### 11-21-2021

3 Stick Arrangements

We previously explored simple stick arrangements that were made with rods positioned in only 2-directions, now we will be adding some complexity by moving onto rod configurations with 3 possible directions. To start we will look at the different ways 3 sticks can be arranged with cubic symmetry, with all components positioned at right angles to each other (rectilinear).

Since we are already familiar with simple 2-directional arrangements from the previous post, let’s start by placing 2 sticks so they cross, touching on their midpoints at 90° (figure 4).

Next, adding a 3^{rd} stick, perpendicular to the 2 crossing sticks, there are 4 different placement options e.g. Figure 4. If you flip these around, 2 of these positions make the same arrangement as the opposite 2 (Equal through 180° rotation) and thus, there are a total of 2 possible placement options; the left and right-handed 3 stick arrangements, Figure 5.

Figure 4: 2 Crossing sticks and the 4 possible placements of a 3^{rd} Stick.

Figure 5: Left and right-handed 3 Stick arrangements.

Next, if the sticks used are pointed/staked, instead of identical (congruent), there are more placement options. Below are the 4 possible ways to arrange 3 pointed sticks.

Figure 6: 4 Ways of arranging 3 pointed (directional) sticks.

But wait! If it mattered what ends of the sticks were touching the ground, there are even more placement options. For the first 2, left and right-handed arrangements, there are 2 ways each can lay on the ground (Figure 7), which makes 4 unique placements.

Figure 7: 4 Ways of grounding 3 sticks.

For the 4 arrangements of 3 staked sticks, there are 24 ways they can be placed on the ground (figure 8).

Figure 8: 24 Unique orientations of 3-pointed Sticks on a ground.

Figure 9: 24 3-pointed stick arrangements.

The 3-pointed stick arrangements shown above, also contain the foundational tripod form that is used in the construction of architectural structures such as teepee and lavvu (excluding quadrapod versions). The use of trees with the larger end on the ground and the pointed ends facing up, have only 4 possible tripod stacking options ( 1^{st} and 3^{rd} rows in Figure 8).

Stay tuned, in the next blog post we finally make our way to tetrastix!

copyright 2021 Anduriel Widmark