Graph theory details how things are connected. There are nodes that are connected by edges. The edges can either be directional or undirectional. Here is the graph for real number multiplication:

One times one makes a loop bringing you back to one. This may be the most important loop in all of mathematical physics since it provides a mechanism for particles to keep on being themselves, just getting a little older. The same type of number (reals) can loop with itself.

Contrast this with the graph for complex numbers. The only edges out to the imaginary node and back are *directional*. 1 times i gets to i, but i must be multiplied by -i to get back to one, a different road. Graph theory provides a reason why real numbers are different from imaginary numbers (I wish this had been pointed out at the start). There is a loop, but involves the imaginary vertex working with the real edge.

The graph for Hamilton's quaternions continue the theme of imaginary numbers. The undirectional edges are all loops using real numbers. At least on this site, the real number *is time* so this graph was animated to make the suggestion more concrete. The three imaginary nodes are what is needed for the three dimensions of space. Things out in space can be reflected in mirrors, which flip handedness, a signal for directional edges. Every node has the four types of edges, but only the real number has a loop which matches the color of the node.

If all the edges are undirectional, the graph applies to hypercomplex numbers, or what I call the California representation of quaternion multiplication.

When you do graph theory with clay and pipe cleaners, the simplicity of the California representation is clear: only one pipe cleaner goes between each node instead of two. Nature must use this sort of math tool often, which I think it does for inertia.

Everything is a simple harmonic oscillator, so if we pay attention to the details, we may learn something from these animations.

The 2 images on the right hand side of these animations may be a way to visually represent quantum mechanics. The wave function is the superposition of all possible states a system governed by a complex-valued wave function can be in. The story is complete, even if uncertain due to the complex numbers which are not a totally ordered set. The upper right image is literally a superposition of every frame that appears in the animation that is front and center. Below all that is possible is a random sampling of those what is possible. All that can be is contrasted with what happens to be, a core issue in the foundations of quantum mechanics.

But why must this be so? We can write complex valued equations which are accurate representations of measurements we make of Nature, yet those equations blissfully ignore if one collection of events could *cause* another set of event to occur. The equation in fact describes clusters of events which are independent of each other due to a spacelike separation. Such an equation is valid math. Yet to make it apply to Nature, we need to take the norm of the expression. When we take this step, the resulting expression is not a description of one event that appears after another as happens in classical physics. Instead the norm is the probability that an event will be seen at a location in spacetime.