Laird Robert Hocking

This fractal octagram bracelet grew out of an attempt to generalize the Menger-slice, which features a hexagon with fractal hexagram holes, into an octagon with fractal octagram holes. At first this seemed promising as I was able to generate an closed fractal family similar to the one used for the Menger-slice - please see my 2024 Bridges paper "Menger-Slice Inspired Fractals based on the Pentagon, Dodecahedron, and 120-Cell" for details on what these words mean and how this works in the case of the Menger-slice. This was done based on a decomposition of the octagon into eight smaller octagons, eight right angle triangles, and an octagram-shaped hole in the middle. However, when I ran the code the result didn't look nice. With a bit of playing, I found that a much nicer look was obtained if you instead applied this decomposition to the unit square, rather than an octagon. You naturally end up with a shape obeying period boundary conditions, meaning that you can then take the result and wrap it around into a cylinder, a torus, a Möbius strip, or a Klein bottle. This bracelet uses the cylindrical wrapping, but I have also done a torus. Like the Menger-slice, this can be run to any recursion depth, meaning you can obtain a pattern with as many different sizes of octagram-shaped holes as you like. Here I only have two different sizes because recursing further made 3D printing impractical. In addition to being worn as a bracelet, this shape produces interesting shadows when a light source is placed inside. Work is underway to turn it into a lamp shade.

My 2023 Bridges project was concerned with 3D diagonal cross sections of 4D Menger sponges. While there are infinitely many different sections (and multiple types of 4D Menger sponge), there is one section that I find particularly striking - it consists of a truncated tetrahedron with six hexagonal faces and six isosceles triangle faces. Moreover, each face has a fractal pattern of hexagram star shaped holes in it. In my 2023 Bridges project, I showed that the six hexagonal faces are precisely the central diagonal cross-section of the 3D Menger sponge, while the six isosceles triangle faces are also a diagonal cross-section of the 3D Menger sponge. The interior exhibits four sharp spikes, and the overall shape has tetrahedral symmetry. While I initially 3D printed this as a large sculpture in gold plated brass with recursion depth three (meaning three sizes of hexagram stars), I later tried shrinking the shape down to a size that was affordable to 3D print in pure gold. This meant reducing the recursion depth to two, but resulted in a wearable pendant.