After looking through some old notes, I came across an idea that, thinking back, was an obsession of mine in high school. In my final year before going to college, I was interested in clockwise and counterclockwise flows and their “sums”. I was particularly interested in adding flows and seeing how the sums appear from the prospective of an observer on the left and an observer on the right of the sum. Here is an example of what I mean:
Consider the following set of flows:
We follow the rules of flow addition, which are simple and meant to be intuitive. Let’s define a binary variable f (so it can only take two values, which can we call 0 (for clockwise) and 1 (for counterclockwise)). As we might guess, f represents the state of a single flow, whether left or right. However, a single flow can be either, if one simple changes perspective. Whenever we perform a “flow sum”, we must define the direction/perspective. We will see how this is done by considering the example given above, with three flows. But first, the rules of flow addition are as follows: f + f = f , f + f* = null, f + null = f , f* + null = f*. I should mention that f* is the opposite flow state of the variable f.
L: f + g + h = 0 + 1 + 1 = null + 1 = 1
R: f *+ g *+ h* = 1 + 0 + 0 = null + 0 = 0
The total sum is either 1, 0, or null, so what L and R observe is a single flow state. Notice that flow sums are not associative. We can see this in this example. The flow sum for L was 0 + 1 + 1. If we did the first sum first we got null + 1 = 1. Second sum first we get 0 + 1 = null. We could do this with R as well. What this says is order matters.
If we streamline the previous example, getting rid of the sums, we can write the following:
L: 011 = 1
R:100 = (011)* = 0*1*1*= 0
We implicitly defined a property of the * operator, namely that operating on a string of flow states is equivalent to applying the operator to the elements, each state, in the string.
For fun, we may see that these representations of flow sums are binary. Thus 011 = 2+1 = 3 and 100 = 4, where + is here the usual operation of addition. We may call 3 and 4 a “flow pair” because, in binary, they are equivalent to the system of flows we visualized earlier just viewed from different perspectives. Another “flow pair” is 101 and 010, or 5 and 2. Here are all the flow pairs for a system of three flows:
000, 111 -> 0, 7; 001,110 -> 1, 6; 010,101 -> 2,5; 100,011 -> 4,3
These are just the different ways of summing integers to 7.
000 or 0 = 0; 111 or 7 = 1
001 or 1 = null; 110 or 6 = null
010 or 2 = 0; 101 or 5 = 1
100 or 4 = 0; 011 or 3 = 1
