The Greek mathematician Theodorus of Cyrene, teaching in the fifth century BCE, left behind one of the most visually elegant pictures in all of elementary geometry. He proved that \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7}, and so on (skipping perfect squares like \sqrt{4} and \sqrt{9}) are irrational — and the picture he drew in the process became known as the spiral of Theodorus.
The spiral builds the hypotenuses \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}, \ldots one right triangle at a time, and each triangle is attached to the previous one by sharing an edge. The result is a spiral where every spoke from the centre has a length that is the square root of an integer, in order. It is one of the few places in mathematics where a sequence of irrational numbers curls into a visible, physical shape.
The construction
Start with a right triangle whose two legs are both 1. By Pythagoras, its hypotenuse is \sqrt{1^2 + 1^2} = \sqrt{2}.
Now attach a second right triangle to that hypotenuse: one leg is the \sqrt{2} from the first triangle, the other leg is a fresh segment of length 1 stuck on perpendicularly. The new hypotenuse has length \sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{2 + 1} = \sqrt{3}.
Continue. At step n, the triangle has one leg of length \sqrt{n} (the hypotenuse from step n-1), a perpendicular leg of length 1, and a new hypotenuse of length \sqrt{n+1}. Every new hypotenuse becomes the next leg, and each successive root shows up on cue.
The picture
The spiral has a haunting quality when drawn carefully: the spokes are incommensurable (no two are rational multiples of each other, except the trivial pairs like \sqrt{4} = 2), but they line up like teeth on a gear.
The algebra behind each step
The Pythagorean theorem does all the work. If a triangle has legs of length \sqrt{n} and 1, its hypotenuse h satisfies
So by attaching a unit leg perpendicular to the \sqrt{n} hypotenuse, you produce the next square root in the sequence automatically. The spiral's entire structure follows from this one move, repeated over and over.
Why perpendicularity matters: if the new leg were not perpendicular, the Pythagorean theorem would not apply, and the new hypotenuse would be given by the law of cosines instead — not a clean square root of n+1. The right angle is what makes every new hypotenuse land on exactly \sqrt{n+1} with no mess.
What Theodorus was really proving
Theodorus's original purpose was not pretty pictures — it was a rigour exercise. The story, recounted in Plato's Theaetetus, is that Theodorus demonstrated the irrationality of \sqrt{3}, \sqrt{5}, \ldots, \sqrt{17}, one root at a time, and then (mysteriously) stopped at \sqrt{17}. Generations of scholars have debated why he stopped there; one likely reason is just that the spiral starts to overlap itself after \sqrt{17}, and a clean geometric argument breaks down.
The irrationality proofs themselves work because each triangle in the spiral has legs that are rational and irrational in combination, and if any hypotenuse were rational, the ratio it would create with the unit leg would contradict number-theoretic results Theodorus had available. The construction is not just a diagram; it is a tool for proof.
Patterns you can spot
The spiral has several surprising regularities.
- Winding rate slows. The first few triangles cover big angular arcs — the first triangle spans 45°, the second about 35°, the third about 30°. By the tenth triangle, each new triangle only adds about 18° to the total angle. The spiral slows down as it winds, which is why 17 triangles fit in one revolution.
- Angular sum is \sqrt{n} in disguise. The angle at the centre from the first spoke to the nth is the sum \arctan(1) + \arctan(1/\sqrt{2}) + \arctan(1/\sqrt{3}) + \cdots + \arctan(1/\sqrt{n-1}). As n grows, this sum diverges — meaning the spiral never stops winding — but it diverges extremely slowly, growing roughly like 2\sqrt{n} radians. This is a classical result in the study of the spiral.
- Limit curve. The smooth curve that connects the tips of all the spokes (the Archimedean spiral-like outer edge of the construction) is an approximation to a continuous spiral whose radius scales with the square root of the angle — sometimes called the "Theodorus spiral" in differential geometry.
Why it shows up in modern mathematics
The spiral is the simplest demonstration of a discrete-to-continuous bridge. A sequence of integer roots \sqrt{n} is discrete — one number per integer. But stacking the triangles produces a continuous shape (the spiral) that has well-defined slope, curvature, and winding at every point. The discrete and the continuous meet in the same picture. Many techniques in modern analysis — from Fourier series to numerical integration to computer-graphics path algorithms — follow the same pattern: discretise a continuous problem by stepping through integers, and watch a shape emerge.
The spiral also makes a clean pedagogical point: every square root of an integer, not just the perfect-square ones, is a legitimate length. You can draw \sqrt{7} with nothing but a ruler and a set square. Irrational numbers are not ghostly — they are lengths you can produce geometrically, and the spiral shows the mechanism that produces them one after another.
The takeaway
Stack right triangles with legs \sqrt{n} and 1, and their hypotenuses march through \sqrt{2}, \sqrt{3}, \sqrt{4}, \ldots in order. The resulting spiral is one of the most elegant artefacts of Greek mathematics, proving irrationality one triangle at a time and unfolding the whole infinite family of square roots into a single curling picture.
Related: Roots and Radicals · Exponents and Powers · Number Systems · Construction of Root-2 on the Number Line