sounds like it might be. 3. The solutions lend themselves to a pleasingly graphical output of twisting lines arranged in space.

trivial Fogleworm configurations for N=1 through N=4

I did a little sketching of these worms over a beer that evening, then woke up in the morning still thinking about them, and that was the start of a couple of weeks of obsession where I played with the problem in a few ways and also enjoyed mucking around with digital renderings of these objects. I also offhandedly named them Fogleworms after Michael, which I'm happy caught on a bit with other math nerd types on twitter.

(These worms would in mathematical literature be more typically described as "lattice paths", a subject with which I wasn't more than very glancingly familiar before diving into this. I am still only lingering-eye-contactingly familiar now, but it's been fun digging in a little. It's interesting to note that while there are many sequences in the Online Encyclopedia of Integer Sequences related to lattices and lattice paths, including some closely related to subsets of this problem, there doesn't seem to be any sequence corresponding to Michael Fogleman's specific proposition of which the print work here represents one set.)

I tweeted a lot about this stuff for the next couple weeks, like really just a great deal of fogleworm content at length, learning to program in Mathematica to automate my work on the problem and chasing down random tangents. In all that the idea of doing a set of linocut prints kept growing in the back of my head.

I documented some of this previously, and talked a little bit more about the math and programming aspect of it, in this blog post from early February.

an ordered set of Fogleworm configurations

In order to create this set of prints, I used my GlowForge laser cutter to cut out squared-off blocks of linoleum in various worm shapes and etch a subtle outline of the circular curves of the worms themselves which I then went at with carving gouges to create my individual worm modules from which each print was assembled.

Working from a reference sheet I created (via the programming work mentioned above), I then laid out each print by inking four separate pieces of linoleum in four colors, assembling them within a frame on my etching press, and pulling a single print of each configuration. I documented the printing process start-to-finish in this twitter thread of pictures and notes.

Ghost XLIX - LXXII

In addition to the 96 prints of the main series, I also created four "ghost" prints, one for each consecutive block of 24 prints. I did this by setting a fresh sheet of good print paper on the bed of the etching press before working through each subset of prints; normally I'd use waste paper for this sort of thing because it's there just to keep ink from getting on the press itself, but I have found by accident in the last year that the patterns left behind by a run of prints (and especially modular prints like this where there are multiple colors and gaps between distinct pieces of linoleum) can be beautiful in their own right. Furthermore they act as a kind of additional record of the process of art-making itself.

And so there are four Ghost prints along with the 96 main worm configuration prints, each ghost capturing the repeated inking and shifting position of the worm modules over the course of a couple dozen prints, imprinting on the paper the accumulation of ink on the burlap backing of each linoleum module. Each ghost is different from the others because of those differences in permutations and the way that ink builds up on the burlap. The first ghost is almost entirely four separate constant fields of color, as the configurations in that first set were almost entirely made of square-ish worms; each further ghost becomes more complicated in its distribution of ink as the permutations of worms moved out of the dominant four-square configurations at the beginning into a variety of more complex and labyrinthine layouts.

Fogleworm XCVI

As with all my mathematical or geometric work, part of what I get out of the process of physically crafting and manipulating and working through an erstwhile pure mathematical concept is a sense of surprise and complication and unexpected trouble where theory meets practice. Fogleworms are pure concepts, and even the tubular form of the design idea here is geometrically very well-behaved. But my actual carvings are uneven, my cross-hatching imperfect; my lino modules don't lay perfectly parallel; my inking is inconsistent. Stray ink accumulates at the edges of the carved-down lino and infiltrates the negative spaces of the prints, adding a very human visual noise to the pure layout concept.

Certain of these prints, involving "s" type worms, are reversed compared with the rest of the set in terms of the layout on my reference sheet because I made one or another incorrect assumption about how many spare or mirrored versions of those linocut shapes to create and had to improvise in order to not shut down the whole printing process.

Even my choice of Roman numerals for numbering of the series surprised me in a couple ways: that I wish I'd omitted the top and bottom bars once I reached the "L" character and saw that it would be illegible in my block printing, and that I had to double check myself more often than I'd expected to be sure that I was rendering the larger numbers correctly.

I like for art to be a record of its own making, and that's both a record of the physical processes involved and a record of the decisions and mistakes and human inconsistency which I bring to the work in the process of making it.

linocut modules inked up for a print

I've assembled the full collection of prints below, shown in order left to right and then top to bottom and broken into sets of 24 as an aid to browsing, with each thumbnail a link to the full image. The Ghost prints follow below the main set. I don't consider the 96 prints to be naturally partitioned in any particular way, and in fact even the order in which they're presented and named in this set is the result of an arbitrary decision about how to work through the whole set mathematically. There's an order to it, certainly, but not one that is somehow inherent to the idea of enumerating these shapes in general.

Each of these prints is an edition of 1, both the 96 main prints and the 4 ghost prints. They're available for purchase at my online store.

I - XXIV