30 Aug

Whence 2/(3+√5) ?

In both the Woven SSD and the Woven GSD, you calculate the inset (from the points where the rods meet near their ends) for putting marks on the rods by multiplying their lengths by 2/(3+√5). Where does that strange-looking number come from?

The key is that both figures consist of (regular) pentagrams, just interlocked in different ways. In other words, in both structures, looking only at any 5 rods in the same plane, we see:

What we’re interested in the fraction of the length of one rod (that is, the length between the points of the star where the rods cross; we’ll ignore the small overhangs beyond that from here on) represented by the distance from a point to the first internal intersection. In other words, we want the ratio of the red segment to the green segment in this diagram:

Now imagine that these segments are roads, and you start driving in a car from the red-blue point, and you follow all five of the roads until you end up back where you started, pointing in the same direction. You make five turns through the angle $\alpha$ indicated in the diagram below, but end up overall making two full turns all the way around (i.e., overall your car turns through an angle of $720^{\circ} = 4\pi = 2\tau.$) Therefore, $5\alpha = 2\tau,$ so the angle $\alpha = 2\tau/5,$ or 2/5 of a turn. We can then conclude that the point angle $\beta$ in the following diagram must be $\tau/2 – 2\tau/5 = \tau/10,$ or one-tenth of a turn (36 degrees).

The same argument applied to a regular pentagon shows that its interior angles are $3\tau/10$ (or 108 degrees). In other words, in the following diagram, the edges of the pentagram trisect the vertex angles of the circumscribed pentagon:

This angle relationship in turn means that the orange-pink-green triangle is similar to the brown-purple-green triangle. But it’s also clear that the long green is the sum of the brown and the purple. This similarity tells us that the sum of the purple and brown is to the orange as the purple is to the brown. But the orange is clearly the same as the purple, so the sum of the purple and brown is to the purple as the purple is to the brown. And this is exactly the definition that the purple cuts the long green in the golden ratio $\phi,$ i.e. that the purple is $1/\phi$ as long as the long green.

On the other hand, the brown-purple-green triangle is also similar to the acute pink-blue-red triangle. Applying the exact same argument as before shows that the red segment is $1/\phi$ as long as the purple segment.

Combining these two facts tells us that the red segment is $1/\phi^2$ as long as the long green segment. By solving the equation $(\phi+1)/\phi = \phi/1$, you can find the value $\phi=(1+\sqrt5)/2$. Using this value and some algebraic manipulations lets us calculate that the red segment is $2/(3+\sqrt5)$ as long as the green one, as used in the posts linked above.