# Mathematical Impressions: The Surprising Menger Sponge Slice

The Menger Sponge, a well-studied fractal, was first described in the 1920s. The fractal is cube-like, yet its cross section is quite surprising. What happens when it is sliced on a diagonal plane? Try to predict the solution to the puzzle proposed in this video.

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Photo by Sébastien Pérez-Duarte

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More videos from the Mathematical Impressions series.

• Very nice! Could you give some detail on how you made the model (e.g. which 3D printer and software you used, any tips, etc.)? Thanks.

• Simply beautiful! mathematics and universal order reveling each other (at least this is how I see it).

Great work

• felix says:

heh, my first instinctive guess was correct, but then I decided it couldn’t be the right answer because it wasn’t “surprising”, and I rationalized my way to a wrong answer.

• Alejandro Urzúa Pineda says:

Simplemente genial, ¡maravilloso! Tengo una pregunta, ¿en qué programa se hizo la simulación? Gracias.

• George says:

Nice! I would also like to know what 3D printer you used and a link to the file since I have my own Replicator 1 which I would like to attempt this on.

• To answer some of these questions: I used at least a half dozen software tools in generating the animations and the 3D models, including Mathematica, Maya, and Rhinoceros.

I fabricated the 3D-printed models by selective laser sintering on a DTM2500+ machine at Stony Brook University, but other high-resolution machines are equally capable of producing such models. If you have access to a 3D printer, the stl file for the sliced 3rd-order Menger sponge can be downloaded from here:
http://georgehart.com/rp/rp.html

• Clara Grima says:

It’s amazing like everything you do …

A big hug from southern Europe,

Clara

• Joe Thacker says:

I’m not sure when I came up the the same results in Sketchup, It may have been even before 2007. But I’m not a studied mathematician or someone in the science field. I figure if someone like me found this, a professor or even a grad student would have already found it. Shows that discovery can come from anyplace.

• Max Mastandrea says:

Very interesting. I have seen this in a mo-math presentation but this helped me understand it more. Everyone I know who has seen this was amazed. Amazing!!

• This is utterly superb work!! Problem is, you have now cost me a lot of money as I now have to go out and buy a 3D printer. I have wanted to do precisely this sort of thing for a few years now but when I looked at this around 5 years ago it was nowhere near as advanced as it is now. First project for me will be to see if I can print out a Roman surface.

• Jim Wilson says:

I teach an Introduction to Fractals and Chaos course eery other year to secondary math teachers. The in-between years I look for new twists on the stable content in the course and your video brings a fresh and exciting extension to the Menger Sponge. Thanks!