The Menger sponge
The Menger sponge starts with a cube, divides it into 27 smaller cubes and removes the center cube plus the center of every face. The rule then repeats on the 20 survivors.
Twenty copies at one third scale
After n steps, 20ⁿ cubes of side 3⁻ⁿ remain. The similarity dimension is log 20 / log 3, between a surface and a solid.
Divide the cube into 27 subcubes and keep 20, removing the center and the six face-centered cubes. Repeating the operation inside every survivor gives 20ⁿ cubes of edge length 3⁻ⁿ. The similarity dimension is therefore log 20/log 3, approximately 2.727. Geometrically the sponge lies between a surface and a volume even though every finite stage is built from ordinary three-dimensional cubes.
Volume disappears
Each stage retains 20/27 of the previous volume, so the limiting volume is zero. At the same time new passages and surfaces appear at every step, and simple surface-area intuition no longer behaves like that of a conventional solid.
Volume is multiplied by 20/27 at each stage and tends to zero. Surface area grows because every removal exposes new interior faces. A physical model stops at material thickness and manufacturing resolution; the mathematical limit continues. A printed Menger sponge is therefore not the limiting sponge itself, but a carefully chosen finite stage with struts thick enough to survive.
A universal curve with a sculptural life
The ideal Menger sponge has deep topological properties, while printed models are finite approximations. Architecture and design borrow its visual porosity, but structural usefulness must be evaluated as engineering rather than inferred from the mathematical limit.
The Menger sponge is topologically rich: it contains copies of broad classes of one-dimensional curves and is discussed as a universal curve. This is an embedding statement, not a claim that every curve appears as an obvious line on its surface. The sculpture helps reveal the interlocking holes, while universality requires a more abstract view of connectedness and dimension.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Menger SpongeWolfram MathWorld
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
Continue the idea
Sierpiński triangle and carpet
How repeated removal creates exact self-similarity, vanishing area and a bridge from the Cantor set to higher dimensions.
Fractal dimension
What fractal dimension measures, how box counting works and why one object can have more than one meaningful dimension.
Space-filling curves
How a one-dimensional interval can map continuously onto a square and why finite Hilbert curves help order multidimensional data.