6-21-2023
Polystix Update
The hexastix continue.
Figure 131: Sigma Rod Packings.
There are two invariant cubic rod packings that can be made by systematically removing rods from a hexastix arrangement. The +Σ and the Σ* / Sigma rod packings, shown in Figures 131-142 (where + or – indicates handedness and * indicates a combination of the two enantiomorphs) can be made by systematically removing 8 out of the 9 or 7 out of the 9 rods from the infinitely repeating hexastix arrangement.
Figure 132: +Σ Rod packing, 3-fold.
Figure 133: +Σ Rod packing, 4-fold.
Figure 134: +Σ Rod packing, grid with unit cell.
The +Σ packing (figures 132-137) is the least dense of the invariant cubic rod packings. An infinitely repeating +Σ rod packing would fill exactly 12.5% of space if made with triangle prisms, and approximately 7.6% of space when made with cylindrical rods.
Figure 135: Hexastix with +Σ Rods.
Figure 136: Simple +Σ Rod packing.
Figure 137: +Σ Rod packing.
The Σ* arrangement contains both +Σ and -Σ (left- and right-handed) arrangements (figures 138-142). The two enantiomorphs do not touch each other, this can make the structure less stable when modeling compared to the other invariant rod packings. I prefer the two frames in the Σ* model to remain separate, so they can move around a little. It is important when modeling the Σ* structure to not let glue join all the separate parts, keeping the left and right-handed frames held apart until the glue drying is complete.
Figure 138: Σ* Rod packings.
Figure 139: Σ* grid with unit cell.
Figure 140: Σ* rod packing with +Σ and -Σ colorations.
Figure 141: Σ* rod packing.
Figure 142: Σ* rod packing model.
The percent of space Σ* occupies is twice that of the +Σ packing. The Σ* packing fills exactly 25% of space when made with triangle prisms, and fills approximately 15.2% of space when made with cylindrical rods, or 16.7% of space when made with hexagonal rods.
Polystix geometry keeps evolving in many interesting ways. More updates will be coming on a periodic basis. Stay tuned.
copyright 2022 Anduriel Widmark