Intuitive reason for periods of 2 and 8 in Bott periodicity?

Question

Is there a reasonably simple explanation for why Bott periodicity for $U$ and $O$ have periods 2 and 8, respectively? For example, in the $h$-cobordism theorem the requirement that $n \geq 5$ has the intuitive explanation that you need enough space in the ambient manifold to use the Whitney trick to push a disk off of itself, which takes ambient dimension at least $2\cdot 2 + 1$. Is there a similar idea or fact that might convey to a (relative) layperson why the numbers 2 and 8 come up in Bott periodicity?

Answer 1

Here is one source of intuition for the 8-fold and 2-fold Bott periodicities, from the physics of topological states of matter. See this online lecture.

In physics, Bott periodicity governs the classification of the fundamental symmetries of free electrons: the anti-unitary symmetries ${\cal T}_\pm$ (time-reversal symmetry) and ${\cal P}_\pm$ (particle-hole symmetry), and the unitary symmetry ${\cal C}={\cal PT}$ (chiral symmetry). The subscript $\pm$ distinguishes whether the anti-unitary symmetry squares to $+1$ or $-1$.

There are 8 symmetry classes with at least one anti-unitary symmetry: $${\cal P}_+;\;\;{\cal P}_-;\;\;{\cal T}_+;\;\;{\cal T}_-;$$ $${\cal P}_+,{\cal T}_+;\;\;{\cal P}_+,{\cal T}_-;\;\;{\cal P}_-,{\cal T}_+;\;\;{\cal P}_-,{\cal T}_-;$$

There are 2 symmetry classes with no anti-unitary symmetries, one with and one without chiral symmetry: $${\cal C};\;\;\times;$$

A gapped system on a $d$-dimensional torus can be raised to dimension $d+1$ by adding one momentum component, in such a way that the gap does not close on the torus. This operation transforms the symmetry class in a way that reflects the Bott periodicity, as indicated in the diagram below (the "Bott clock").

The arrows indicate the change of symmetry if dimension $d$ is raised to dimension $d+1$. Without any anti-unitary symmetry the cycle has period 2, with at least one anti-unitary symmetry the cycle has period 8.