Inverting a Menger Sponge

So here’s the question: if a Menger sponge is what you get when you keep removing successive chunks of innards from a cube, what do you get when you keep the innards and throw away the sponge?

You get an inverted Menger sponge, is what.  What does that look like?


Here’s three iterations, illustrated.

A zero-iteration Menger sponge is literally just a cube; you remove nothing, and the inverted sponge is just empty space.

A one-iteration Menger sponge has a tri-axial cross shape removed from it, as if it were a 3x3x3 stack of smaller cubes and you removed the center cube and each of the six cubes adjacent to that center cube on the six respective sides of the original large cube.  The Menger sponge has holes in the middle; the inverted Menger sponge is a blocky 3D cross.

A two-iteration Menger sponge removes the same arrangement of cross-shaped blocks, at one third the scale, from each of it’s remaining smaller cubes; the corresponding inverted Menger sponge gains those, as miniature crosses glued onto the bigger one from the first iteration.

The process continues from there indefinitely, in principle, as the sponge itself gets emptier and emptier, and the inversion gets increasingly bumpy and full of discarded, ever-shrinking cross shapes.


I liked putting this chart together — it started as some sketches last night, see above, because I’ve thought often about the flipside object to my pet fractal — but I was surprised to discover that I don’t actually find the resulting inverted Menger sponge very visually appealing.

It’s got a kind of visual dazzle that I can appreciate, but it doesn’t communicate as clear a sense of shape; all those greebly outcroppings obscure, rather than illuminate, the essential form of the fractal shape, and it’s hard to get a clear sense of where the holes are.

But that may be in significant part an artifact of this flat-shaded isometric approach; that serves the flat-surfaced foundation of the Menger sponge well aesthetically, but a proper 3D model of the inversion with some proper lighting and shadows would probably go a long way toward making it a more interesting specimen to look at and especially to interact with via rotation, etc.

The Abominable Snowmeng, and other winter fractals


It snowed quite a lot in Portland, yesterday and overnight.  About a foot altogether, which for this town is if not historic at least very rare; a typical Portland winter sees no snow at all, or some brief flurries of fat flakes that don’t survive contact with the wet-from-recent-rain ground.  Every two or three years we’ll get a nice blanket of 2 or 3 inches and the city will shut down for a day or two while everyone panics.

This isn’t the first snow we’ve had this winter, so it’s been a weird one on that front already; last time we got some decent coverage, I scraped a Sierpinski Carpet in the driveway with a chunk of scrap 1×6 we had sitting around:


But this was something else, and so it demanded something else: increased dimensionality. Continue reading “The Abominable Snowmeng, and other winter fractals”

Pipe Minimal


I’m calling the typeface Pipe Minimal, at least for now; easier than calling it “the typeface”.  It’s aggressively minimalistic in terms of the core design elements, and those elements could be commodity bits of piping, just a bucket full of straight bits and an elbow joints in PVC or copper.

The above demo plate is what I put together as soon as I’d crossed the threshold of having an actual usable font file; I managed to sort out a few problems that only came up once I tried to get FontForge to export, and got a set of lowercase characters into an .otf file and Photoshop was happy to use it.

But when I went back to FontForge to start working out upper case glyphs, I found myself frustrated by a couple things: Continue reading “Pipe Minimal”

Self portrait in pencil


Sketch based on my passport photo.  Originally I’d just meant to get basic shapes in to try inking over, since that’s something I need a lot more practice with, but I ended up digging in a little bit with pencil.

I cam out looking more like Alan Ruck than I’d meant to; something in the set of the mouth and the long face.  Like Cameron Frye scraggled up for a bohemian Ferris Beuller sequel.

This is in a 5×5 inch notebook, drawing while holding the notebook with my left hand and looking at computer screen for reference.

I’m not unhappy with it as a drawing per se but as always happens with portraiture I’m unhappy with how it fails to feel like I got the key proportions right.  Working from my own face makes this worse in some ways because I’m fighting not just my eyeball take on subject vs. portrait but also some deeply wired self-identification stuff.

Here’s the actual photo, for reference: Continue reading “Self portrait in pencil”

Fractal Mailbag #1

Lego Sierpinski carpet, Kacy.

Blanketing my friends and family and social media network with fractal imagery for months on end is paying dividends: I get people throwing found imagery and straight up craft projects my way now, which basically always makes my day.

And so I’m gonna celebrate a lazy, snowy (in Portland, again, somehow) Saturday by showing off other folk’s stuff instead of my own. Continue reading “Fractal Mailbag #1”

Isometric graph paper!

Simple isometric Menger sponge with orange striped shading on one face.

Little things make me happy a lot of the time, and fancy graph paper is a pretty little thing — a few bucks for a pad of 50 sheets of the stuff feels like a big pile of promise.

And with the stuff I’ve been doing lately with fractals, grids are a handy thing to have. But a standard square grid doesn’t help as much as I’d like with things like isometric views and triangle-based designs.  You can wing a pseudo-equilateral design on square grid paper by centering a triangle in a 2×2 square, like so: Continue reading “Isometric graph paper!”

Large die-cut Sierpinski carpet

12″x12″ paper Sierpinski carpet, with shadow and background Menger sponge for maximum visual argh.

I bought a consumer die-cutting machine late last year, after thinking a lot about the possibilities of using machine-driven cutting to do elaborate fractal pieces that’d be difficult to execute with an x-acto knife.

The machine’s a Cricut Explore Air 2, and I’m very happy with it and will write a bit more about how it operates and what I’ve been doing with it at some point.

But what I was doing with it yesterday evening was cutting out an approximately one foot square Sierpinski carpet.  (Slightly less than, because the cutting material itself was a 12×12 sheet and leaving a small margin at the edges is a safe bet.) Continue reading “Large die-cut Sierpinski carpet”

Lessons From a Crappy Sierpinski Carpet

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: Continue reading “Lessons From a Crappy Sierpinski Carpet”

The Hardest Part Is Not Drinking It

Tackling another project:


I wanted to do something with some serious specular highlighting, and some transparency, and distinct shapes that I’d have to not try and cheat my way around too much.

Beer! Beer will do. And while it will go flat, it won’t literally rot away over the course of several days left out, so I can keep coming back to this one. But it’s a bit weird in retrospect to have a beer just sitting there, not for drinking. And a really nice Nut Brown, too. Shame.


I started with an underpainting with thinned out orange, same basic deal as the last couple, and it’s starting to feel a little more natural; I spent more time getting stuff outlined this time, partly because there was just more to do but also because I wanted to try harder to get the shapes and proportions close to correct.

That feels like it paid off reasonably well so far. But the new ideas in this — getting that subtle shading of brown light in the beer to come out, getting the play of direct and indirect shadows right, getting the relatively cool and warm and light and dark colors in the room — have all been a challenge.

The deceptive nature of relative hue and value and intensity is something I’m appreciating a lot more with this one where, unlike the self-portrait I did previously, there’s not really any anthropomorphic or emotional engagement to be distract from all the stuff that’s off the mark.

More work to do on the first pass, get the floor and walls and green piping on the cloth in place and some of the rim/highlight stuff for the glass, and beyond that the paint on the already painted bits is full of gaps because I keep mixing not nearly generous enough amounts of each color when I’m working.

This is on rougher canvas than I’ve been using, too, owing to me grabbing supplies at the store without really looking. It’s an interesting change and I might like it, but I was expecting to be a bit more fiddly with details on this one and the rough goes against that a bit it feels like.