The Spiral of Theodorus — Square Roots Geometrically

What just happened?

By adding to the spiral successive right triangles in which one leg always has length $1$, you get hypotenuses of lengths $\sqrt2,\sqrt3,\sqrt4,\sqrt5,\dots$ — the square roots of the successive natural numbers, "drawn" with compass and straightedge.

How the spiral is built:

We start with an isosceles right triangle with both legs $1$ — its hypotenuse has length $\sqrt2$ (because $1^2+1^2=2$). To that hypotenuse we attach a new triangle with a leg $1$ — its hypotenuse is $\sqrt3$. And so on.

The invariant — one step is one Pythagoras:

Each successive hypotenuse is formed from the previous one $\sqrt{n}$ and an attached leg $1$, so it has length $\sqrt{n+1}$. Hence the sequence $\sqrt2,\sqrt3,\sqrt4,\dots$ appears automatically.

Who was Theodorus?

The spiral of Theodorus (also called the Pythagorean spiral or the square-root spiral) is a construction attributed to Theodorus of Cyrene (c. 465–398 BC), a Greek mathematician and Pythagorean, a teacher of Plato. Theodorus used a similar construction to prove the irrationality of the roots of non-square numbers — $\sqrt3,\sqrt5,\sqrt6,\dots$ up to $\sqrt{17}$.

Why does it matter?

The construction shows how abstract irrational numbers can be represented as concrete, measurable geometric lengths. The successive triangles wrap around a common centre with a constant "increment" of the angle $\arctan\!\big(1/\sqrt{n}\big)$, and at $\sqrt{17}$ the spiral completes a full turn and begins to overlap itself.

Why you need this for the exam:

It is a model example of the Pythagorean theorem in action and of the geometric meaning of a square root. Remember the formula $(\sqrt{n})^2+1^2=n+1$, which gives the hypotenuse $\sqrt{n+1}$ at each step.

Remember: the spiral of Theodorus "draws" the roots $\sqrt2,\sqrt3,\sqrt4,\dots$; each step is $(\sqrt{n})^2+1^2=n+1\Rightarrow\sqrt{n+1}$.