6-21-2022
Advanced Hexastix Geometry Cont.
The hexagonal prisms that make up a hexastix occupy 75% of space when extended infinitely. The space that is not occupied by the prisms of a hexastix consists of face touching oblate tetrahedron in a lattice, that makes up the 25% of empty space of hexastix. A small cut out of the interior of a hexastix is shown in figure 86.
Figure 86: Hexastix interior cut out.
Calculating the density of a hexastix 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 87). The unit cell shown in figure 87 also shows a triangular grid that can be used to count the triangles that are used by hexagonal rod and dividing by the number of total triangles in the unit cell, or 6/32 = .1875 x 4 to = 75% of space.
Hexastix made with cylindrical rods rather than hexagonal prisms has its density calculated in a similar way, but instead uses the area of an inscribed circle that is divided by the area of the overall unit cell. The area of the circle, in relation to the unit cell, is found by using π r² where r is equal to quarter the height of the unit cell by way of a* √3/2 (equilateral triangle hight, with a as the side of a triangle), then multiplied by 4 and divided by the area of the entire unit cell, to find that hexastix made with cylinders fills approximately 68.0175% of space.
Figure 87: Hexastix unit cell.
Hexastix can also be reduced to a repeating 3-dimensional unit cell e.g., figure 88. Each rod in the infinite hexastix arrangement can be considered to have the same symmetric relationship to each other rod. There are only 6 known arrangements of non-intersecting rods packings with cubic symmetry that have cylinder axes in invariant positions (locations completely determined through symmetry), but as homogeneous cubic packings there are 59 possible arrangements, defined by M. O’Keeffe in 2002 [1].
Figure 88: Hexastix unit cell.
Modeling hexastix always causes some loss of symmetry from the infinite pattern it represents [2]. Physical models of chiral hexastix can be built and centered around a rod as the central axis (figure 89) or centered around an empty space/ in a polar configuration where the center is surrounded equally by 2 crossing x’s (figure 90) resting against each other in a regular way (figure56).
Figure 56: Hexastix, centered around empty space.
Figure 89: Hexastix, centered around rod axis.
Another combinatorial problem in hexastix is deciding colorations. Hexastix made with 4 colored axes in which colors do not repeat have 12 possible colorations, seen in figure91 (made with the hexagon pattern in an empty space centered arrangement). If hexastix is made with a rod centered axis, there are only 4 possible colorations. More colors and patterns can be added into a hexastix model not only to increase the complexity but also as a fun way to introduce some creative opportunities and add beauty to the world.
Figure 90: Hexastix color options simplified.
Figure 91: Hexastix 72sticks, 12 colorations.
For now, we will pause our hexastix investigations and in the next blog post, look at a different polystix, tristix.
copyright 2022 Anduriel Widmark
[1] M. O’Keeffe, J. Plevert, and T. Ogama. “Homogeneous Cubic Cylinder Packings Revisited” Acta Cryst. A58, 2002.
[2] A. Widmark. https://archive.bridgesmathart.org/2022/bridges2022-379.pdf.
[3] A. Widmark. https://archive.bridgesmathart.org/2021/bridges2021-293.pdf.
Posts:
1- Polystix blog introduction.
5- Polystix modeling materials.
9- Tetrastix shadow projections.
15- Hexastix design variations.
17- Advanced hexastix.