Can a drawing contain a kind of “absolute” truth? With exhaustive combinatorics, yes.

Consider this problem:

This *really enjoyable* 9-minute video takes us through the combinatorics for such circles. (Note that the video creator has chosen to use slightly different rules; he does not consider the one-point kissing circles to be valid; and that’s OK!)

Now consider graphs that connect points. From mathematician Arseny Khakhalin, here is a set of “all graphs with 3 nodes”. These are the **only two possible topologies** by which 3 things can be connected (not counting configurations in which a point is disconnected):

Here are all possible graphs of 4 nodes:

All graphs of 5 nodes:

All graphs of 6 nodes:

All graphs of 7 nodes:

In such depictions, generative rules are executed to their logical conclusion. The visual forms that result are no longer the arbitrary product of human imagination, but something more like a fundamental, irreducible property of the universe itself.

It may surprise you to learn that such rule-based images can also be considered *artworks*. “In 1974, American artist Sol LeWitt created one of his major works, a seminal piece on the themes of seriality and variation, the series entitled “*Variations of Incomplete Open Cubes*”. The work is a collection of 122 frame structures presented together with the corresponding diagrams arranged on a matrix. Each sculpture is the projection of a three-dimensional cube with some of the edges removed in a way that the structure *stays three-dimensional* and the *edges stay all connected*. The minimum number of edges kept is three and the maximum is eleven.” [Source]. Lewitt’s artwork depicts all possible combinations of cube-edges that meet these conditions. There are *no other possibilities*: all possibilities are included and none are missing.

Here’s a simpler and more idiosyncratic example. Lewitt’s *Geometric Figures Within Geometric Figures* (1976), which shows all possible pairings of six basic shapes, hints at how conducting such studies can be a tool for *design exploration*, while retaining the power of a statement of mathematical fact.

Here’s a more recent example, which illustrates the risk of combinatoric explosion. *Arc Forms* by Christopher Carlson (2009) shows all possible combinations of semicircles joined at 3 connection points evenly distributed along a vertical line. (He includes the cases in which they *aren’t* there, too.)

But be *careful*. Change the number of connection points for such arcs, and the possibility space balloons dramatically:

Clearly, you can choose to design a visual system for which it is practical to enumerate all possible combinations — or you may design one for which it isn’t. One can impose additional criteria to narrow the space once more. In* curating the rules* by which such gargantuan spaces are filtered and culled, the voice of the ‘artist’ reemerges, producing an idiosyncratic design language through sub-selection:

All of the systems described on this page are:

*instances of “ minimum inventory/maximum diversity” systems, a term coined by Peter Pearce in his book, Structure in Nature Is a Strategy for Design (MIT Press, 1978). A minimum inventory/maximum diversity system is a kit of modular parts and rules of assembly that gives you maximal design bang for your design-component buck. It’s a system that achieves a wide variety of effects from a small variety of parts. Nature excels at this game: every one of the many millions of natural proteins is assembled from an inventory of just 20 amino acids. Snowflakes are all just arrangements of the humble water molecule, H_{2}O. *[Source]

In other words: using very simple rules/constraints, we can get profound and surprising diversity. Here are a few more such systems:

Michael Fogleman, *SQUARES IN A SQUARE* (2021). Fogleman asks: “How many ways can you chop up an NxN square into integer-sized squares? 1, 1, 2, 6, 40, 472, 10668. Here is N=5 (but rotation-invariant)”:

Michael Fogleman, *MOWING A LAWN* (2021): The number of ways to mow a square of size N (N=7):

By the way, here’s a case in which Fogleman is *failing in public*. He thought his algorithm for computing combinatorics was correct… and then concedes he had a bug. It can be tricky to verify that these systems are correct,* even for experts*!

One last one. Here’s Michael Joaquin Grey’s *Erosion Blocks* (c.1990), a sculpture which shows progressive combinatoric removals of the sides of a rectangular prism, leaving (at the end) the potato-like Philosopher’s Stone. These 43 blocks illustrate the following truth: they are *the only possible ways* of removing 0,1,2,3,4,5, and 6 sides from a rectangular prism with a square cross-section.

**This stuff is difficult for me. What’s the easiest place to start? **

Try asking yourself a question about simple shapes that has a discrete, countable answer. Start with small numbers of things. Here’s an example, in the form of a question: “How many logically distinct ways can I arrange 1 hexagon? 2 hexagons? 3 hexagons? 4 hexagons?” (These are called ‘polyhexes‘ by the way). (Also: “Logically distinct” is different from “visually distinct”; for example two visually-different configurations might be considered logically similar if you can obtain one from the other by rotating it or flipping it.) So, coming up with such a question and showing all of the tetrahexes would be a perfectly legitimate response to the homework prompt. __But note: the homework is much more concerned with your ability to come up with a question like that, than to actually correctly work out all of the permutations!__