Knots and the Menger sponge

The Menger sponge is a fractal formed by iteratively subdividing a cube into 27 equal cubes, and then removing the central cube of each face and the interior central cube. Doing this once results in the shape of a Rubik’s cube with seven missing blocks, and doing this four times results in the shape above.

The Cantor set is a 1-dimensional relative of the Menger sponge, formed by iteratively splitting a line segment into 3 equal segments and removing the middle segment. The picture above shows the first six iterations of this process.

The Sierpiński carpet is a 2-dimensional analogue of the Cantor set. It is defined by iteratively dividing a square into 9 equal squares and removing the central one. The first six iterations of the Sierpiński carpet are shown below.

The Cantor set can be defined in a non-recursive way using base 3. More precisely, a number between 0 and 1 (inclusive) is in the Cantor set if it can be written as a “decimal”  in base 3 without using the digit “1”. Recall that, in base 3, the “tenths” and “hundredths” places are replaced by “thirds” and “ninths”. It follows that the fractions 2/9, ⅔, and 8/9 are in the Cantor set, because they can be written in base 3 as 0.02, 0.2, and 0.22, respectively. The number ½ is not in the Cantor set, because its only representation in base 3 is as 0.1111… recurring, which uses the digit “1”. Less obviously, the number ⅓ is in the Cantor set, because although the natural way to express ⅓ in base 3 is as 0.1, which uses the digit 1, it is also possible to express ⅓ in base 3 as 0.02222… recurring, which avoids the digit 1.

A knot in mathematics refers to an embedding of a circle in three dimensional Euclidean space. Knots can be thought of as closed loops of string, such as the figure-eight knot shown in the picture above. 

The main result of the recent paper Knots inside Fractals by Joshua Broden, Malors Espinosa, Noah Nazareth, and Niko Voth is that it is possible to embed any knot into the Menger sponge, avoiding all the holes. The first step in doing this is to express a knot as an arc diagram on a square grid, and the above picture shows how to do this for the figure eight knot. When two strands in the arc diagram cross each other, things are always arranged so that the vertical strand passes over the horizontal strand.

The next step is to stretch the arc diagram so that all the points where arcs cross have coordinates in the Cantor set. In the example shown, the red crossings occur at the points (2/9, 2/9), (⅔, ⅔), and (8/9, 8/9), and as we noted earlier, the numbers 2/9, ⅔, and 8/9 all lie in the Cantor set. A key property of the Menger sponge is that the straight line between the point (2/9, 2/9) on the front face of the sponge and the point (2/9, 2/9) on the back face of the sponge avoids all of the holes in the sponge, and this happens because the number 2/9 lies in the Cantor set. 

Finally, it is possible to inscribe the red arc diagram in the Menger sponge, avoiding all the holes, so that the vertical line segments end up on the front face, the horizontal line segments end up on the back face, and the corners in the red diagram correspond to straight lines from the front face to the back face. In fact, it is possible to do this in such a way that the red arc diagram follows the edges of one of the finite iterations of the Menger sponge. For example, the arc diagram in the previous paragraph follows the edges of the second iteration of the Menger sponge, a physical model of which is shown above.

There is also a triangular version of the Sierpiński carpet, called the Sierpiński triangle, and a tetrahedral version of the Menger sponge, caled the Sierpiński tetrahedron. The picture above shows the 8th iteration of the Sierpiński tetrahedron. The authors of the paper conjecture that it should also be possible to inscribe any knot in the Sierpiński tetrahedron, and they prove that this is indeed the case for a particular class of knots known as “pretzel knots”.

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Picture credits and relevant links

The picture of the Menger sponge is by Niabot and appears on Wikipedia.

The picture of the Cantor set is in the public domain and appears on Wikipedia. 

The animation of the Sierpiński carpet is by KarocksOrkav and appears on Wikipedia.

The picture of the Sierpiński tetrahedron is by PantheraLeo1359531 and appears on the Wikipedia entry on the Sierpiński triangle.

The picture of the figure-eight knot is by Jim.belk and appears on Wikipedia.

The two pictures of the arc diagrams come from the paper by Joshua Broden, Malors Espinosa, Noah Nazareth, and Niko Voth.

The paper model of the Menger sponge is on display in the Mathematics Institute at the University of Warwick, and the picture of it is my own.

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