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While building fractals in minecraft I built this fractal enter image description here with the intent of making a Sierpiński carpet but I made a mistake and created this (I also built this in 3d).

The procedure I used to create this fractal is as follows. Start with a square with the inner third removed.

square with inner third removed

Take 12 copies of this and arrange them to form a square void in the middle. Any sides that are next to each other get overlapped.

12 voids around a single larger void

Then rescale the shape so it is bounded within the previous shape.

Repeat this action. Noting that you now overlap not just the solid part but also the voids around the middle.

enter image description here

The limit of this sequence is the fractal I created. So what is this shape if it known?

Q the Platypus
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  • It looks like the first iteration is different from the second iteration: the medium-sized squares look like 2/5 of the side length of the big square, but the small squares look to be about 1/6 of the side length of the medium squares. You need to disambiguate what the pattern is. – Dennis Sep 05 '23 at 03:53
  • I will upload a better set of pictures explaining the iteration. – Q the Platypus Sep 05 '23 at 08:43
  • Fixed the problems and made it clearer how this fractual is created. – Q the Platypus Sep 06 '23 at 00:27

2 Answers2

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This shape, strangely enough, is in fact a Sierpiński Carpet. It's just not the "standard" carpet. But it is topologically equivalent to it.

In fact, the mathematician Whyburn showed that as long as the diameters of the holes go to zero, the boundaries of the holes are disjoint, the boundaries of the holes are simple closed curves (loops that don't cross themselves), and the union of boundaries of the holes is dense (meaning any point of the end fractal is arbitrarily close to some hole), then your fractal is topologically equivalent to the standard carpet.

M W
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As MW points out this carpet is topologically equivalent to a Sierpiński Carpet; moreover it has been described by Peter Karpov in the OEIS as A285391.

A way to describe how to generate this carpet is to follow the following procedure.

Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 2.

J. W. Tanner
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Q the Platypus
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