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Figure 92: Tristix. 

Tristix is a symmetric chiral arrangement, consisting of an infinite number of non-intersecting prisms that are parallel to only 4 directions, and lie on the body-centered cubic lattice, shown in figure 92. Tristix when made with triangle prisms, fills exactly 1/2 of space, and leaves an empty space of vertex touching rhombic dodecahedrons. The arrangement is invariant, with each cylinder being equal, and their positions can be established using symmetry. 

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

Tristix arrangements that are made with either cylinders or triangle prisms can be defined by the same space groups and symmetries. If tristix is made with cylinders it occupies 30.22% of space, instead of the 50% of space occupied by triangle prisms. The name tristix is also used to describe the geometry of finite arrangements or subsets that lie on the infinite tristix lattice. Tristix arrangements can be found in surprising places such as; architecture, the structure of crystals, biology, math, puzzles, and of course in art.

Figure 93: Tristix 72 pencils. 

Figure 94: Tristix 72 pencils. 

In the next blog post, we will find out how to make a tristix.  

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 copyright 2022 Anduriel Widmark


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

[2] M. O’Keeffe, J. Plevert, and T. Ogama. “Homogeneous Cubic Cylinder Packings Revisited” Acta Cryst. A58, 2002.