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.

Author: Josh Millard

I manage and help moderate the community website MetaFilter, where I go by "cortex"; in my spare time I get up to all sorts of creative nerdery on the internet and in Portland, Oregon.

2 thoughts on “Inverting a Menger Sponge”

  1. Heh, very nice. And yeah, I haven’t dabbled in 3D printing directly myself yet but the impression I’ve gotten second-hand is that if you can ask the question “I wonder if anyone’s 3D printed [x]”, the answer turns out to be “yes”.

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