Menger sponge

The Menger sponge is a mathematical object formed by starting with a cube, and then through each direction drilling a hole down the center. The result can be thought of as 20 cubes put together and now we repeat the process with each of those cubes, and so forth and so on. This makes a simple example of what is known as a fractal (or fractional dimensional object, in this case the dimension is 2.7268...=log(20)/log(3)).

I found a way to build some smaller Menger sponges via LEGO, and in particular using only one part (4070 commonly known as "headlight"). Using ~32,000 pieces and the help of the ISU REU 2022 students we were able to successfully build a level 3 Menger sponge.

The three scales of the constructed Menger sponge.

  • Level 3 requires ~32000 LEGO pieces

  • Level 2 requires 1968 LEGO pieces

  • Level 1 requires 120 LEGO pieces

Looking through the center of a level 3 Menger sponge.

The construction is so symmetric that the center of mass is in the exact middle of the cube. This allows the Menger sponge to perfectly balance on a corner. (Also can double as an earthquake detector.)

If we combine the Menger sponge together with the compliment of the Menger sponge (=what we take out to make the Menger sponge), we get the whole cube back.

LEGO banana man to help give a sense of the scale of the level 3 Menger sponge.

The two different approaches to building a level 2 Menger sponge. On the left is one built directly without all of the internal parts (takes 1968 pieces, though if trapped on a desert island you can make do with 1944 pieces). On the right is one built by combining 20 level 1 Menger sponges together (takes 2400 pieces, and is a bit blockier (hurts more if you drop it on your foot).

A level 1 Menger sponge that has exploded.

A level 2 Menger sponge that has exploded.

Menger sponge built out of transparent bricks.

The assembly line crew for building a corner of the level 3 Menger sponge.

Having fun with building a corner. No need to stress, it is just LEGO.

The 2022 ISU REU group that helped build the level 3 Menger sponge. Special thanks to Joel Jeffries, Elena D'Avanzo, Rachel Heikkinen, Alyssa Kruczek, Harper Niergarth, and Laura Gamboa Guzman who went above and beyond in putting the Menger sponge together.