Geometric Construction of the Spiral of Theodorus

This GeoGebra applet demonstrates the geometric construction of the Spiral of Theodorus, a classic mathematical spiral composed of a sequence of contiguous right triangles. In this construction, each right triangle shares its hypotenuse as one leg of the subsequent triangle, while the other leg maintains a constant length of 1. As a result, the hypotenuse of the nth triangle equals the square root of n, producing the elegant progression of lengths 1, sqrt(2), sqrt(3), sqrt(4), and so on. The applet employs several powerful GeoGebra features to bring this construction to life dynamically. IterationList and Sequence commands generate the repeated geometric pattern automatically, while Rotate and Dilate commands apply planar transformations that position each successive triangle precisely. Students can interact with the model to observe how the spiral emerges step by step, watching as the right angles accumulate and the figure gracefully unwinds outward from its origin. At its core, the applet provides a vivid, visual illustration of the Pythagorean theorem in action. Each triangle individually satisfies the relationship a squared plus b squared equals c squared, and the cumulative effect of chaining these triangles together produces a continuous spiral that has fascinated mathematicians for over two millennia. Theodorus of Cyrene is traditionally credited with studying this figure, and he reportedly used it to demonstrate that the square roots of integers from 2 through 17 are all irrational. Educators can use this interactive model in a variety of classroom contexts. Students exploring geometry will gain intuition about how iterative processes and geometric transformations combine to create complex curves from simple rules. Those studying the Pythagorean theorem will see its recursive application across an entire sequence of triangles rather than in isolation. Learners interested in the history of mathematics can appreciate how ancient discoveries about irrational numbers manifest in a beautiful and accessible visual form. By adjusting parameters or stepping through the construction at their own pace, students develop a deeper understanding of both the computational and geometric aspects of the Spiral of Theodorus.