And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about. – 1 Kings 7:23
Some fans of mathematical lore quite enjoy citing this "miscalculation" (more reasonably called an approximation) of $\pi$ in the Bible, as well as the "Pi Bill" fiasco, where the Indiana Legislature very nearly codified $\pi=3.2$ into law. But what's so special about this constant $\pi=3.14159265358979323846264\dots$ anyways? Sure, it describes circumference/diameter of classical circles, but we will see that an appropriate generalization allows pi to take any value in the interval $[3,4]$. Fascinatingly, the classical constant $\pi$ retakes center stage when we impose various symmetry conditions, which bound pi by $\pi$ (above or below). For each of these bounds, the extreme values are achieved by (essentially) unique norms! I will discuss these cases and describe my proof for one of them. Given time, we will also look at symmetries involving duality of norms and/or see how these questions propagate into higher dimensions.