The Spiral of Theodorus is a beautiful illustration of the Pythagorean theorem.
The Pythagorean theorem states: a2 + b2 = c2 where a and b are two sides of a right triangle, and c is the hypotenuse.
The Spiral of Theodorus begins with a right triangle where sides a and b are unit length 1 (the red triangle above). The value of the square root of 1 is 1. Since 12 + 12 = 2, the hypotenuse must be the square root of two.
In the next (red-orange) triangle, one side is length 1 and the other is the square root of 2. Since 1 + 2 = 3, the hypotenuse is the square root of 3. And so on.
Here we stop at the square root of 17, but the spiral can continue indefinitely, with triangles overlapping each other, and it has been proven that no hypotenuse will ever precisely overlap another.
Here is the spiral without the numbers:
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