I was doing some geometry with circles locked on a grid, and I noticed that the distance between a unit circle and the corner of a square that it is inscribed in is of length $\sqrt{2} - 1$, or roughly $.41421356237$. Does this number have a widely-recognized name, and does it pop up in seemingly unrelated situations in the way that pi or the golden ratio do?
Granted, the primary purpose of such names are to create shorthand, and $\sqrt{2} - 1$ isn't very long to begin with. Still, it's not much less cumbersome than the mathematical expressions that define the first few metallic ratios, which are nonetheless named. Furthermore, if we re-scale the geometry to involve a unit square instead of unit circle, then the similar line segment is of length $\frac{\sqrt{2}-1}{2}$, which is closely reminiscent of the metallic ratios. I can easily imagine that constant being named rather than the first one.
There is some significance to this constant in the theory of quasilattices. While the best-known cases involve fivefold symmetry, eightfold examples have also been realized in quasicrystalline materials in the laboratory. So this eightfold quasilattice concept is a thing indeed.
We may then render the vertices of a regular octagon as quasilattice points, and then identify a quasiperiodic ratio by comparing the length of a side of the octagon with that of a parallel diagonal. If we divide the diagonal by the side, we get a ratio of $\sqrt2+1$; with the reverse division it comes out to $\sqrt2-1$ (in the fivefold case we would use a pentagon instead, in which case the ratio between side and parallel diagonal would be $(\sqrt5\pm1)/2$). Both period ratios are consistent with the same quasilattice structure because they differ by a whole number.