Mathematical Impressions: The Surprising Menger Sponge Slice

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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.


The Menger Sponge (Wikipedia page)

Photo by Sébastien Pérez-Duarte


More videos from the Mathematical Impressions series.

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  • 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.

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

  • 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:

  • 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.

  • 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.

  • 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!

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