√2-1 and Octagons

A short while ago, it was 4/14. Also known as √2-1 day, as $\sqrt{2} - 1\approx 0.414$.

There’s the golden ratio, $\frac{1}{2}\sqrt{5} + \frac{1}{2}$, but there’s also a silver ratio: $\sqrt{2} + 1$, which is just $\frac{1}{\sqrt{2} - 1}$¹. As you may know, if you take a golden rectangle (aspect ratio $\frac{1}{2}\sqrt{5} + \frac{1}{2} : 1$) and chop off a square from the end, you get another golden rectangle. A silver rectangle (aspect ratio $\sqrt{2} + 1 : 1$) is similar, but you need to chop off 2 squares:

Silver Rectangle — A silver rectangle. Chopping off 2 squares from the side reveals a smaller silver rectangle. Note that as mentioned before, $$\frac{1}{\sqrt{2} - 1} = \sqrt{2} + 1$$.

Here’s where things get interesting. The golden ratio shows up a bunch in regular pentagons and pentagrams, but the silver ratio’s favorite regular polygon is the regular octagon. If we take just 3 sides of a regular octagon and draw the diagonal connecting the endpoints, we can see that this ratio shows up:

Octagon Segment — 3 sides of a regular octagon, along with a diagonal connecting the endpoints. The diagonal's length is the side length times $$\sqrt{2} + 1$$.

If we take an octagon and draw all the diagonals parallel to some side, this ratio shows up all over the place!

Octagon with Several Silver Ratios — An octagon, along with several lengths shown. $$d_S = \sqrt{2} + 1$$.

This diagram, the octagon-and-parallel-diagonals², is so pretty, I decided to make a Blender render and a short animation about it.

Octagon + Parallel Diagonals Blender Render — A Blender render showing the octagon and its parallel diagonals. Power-of-(√2-1) ratios are emphasized.

Constructing the octagon + parallel diagonals diagram with side lengths involving √2-1.

Note that $\sqrt{2} + 1 = 2.414...$, which would not make a nice day of the year. ↩
Does this have an actual name? ↩