"Binary Menger" Space-Filling Fractal Sponge

3D Model

Binary Menger Sponge. ** At first sight, the Binary Menger Sponge is nothing special - it looks like a fairly uninspired rip-off of the standard Menger, but using a grid that’s 4×4×4 rather than 3×3×3. However, the “Binary” version is special - the enclosing cube is 64 blocks, removing the central eight-cube block and six surrounding four-cube clusters makes the volume of the fractal exactly halve at each iteration. And the resultling frames then interlock, perfectly, so this shape perfectly fills 3D space when tiled (and interleaved), even though its 2D counterpart doesn’t fill a plane. it’s a space-filling solution that requires thinking in three dimensions.

From page 61 of “Alt.Fractals: A Visual Guide to Fractal Geometry and Design”, ISBN 0955706831 (2011). https://www.amazon.com/Alt-Fractals-Visual-Fractal-Geometry-Design/dp/0955706831/