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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.  

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