What is the relation between the Reuleaux triangle and the imaginary golden ratio?

Question

The imaginary golden ratio is given by

$$\varphi_i = \frac{1+\sqrt{3}i}{2}=e^{i\pi/3}$$

This has many properties in common with the golden ratio and has been adequately described here (Imaginary Golden Ratio) and here.

If we look at the normalized Reuleaux triangle (i.e., of unit width) as in the figure below, we’ll see that the corners are given by $\varphi_i,\ \varphi_i^2,\ \varphi_i^2-\varphi_i+1=0$. It’s also worth noting that the centroid is given by the average of these three points, to wit, $z_c=i/\sqrt{3}$.

I have looked for equations for the Reuleaux triangle, but have found only one that was given by Jan M, reproduced here, albeit in complex form

$$z=2\cos(\pi/2n)\cdot \exp\bigg(\frac{i}{2}\bigg(t+\frac{\pi}{n}\big(2\lfloor nt/2\pi\rfloor+1\big)\bigg)\biggr)- \exp\biggr(\frac{i\pi}{n}\big(2\lfloor nt/2\pi\rfloor+1\big)\bigg)$$

This is an equation for all the Reuleaux polygons ($n\ge3,\ n \text{ odd}$). This equation works for even $n$ as well, but these polygons do not satisfy the Reuleaux requirement of uniform width. There is nothing (that I can see, at least) that gives any hint of the imaginary golden ratio.

In the answer that follows, I will demonstrate the Reuleaux triangle, indeed, all the Reuleaux polygons, as degenerate pseudospirals, with known analytic solutions.

Answer 1

The answer to the question of the relation between the Reuleaux triangle and the imaginary golden ratio is rooted in the pseudospiral. That term, as used here, refers to spirals that are composed of circular arcs connected at regular arc spacing and whose radii are given by a sequence of positive and negative real numbers. The pseudospiral was developed about ten years ago while searching for an analytic solution for the Fibonacci spiral. The work is described here and in the linked pdf file. Briefly, the equation for the pseudospiral is given by

$$z(\theta)=S_1 e^{i\theta}+\sum_{n=2}^{N}(S_n-S_{n-1})( e^{i\theta}- e^{i(n-1)\theta_r})u(\theta-(n-1)\theta_r)$$

where $S,\ \theta_r,\ \text{and } u$ are the sequence, rotation angle, and Heaviside step function, respectively.

The sequence, $S$ can be just about anything, including positive and negative real numbers. Complex sequences have been tried as well, but it’s hard to ascribe any meaning to them. Until now.

It was no trouble to set up a sequence of the imaginary golden ratio, say $S=\varphi_i^n$ to test it as a pseudospiral. However, the expectation is that the spiral will not grow, per se since the absolute value of $\varphi_i^n$ is unity. The first figure below shows such a spiral with $\theta_r=\pi/5$. Now, we know that $\varphi_i^n$ repeats in series of three, that is, $\varphi_i^4=\varphi_i^1$ and so on, So, if we set $\theta_r=\pi/3$ we’ll only need three terms to complete our 'spiral,' as shown in the second figure below. And that is the Reuleaux triangle, which can be expressed analytically as

$$z=\varphi_i e^{i\theta}-( e^{i\theta}- e^{i\pi/3})u(\theta-\pi/3)- \varphi_i ( e^{i\theta}- e^{i2\pi/3})u(\theta-2\pi/3);\quad \theta\in[0,\pi]$$

where we have used the identity $\varphi_i^2-\varphi_i=-1$.

From here it’s an easy stretch to see that the remaining Reuleaux polygons (and the even ones, as well) can be had by setting $\varphi_j=e^{i\pi/j}$, $\theta_r=\pi/j$, and $\theta\in[0,\pi]$. A sampling of results for $j=(3:11)$ is shown in the third figure below.