Sliced Fractal Sponges

Start with a cube.

plain cube

If you slice off a corner of the cube, the exposed surface from the cut is a triangle. No surprise there.

cube with corner cut off, showing triangle

If you make a deeper diagonal slice that intersects all of the cube’s faces, the exposed surface is a hexagon. Also not much of a surprise.

half cube, showing hexagon

Now start again with the first iteration of the Menger sponge.

Menger sponge

If you slice off a corner but miss the square holes, the exposed surface is still a triangle.

Menger sponge with corner cut off

But if you make the slice deeper, the exposed hexagon surface contains a star-shaped hole. That is a surprise.

Half Menger sponge, showing hexagon with six-pointed-star-shaped hole

To help see why this happens, consider the inverse of the Menger sponge.

inverse of Menger sponge

Draw dashed lines where the diagonal slice will go. You can kind of see that the cubes making up the arms of the shape will alternate between having the top front corner sliced off and having all but the lower back corner sliced off, and you can kind of guess that the hidden center cube will show the hexagon face.

inverse of Menger sponge with preview of slice

Sure enough, the combination of the exposed inner hexagon and the surrounding exposed triangles makes up the star shape.

half of inverse Menger sponge showing star-shaped surface

(End of detour.)

If you do the same thing with the next iteration of the Menger sponge, the exposed hexagon surface shows even more star-shaped holes, where the slice meets the different sizes of square holes.

sliced Menger sponge, iteration 2

And the next iteration shows even more.

slice through level 3 of Menger sponge

Nice, isn’t it?

Inspired by this page and links:

Work in progress, but how about this asterisk-shaped hole in a slice through the cross Menger cube?

asterisk-shaped hole in diagonal slice of cross Menger cube

And again.

slices through iterations zero, one, and two

Designed and rendered using Mathematica.

© 2014–2024 by Robert Dickau.

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