The Polar Shell Renderer: Equal-Area Denominator Shells for Reduced Fractions and Primitive Roots of Unity

We describe a polar rendering of the reduced fractions. Each reduced fraction $k/n$ with $\gcd(k,n)=1$ is placed at radius $\sqrt{n}$ and angle $2\pi k/n$. Under this map four classical structures become visible together in a single figure: each concentric shell at radius $\sqrt{n}$ carries exactly $\varphi(n)$ points (Euler's totient); the angular positions on that shell are exactly the primitive $n$-th roots of unity (the roots of the $n$-th cyclotomic polynomial); the $\sqrt{n}$ radial law makes every denominator shell an equal-area annulus; and the cumulative point count out to denominator $N$ is the summatory totient $\Phi(N)$, whose leading asymptotic $3N^2/\pi^2$ is the classical Mertens result (with Walfisz's later refinement of the error term) and whose value $1+\Phi(N)$ is the length of the Farey sequence of order $N$. Because shell $n$ carries $\varphi(n)$ points rather than one, the rendering is not a constant-density filling of the disk; the equal-area claim is instead denominator-local, so each assigned annulus has area $\pi$ and its annularly normalized population is $\varphi(n)/\pi$. All four facts are classical; the contribution is the specific unified representation and the equal-area-by-denominator choice that ties the number-theoretic content to the phyllotaxis (sunflower) tradition. We position the rendering among existing fraction- and root-visualizations --- Ford circles, Farey-sunburst constructions, Vogel's sunflower, roots-of-unity diagrams, and the Ulam--Sacks prime spirals --- and within the author's Triangular-Fractional Grid series, giving the exact denominator-collapse map that realizes the rendering as the reduced skeleton of that grid. No prior instance of the specific unified rendering was found, but the absence of a found instance is not a proof of novelty, and no arithmetical claim is made beyond the classical facts recalled.