Motives are in a sense the fundamental building blocks of polynomial equations, in the same way that prime factors are the elemental pieces of larger numbers.
If the period of a motive arising in one system of polynomial equations is the same as the period of a motive arising in a different system, you know the motives are the same.
“Once you know the periods, which are specific numbers, that’s almost the same as knowing the motive itself,” said Minhyong Kim, a mathematician at Oxford.
Pi shows up in many guises in geometry: in the integral of the function that defines the one-dimensional circle, in the integral of the function that defines the two-dimensional circle, and in the integral of the function that defines the sphere.
“The modern explanation is that the sphere and the solid circle have the same motive and therefore have to have essentially the same period,” Brown wrote in an email. Every route a particle collision could follow from beginning to end can be represented by a Feynman diagram, and each diagram has its own associated integral.
The entering and exiting particles are the same as previously described, but the fact that those unobservable collisions happen can still have subtle effects on the outcome.