Figure 56: Hexastix.

Hexastix is a symmetric arrangement of non-intersecting prisms that are all parallel to 4 directions. Constructing hexastix with hexagonal prisms, the arrangement fills exactly 75% of space when extended infinitely. In 2008, hexastix and the larger polystix family were named in The Symmetries of Things by Conway, Burgiel, and Goodman-Strauss [1]. This arrangement has also been described in depth by Michael O’Keeffe and associates, where it is known as the “Γ rod packing” and is classified as the densest of the 6 possible invariant cubic rod packings [2].

Figure 57: Hexastix 72 pencils.

I first became interested in hexastix after seeing the sculpture “72 Pencils”, made by the amazing math artist George Hart [3]. After a quick web search, I found several tutorials on how to recreate the complicated structure. The hexastix building instructions presented on Alejandro Erickson’s website[4] were very clear and easy to follow. I was most impressed that the complex hexastix geometry did not require any precise measurements or complicated fixtures to construct, but instead could be created and assembled simply with only sticks and string. After making a few hexastix, I was hooked on exploring the endless array of different possible shapes and combinations. Over time, my polystix models, and the mathematics describing them, grew in complexity and guided the creation of intricate glass knot sculptures. *

Figure 58: Hexastix group.

Hexastix arrangements can be found in surprising places such as: architecture, crystallography, biology, puzzles, mathematics, and art. Making models of hexastix is not only a useful tool for learning mathematics, but also a fun project that results in a beautiful sculpture.

Next, we will find out how to make a hexastix.


[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.

[3] https://www.georgehart.com/sculpture/pencils.html

[4] https://alejandroerickson.com/2011/09/27/090-how-to-make-hexastix-detailed-photo-instructions.html

copyright 2022 Anduriel Widmark