Tetrix

The tetrix is the three-dimensional analog of the Sierpiński sieve illustrated above, also called the Sierpiński sponge or Sierpiński tetrahedron.

The th iteration of the tetrix is implemented in the Wolfram Language as SierpinskiMesh[n, 3].

Let be the number of tetrahedra, the length of a side, and the fractional volume of tetrahedra after the th iteration. Then

			(1)
			(2)
			(3)

The capacity dimension is therefore

			(4)
			(5)

so the tetrix has an integer capacity dimension (which is one less than the dimension of the three-dimensional tetrahedra from which it is built), despite the fact that it is a fractal.

The illustrations above demonstrate how the dimension of the tetrix can be the same as that of the plane by showing three stages of the rotation of a tetrix, viewed along one of its edges. In the last frame, the tetrix "looks" like the two-dimensional plane.

Broden et al. (2024) proved that all pretzel knots can be embedded into the tetrix (Barber 2024) and conjectured that every knot can be embedded into the tetrix.