## Wednesday, November 28, 2018

### Interaction

Left Side Offset:

Left Side Direction:

Right Side Offset:

Right Side Direction:

Interaction Energy =

The foundation for the dynamics of a system is the way the components interact. The "wrinkles" system that I am exploring is a two dimensional array of cells. Each cell interacts with its four neighbors. The state of each cell is given by a direction and an offset. These change over time. In the thermodynamic simulation that I am building, a cell will tend to change so its direction and offset are similar to those of its neighbors.

Handling the offset gets tricky. The idea is that the dark and light strips should line up on the boundary between neighboring cells. If the cells have the same direction, the offsets should simply match. But when the cells have different directions, it gets messy!

What I'm doing to manage this is to look solely at the midpoint of the edge that forms the boundary between two neighboring cells. The offsets are close if the stripes fit together at that midpoint.

There is some topological strangeness here. The light and dark stripes have a symmetry: if you turn them upside-down, the result is the same pattern as where you started. But at the two extremes of cell direction, increasing offset will move the stripes in the opposite direction. Changing the direction rotates the stripes: every 180 degree brings the stripes back to where they were. Similarly, changing the offset enough will move the stripes back to where they started. So the topology of these two parameters together forms a kind of torus. But because the direction of offset is reversed after a 180 degree change of direction, the topology is actually that of a Klein bottle. This topology is what allows an array such as: