9-7-2022

Polystix Names

Figure 121: Polystix: tetrastix, hexastix, and tristix.

Polystix, and similar symmetric line arrangements, have been independently discovered and described in several instances. The various names, parameters, and applications that have been used to describe these polystix arrangements can be tricky to keep track of. I prefer to use the polystix naming system developed by Conway, but it is unclear how it would extend to include the other two invariant cubic rod packings (*Σ and +Σ [2]), or if it could be extend to include homogeneous 6-way packings [11] as pentastix. Below is a table I’ve made to help keep track of some of the different names used to describe these symmetric non-intersecting arrangements.

To further complicate the polystix naming situation, these arrangements overlap with a closely related, but distinct family of periodic stick arrangements. Rotating the faces or edges of regular polyhedra can create unique sets of vertex transitive links, as well as arrangements of parallel rods, that can be considered as finite polystix sets. Using a process of rotating the edges of polyhedra in regular ways has been independently discovered and described in several instances, some exceptional examples are listed below.

-Alan Holden, Orderly Tangles [6].

-George Hart, Orderly Tangles Revisited [7], Symmetric Stick Puzzles [8].

-Robert Lang, Polypolyhedra [9].

-M O’Keeffe and M. M. J. Treacy, Isogonal Weavings on the Sphere [10].

Figure 122: Polystix.

In the next blog post, we will look at symmetric arrangements made of rods parallel to 6 directions.

copyright 2022 Anduriel Widmark

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, and T. Ogama. “Homogeneous Cubic Cylinder Packings Revisited” Acta Cryst. A58, 2002.

[3] S. Coffin, The Puzzling World of Polyhedral Dissections 1990, https://johnrausch.com/PuzzlingWorld/default.htm .

[4] W. Cutler. https://www.billcutlerpuzzles.com 2022.

[5] J. Kosticks. http://www.kosticks.com 2022.

[6 ] A. Holden. Orderly Tangles: Cloverleafs, Gordian Knots and Regular Polylinks, 1983.

[7] G. Hart. https://archive.bridgesmathart.org/2005/bridges2005-449.html#gsc.tab=0 , 2005.

[8] G. Hart. https://archive.bridgesmathart.org/2011/bridges2011-357.html#gsc.tab=0 , 2011.

[9] R. Lang. https://langorigami.com/article/polypolyhedra/ , 2022

[10] M. O'Keeffe, M. M. J. Treacy. Isogonal Weavings on the Sphere Acta Cryst. (2020_. A76, 661-621.

[11] A. Hizume. http://www.starcage.org , 2022.