### 12-14-2021

Tetrastix

Figure10: Tetrastix models.

Tetrastix is a symmetric arrangement, consisting of an infinite number of non-intersecting prisms that are parallel to only 3 directions, and lie on the simple cubic lattice (figure 10). Tetrastix made with square prisms, fills exactly 3/4 of space and leaves vertex touching cubic voids. The arrangement is homogenous, with each cylinder being identical in shape and symmetry.

Tetrastix and the larger polystix family was named in The *Symmetries of Things* by Conway, Burgiel, and Goodman-Strauss [1]. Polystix have also been defined and explored in depth by Michael O’Keeffe and associates, who name this arrangement the “simple cubic” or “Π* rod packing”. O’Keeffe classifies this as one of the 6 possible invariant cubic rod packings [2].

Tetrasix arrangements that are made with either cylinders or square prisms, can be defined by the same space groups and symmetries. If tetrastix is made with cylinders it occupies 58.9049% of space, instead of the 75% of space occupied by square prisms. The name tetrastix is also used to describe the geometry of finite configurations or subsets that lie on the infinite tetrastix lattice. Tetrastix arrangements can be found in surprising places such as; architecture, the structure of crystals, biology**,** math, puzzles, and of course in art.

Figure11: Tetrastix 72pencils

In the upcoming blog posts, we explore what materials and techniques can be used to make tetrastix.

**References**

[1] J. Conway, H. Burguel, C. Goodman-Strauss. *The Symmetries of Things*, CRC Press, Boca Raton, FL, 2008.

[2] M. O’Keeffe, J. Plevert, Y. Teshima, Y. Watanabe, and T. Ogama. “The Invariant Cubic Rod (Cylinder) Packings.” *Acta Cryst. *A57, 2001.

copyright 2021 Anduriel Widmark