This is a Sierpinski carpet.  Or more precisely this is a dodgy approximation of one using brush-tip marker on post-it note.  It’s got all kinds of problems, and those are interesting to me.  Let me dig in on this a little.

(Disclaimer: I’ve been thinking a lot about, and making a lot of art out of, fractals in the last few months, and I’m working on a long writeup about some of that that will cover a fair amount of artistic and personal ground, but I’m gonna try and write here more often and part of that means remembering to do smaller things quickly and to share them regularly.  So if you’ve been following me on twitter or elsewhere and are thinking This, At Last, Is The Exegesis: nope.  Whether that’s relieving or foreboding probably varies from reader to reader, sorry not sorry as necessary.  More to come.)

Ceci n’est pas une tapis.

So.  This is a Sierpinski carpet, and it isn’t.

What’s a Sierpinski carpet? The short version:

1. Take a square. Cut out a smaller square from the middle, one third as wide and as tall as the original.  Now you’ve got a sort of square donut, eight solid squares surrounding a square hole.

2. Now visit each of those remaining squares, and cut out their middle third as well.  A donut of square donuts.

3. Now visit each of the eight smaller-yet squares in each of those smaller square donuts, and cut out their middle third.

4. And then and thus and so on, into infinity.

That’s how you make a Sierpinski carpet, and that’s why this isn’t one even though it obviously is one.

Because this post-it note drawing is one in a reasonable, practical sense — that’s clearly the pattern being presented here, in enough detail (interesting word when fractals are involved) to be unambiguous and not just some geometric coincidence, some look-alike happenstance of tiles.  It’s a Sierpinski carpet; it couldn’t possibly be anything else.

It isn’t one in a literal and undeniable sense, because the Sierpinski carpet is a mathematical object with some interesting properties involving infinity and infinitesimility.  Because my square donuts stop after four steps.  Because these aren’t perfect thirds and thirds-of-thirds.  It’s obviously only a rough facsimile.  And like most things involving infinity, you’ll only ever get rough facsimiles since even with a steady hand and a very fine tipped pen it won’t be very long before the detail exceeds your skill.  It won’t even be long, cosmically speaking, before you get down to a level of detail that requires making dots smaller than atoms, than electrons, than quarks.  Eventually the Planck constant starts to wield an unsteady hand at the drafting board.

So it’s a Sierpinski carpet, and it isn’t.  It is because it’s a representation of one; it isn’t because it’s not a perfect, impossible realization of one.  But that dichotomy is a mathematical and philosophical one, and it’s interesting but it’s not what I’m interested in this morning.

I’m interested in all the space between that Is and that Isn’t.  I’m interested in art and failure and approximation.

Fucking Up Your Fractals

The basic limits on creating a physical approximation of a fractal object become clear quickly.  As you repeat the steps to produce the object — as you iterate — the details get finer and more numerous rapidly.  Each step in a Sierpinski carpet means drawing things one third the size of the last step, and eight times as many of them.  One big square; eight smaller squares; sixty-four smaller-yet squares; two hundred and fifty-six even-smaller-yet ones.

It gets absurd, rapidly.  It approaches all sort of limits (notwithstanding the literal mathematical concept of the word) and does so mercilessly.  A square of three inches gets divided into one inch, and then 1/3rd inch, and then 1/9th inch, and then 1/27th, and already we’re to the reasonable limit of rulers if you can even find one that counts by fractional powers of 3 instead of the standard 2.  And all of this on an idealized perfect grid where errors become more obvious the smaller the detail is, where an inch-wide square being off by 1/27th of an inch may be imperceptible but a 1/27th-inch-wide square being off by as much is a glaring error.

That absurd trip into merciless territory is the trick, the trap, the nasty attractive pot of honey in all this.  To try to do something that is by definition not going to work.  To figure out just how much you can get away with.  How much you’re willing to try to get away with.  To pit skill and patience and planning and will against an uncaring mathematical god.  To see where you can stand up, and where you fall down.  And to figure out where you surprise yourself in all of that.

So look at this post-it note Sierpinski carpet.  Look at everything that’s wrong with it, and why what went wrong went wrong.

A. The squares aren’t square.

B. The squares aren’t even rectangular.

C. The squares that are the same size aren’t the same size.

D. The things I’ve cut into thirds aren’t cut into thirds quite right.

E. The smallest non-square squares aren’t remotely sticking to the grid they should be on.

F. The ink is uneven and streaked, blotted in spots and filling the larger squares inconsistently.

None of these problems have to be problems, but they’re what happened when I made this little bit of art.

They happened because I did it without marking guidelines, eyeballing distances instead of measuring it out. Because I used plain paper instead of gridded graph paper to lay the foundation. Because I drew freehand instead of using a straight-edge to keep the lines straight or the grid even. Because I used a brush-tip marker, with all its expressive possibilities, instead of a better-behaved fine-tip pen like the Microns that I like for detail work, earning myself slotted side-of-the-brush lines for many of the smallest squares.

And they happened because I rushed, because I worked fast and favored getting it done over getting it accurate.  Because I did it while distracted, drinking tea and moderating MetaFilter and being headbutted repeatedly by a cat who interprets a bent head as a social entreaty.  Because I did it in dim lamplight while the sun was still down.  Because I kept reaching my fingers to make fine dots from greater distances so as to avoid smearing my hand in wet ink, instead of waiting for the ink to dry.  Because my hand isn’t terribly steady.  Because it gets less steady still when making 256 tiny dots in a row, and I didn’t take breaks to let it recover, just motored into hand-cramp territory.

They happened because errors accrue, because each little mistake and mismeasure builds up a debt that gets paid downward with each further level of detail.  Because as the damage is done, it becomes unfixable and easy to pass on and amplify.

All that made it what it is, which is a mess.  A mess that gets the idea across, but a mess.  Sloppy, deviating from the form, full of unforced errors.  Look at that top left corner!  You can guess where I finished up.

But it’s its own thing.  It’s one Sierpinski carpet, and everything that’s wrong with it is what it is.

I could do it again with the same pen and more patience and do it better.  I could do it again with different tools and do it better.  I’ve done it before, better and worse; I’ll be doing it again better and worse.

Iterate.

I’ll do it again, and each time it’ll be a little different, and if I were in the business of platonic ideals that’d be a problem but I’m making art, doing a craft, and doing the same thing and getting different results is one of the big draws there.  Figuring out what “better” even means is part of it, getting away from the narrow idea of being mathematically correct.  Even with a mathematical object.

Making art is a process where you learn about yourself.  You make art to learn about, among other things, how you make art.  I don’t expect to unearth deep truths about myself from brush-tipping a lopsided Sierpinski carpet on a post-it note, but I know I’ll get a sliver of something out of it.  About the physical and mental limits of my patience for any given project.  About how I use a pen, about how I manipulate my paper.  About how I balance the desire to have made something against the tedium of actually making it.  About how I plan out a new project in order to compensate for what I’ve learned about myself from the last one, and the one before, and the one before.

I do it again and again.  I iterate.  I learn.