Posted by doug

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.