The Dottie Number

The Dottie number is the unique fixed point $d$ of the cosine function: $\cos d = d$. It is approximately 0.739085133215 and has no known closed form.

This theory establishes the Dottie number's key properties: the fixed point exists (by the intermediate value theorem) and is unique (because $\cos x - x$ has a strictly negative derivative). Next, the value of $d$ to 12 decimal places is shown using the approximation proof method. Two more properties of $d$ are also shown: first, that it is transcendental (via the Hermite–Lindemann–Weierstrass theorem); second, that it is a universal attractor, in the sense that iterating the cosine function from any real starting point converges to it.