finitely many values of $\theta$. Therefore, we can study the family of noncrossing matchings, equipped with a cyclic order, obtained by letting $\theta$ vary; we call this family the *necklace of matchings* associated to $f$. Notice that the necklace depends only on the polynomial $f$ itself, and not on a choice of angle.

The paper is structured as follows. In Section \[section:FTA\], we give a modern exposition of Gauss’s proof of the Fundamental Theorem of Algebra, and explain how Gauss’s ideas may be used to associate basketballs with complex polynomials. Section \[section:basketball\] is devoted to the combinatorics of basketballs. Section \[section:IBT\] contains the proof of the Inverse Basketball Theorem. We conclude in Section 4 with some brief remarks on necklaces of matchings; there appears to be much more to say here from both the combinatorial and geometric points of view.

Gauss’s proof of the Fundamental Theorem of Algebra {#section:FTA}
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We begin by describing Gauss’s approach to the Fundamental Theorem of Algebra [@Gauss]; for the technical details, see Gersten and Stallings [@Gersten]. Let $f(z)$ be a monic polynomial of degree $n$, and consider the two curves $$\begin{aligned}
    R &= \{z {\;:~}{\mathop{\rm Re}\nolimits}f(z)=0\},\\
    I &= \{z {\;:~}{\mathop{\rm Im}\nolimits}f(z)=0\} \,.
  \end{aligned}$$ In 