Theodorus Spiral Lab

Recursive Geometry & Irrational Sequences

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1. Construction Logic

The Spiral of Theodorus is an iterative construction where each triangle \( T_n \) is a right triangle with:

One leg equal to the hypotenuse of the previous triangle \( T_{n-1} \).
One leg equal to a constant unit length (1).
The unit leg is always perpendicular to the previous hypotenuse.

2. Sequence Verification

Calculate the exact hypotenuse length and the vertex angle \(\varphi_n\) for the first 5 steps.

Triangle \( n \)	Hypotenuse \( h_n \)	Interior Angle \( \varphi_n \)
\( n=1 \)	\(\sqrt{2}\)	\( \tan^{-1}(1/1) = 45^\circ \)
\( n=2 \)	\(\sqrt{3}\)	\( \tan^{-1}(1/\sqrt{2}) \approx 35.3^\circ \)
\( n=3 \)
\( n=4 \)
\( n=5 \)

3. Construction Plot

Sketch the first 10 segments of the spiral here

Convergence and Limits:

As \( n \to \infty \), does the angle \( \varphi_n \) increase, decrease, or stay constant? What does this imply about the "speed" at which the spiral revolves around the origin?

Theodorus Spiral

Theodorus Spiral Slides

Theodorus Spiral Lab