8-7-2022

Advanced Tristix Geometry Cont.

Figure 113: Rhombic dodecahedron vertex touching.

The triangular prisms that make up a tristix occupy 50% of space when extended infinitely. The 50% of space that is not occupied by the prisms of a tristix consists of vertex touching rhombic dodecahedron voids. A repeating section of the interior of a tristix is shown in figure 113.

Figure114 : Tristix with unit cell outlined.

Calculating the density of a tristix can be accomplished by figuring out how much space is occupied by one of the repeating axes in just 2- dimensions, and then multiplying by 4 (figure 114).

The unit cell shown in figure 114 displays a triangular grid that can be used to count the triangles that are used by the triangular rod, and then dividing by the total number of triangles in the unit cell or 1/8 = .125 x 4 to = 50% of space.

Tristix, made with cylindrical rods rather than triangular prisms, has its density calculated in a similar way. However, it instead uses the area of the inscribed circle divided by the area of the overall unit cell. The area of the circle, in relation to the unit cell, is found by using πr² where the radius of the inscribed circle is equal to (a/2)/√3 or πa^{2}/12 (a as side of triangle). Then, the area of the circle is divided by the area of the unit cell = .075575 and multiplied by 4 to find that tristix made with cylinders fills approximately 30.22999% of space.

Figure 115: Tristix unit cell.

Figure 116: Tristix unit cell cylinders.

Tristix can also be reduced to a repeating 3-dimensional unit cell, e.g., figure 115 and 116. Each rod in the infinite tristix arrangement can be considered to have the same symmetric relationship to each other rod in the arrangement. There are only 6 known arrangements of non-intersecting rod packings with cubic symmetry, that have axes in invariant positions (locations can be completely determined through symmetry), as defined by O’Keeffe, J. Plevert, and T. Ogama. in 2002 [1].

Figure 117: Tristix uniaxial.

Figure 118: Tristix cylinders.

Tristix models can be built and centered around a rhombic dodecahedron (figure 118) or centered around a central rod axis (figure 117). If the rods in a tristix have directionality/ are pointed (like the stakes in post 3), the symmetry is altered and the arrangement is known as tristakes (figure 119).

Figure 119: Tristakes.

Figure 120: Tristix detail.

For now, we will pause our tristix explorations, and in the next blog post look at some polystix relationships.

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

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

copyright 2022 Anduriel Widmark