There is a lot of interest in the maximal density of equal circle packing in a circle. And I thought that knowing whether or not the solution is always algebraic or not would be useful.
My original hunch was that almost all of them would be transcendental numbers, but then I saw that the known solutions and popular conjectures are all algebraic numbers from this Wikipedia article: Circle packing in a circle. And I was wondering if that is always the case and if so why?
Yes indeed, they are all algebraic. The idea is that we can describe the critical $r$ as a first-order formula in the language of fields, something like $$\forall r_1\quad ((0 < r_1 \wedge r_1 \le r)\Longleftrightarrow (\exists x_1 \exists y_1 \ldots \exists x_n \exists y_n\,\, \phi(r_1, x_1, \ldots , x_n, y_1,\ldots , y_n))),$$
where $\phi$ is the statement that $r_1 > 0$ and the disks of radius $r_1$ and centers at $(x_1, y_1), \ldots (x_n, y_n)$ all lie in the unit disk and do not intersect, which you can make explicit by the Pythagorean theorem.
Then, run the quantifier elimination on it (using the Tarski--Seidenberg Theorem), you can get that this formula is equivalent to a formula about $r$ without quantifiers. It will look like a bunch of polynomial equalities and inequalities (with integer coefficients, since in the language of fields we only have constants $0$ and $1$, which are integers), joined by various logical symbols. I claim that the optimal $r$ is the root of one of these polynomials. Indeed, if it is not then by continuity all $r'$ which are close enough to $r$ will also satisfy the same formula, but clearly our initial formula uniquely determines $r$.
Thus, $r$ is a root of a polynomial with integer coefficients. In principle, since the quantifier elimination can be done by an algorithm, for each fixed $n$ the above argument can be turned into the construction of said polynomial, given infinite time.
And of course this argument can be applied to fitting any algebraic shape into any other algebraic shape, as long as you can write the formula $\phi$ using only polynomials.
