4-14-2022

Advanced Tetrastix Geometry Cont.

The prisms in a tetrastix occupy 75% of space when extended infinitely. The 25% of the spacethat is not occupied by the prisms in tetrastix consists of cubic voids in a vertex touching arrangement, shown in figure47.

Figure47: Vertex touching cubes, tetrastix empty space.

Tetrastix are defined by the cubic symmetry group Pm3n, or -8° (depending on what notation is used). A related polystix structure can be made using only 1/2 of the sticks in a tetrastix arrangement, this is named “hemistix” in The Symmetries of Things by Conway, Burgiel, Goodman-Strauss, and known as the Π+ cubic packing by Michael O’Keeffe. Hemistix occupies 37.5% of space when made with square prisms (figure 48) and 29.45% of space when made with cylinders (figure 49). Hemistix are chiral arrangements and tetrastix can be thought of as a combination of 2 hemistix enantiomers (figure50).

Figure48 : Hemistix.

Figure 49: Hemistix made with cylinders.

If the prisms used in a tetrastix or hemistix have directionality (like the pointed sticks in post 3), the structure’s symmetry is altered, and the arrangements are known as Tetrastakes (figure 50 and 51) and hemistakes.

Figure 50: Tetrastakes with hemistakes coloration.

Figure 51: Tetrastix made with matches, held together without glue, only friction.

Tetrastix models can be built with the sticks positioned around the center of a cube (figure 52), or centered on the vertex of a cube (figure 53).

Figure 52:Tetrastix cube template of 18 sticks.

Figure 53: Tetrastix cube template of 27 sticks.

Figure 54: Tetrastix with alternating colors.

Figure 55: Tetrastix popsicle sticks.

In addition to the tetrastix distinctions previous discussed in these blog posts, some possible tetrastix projects could include: calculating the density of invented designs, introducing interlock, linking rods, finding optimum rod shape, using bundles of sticks, minimal symmetric arrangements, tension and force analysis, minimal surface relationships, interweaving spiral structures, alternating coloration, and many more.

For now, we will pause the tetrastix investigations and in the next blog posts we move on to a different polystix… Hexastix!

copyright 2022 Anduriel Widmark

[1] A. Widmark. https://archive.bridgesmathart.org/2021/bridges2021-293.pdf.

[2] A. Widmark. https://archive.bridgesmathart.org/2022/bridges2022-379.pdf.

Posts:

1- Polystix blog introduction.

5- Polystix modeling materials.

9- Tetrastix shadow projections.

11- Advanced tetrastix.

15- Hexastix design variations.