Does this spiral, formed from similar right triangles arranged around a point, have a name?

Question

If you start with any right triangle and add contiguous right triangles with the same angles, you form a spiral like this.

The arms of the spiral grow according to the rule $H_n=(\cos\theta)^{-n}$, where $\theta$ is the angle at the centre point of the triangle. The radial arms grow exponentially. I believe that makes it a geometric form of a logarithmic spiral.

The example in the picture has $\theta=30°$.

Besides being a class of logarithmic spiral, does this particular kind of spiral have a name?

Answer 1

Well sort of(?) if the right-angle triangles have unit leg we get a special type - this spiral is variously known as the spiral of Theodorus, or the Pythagorean Spiral, or the square root spiral, etc. It has been extensively studied and has some very interesting properties. For example, it can be shown that the angle of any point is never repeated. After only two turns it closely approximates an Archimedean spiral.

However, your spiral with constant central angle is another matter - I am unaware of any work on that - perhaps others can shed some light on it. It will be closely related to the logarithmic spiral.

Answer 2

This is not just a class of logarithmic spiral, it is exactly a logarithmic spiral. If we call the angle of the triangle $\Delta\theta$, and growth rate, call it $q=1/\cos\Delta\theta$, the flair coefficient of the log spiral is $b=\ln q/\Delta\theta$ and the spiral is given by

$$ \begin{align} &z=e^{(b+i)\theta},\quad \theta\in [0\dots]\\ &z_n=e^{(b+i)n\Delta\theta} \end{align} $$

where $z_n$ are the vertices of the triangles (along with the origin). All you have to do now is to connect the $z_n$ to $z_{n+1}$ and the origin to form the tiling.