Research question: What mathematical properties emerge from the Wheel of Theodorus, and how does it approximate the Archimedean spiral?

Overview: This IA constructs the Wheel of Theodorus by joining successive right triangles with unit bases. It shows how hypotenuse lengths naturally trace irrational square roots (√2, √3, √4, …) and how the spiral structure relates to trigonometric angle sums. By comparing the spiral’s growth to the Archimedean spiral $r = a + b\theta$, the work demonstrates that the Wheel asymptotically aligns with an Archimedean curve while retaining unique discrete geometry.

Highlights

Hypotenuse of the $n$-th triangle follows $R_n = \sqrt{n+1}$.
Cumulative angle given by $\sum_{x=1}^{n}\arcsin\big(\tfrac{1}{\sqrt{x+1}}\big)$.
Winding distance from the $v$-th triangle to its immediate inner $u$-th triangle is
\[\sqrt{v} - \frac{\sqrt{u}}{\cos\Bigg(\sum_{n=u+1}^{v} \arcsin\bigg(\tfrac{1}{\sqrt{n}}\bigg)\Bigg)},\]
which approaches $\pi$, consistent with Archimedean spiral spacing.