Geometric visualization of nested radicals

Question

There are a number of extremely useful geometric techniques for visualizing results from number theory, such as various infinite sums, results from combinatorics, and so on. (One of my favorites is the grid representation of the Euclidean algorithm for computing the greatest common factor of two integers, shown in an animation here.) Are there any analogous visualization techniques for nested radicals--to visualize, for instance, equations of the form:

$$\sqrt{n + \sqrt{n + \sqrt{n + ...}}} = \frac{1 + \sqrt{1 + 4n}}{2},$$

such as studied by Ramanujan and others?

The algebraic solution of that equation (in particular) is simple indeed, by denoting the left-hand side $x$, squaring and solving a quadratic equation. But can one represent it (and others like it) graphically/geometrically?

For example, Viète's expression for $\pi$ is:

$${2 \over \pi} = {\sqrt{2} \over 2} \cdot {\sqrt{2 + \sqrt{2}} \over 2} \cdot {\sqrt{2 + \sqrt{2 + \sqrt{2}}} \over 2} \cdot \cdots$$

and its form strongly suggests there is some graphic involving linked hypotenuses of scaled squares and a circle that illustrates this result. Is there one?

Answer 1

Viète's formula does indeed come from geometry. Starting with a square inscribed in a circle, double the sides to get a regular octagon. The ratio of the areas of these two polygons is $\frac{\sqrt2}2$. Continuing in the same manner, the product of the successive ratios of areas gives Viète's expression.

Here's the picture from the Wikipedia article: