An Averaged Menger Sponge

I’ve been doing a lot more painting than posting-about-painting recently, and got it in my head to line up a lot of Menger objects I’ve made since last summer in one image file and use transparency to roughly average them.  Here’s the result of futzing with that last night:


It’s an imperfect approach since I’m just using some rough napkin math to try (and I am certain fail) to keep the various layers approximately equally contributing to the final image.  It’s 35 layers altogether, I think, a combination of paintings, drawings, sketches, papercraft, and photographs of actual 3D objects carved from soapstone and snow and cork.

I also put together an animated gif that steps rapidly through all of those individual images: it’s by nature a bit flashy so be warned before clicking through: Continue reading “An Averaged Menger Sponge”

The Decline And Fall Of Snowmeng


Portland’s snowy hellscape has gone the way of warmer temperatures and more typical rain, and the snowy Menger sponge I built along with it, but I took some pictures while it was still up and while it was in the process of ceasing to be.

My wife had the excellent idea to stick a colored LED lamp in the interior of the sponge to light it at night; the various colors seen in these images are on a cycle that the lamp runs through, and it spent that whole first night slowly easing from one color to the next. Continue reading “The Decline And Fall Of Snowmeng”

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”

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”