interactive exact map a lens, not a law DOI 10.5281/zenodo.21280464 paper PDF TeX

The Polar Shell Renderer

Equal-Area Denominator Shells for Reduced Fractions and Primitive Roots of Unity

Choose a denominator range, then inspect the exact map. The denominator sets radius, the numerator sets angle, reduction chooses the shell, and Euler's totient counts its visible points. The picture is interactive; the claims remain attached to the formulas.

gcd(k,n) = 1 r = √n θ = 2πk/n #Sₙ = φ(n) Area(Aₙ) = π
reduced fraction selected / prime-denominator overlay hover θ = 0 east · positive θ counterclockwise
color points by
rendered points 604 Φ(44) = 604 selected shell 12

Four classical structures, one view

the map displays, it does not invent

01 · population

Totient shell count

#Sₙ = φ(n)

Shell n contains exactly the reduced residue classes modulo n.

02 · angle

Primitive roots

exp(2πik/n), gcd(k,n)=1

The angular positions are precisely the primitive n-th roots of unity.

03 · radius

Equal-area annuli

Area(Aₙ) = π

The law r = √n assigns every denominator the same annular area budget.

04 · cumulative count

Farey / summatory totient

Φ(N) = Σₙ≤N φ(n)

The total point count is the summatory totient; 1 + Φ(N) is the Farey length.

Density hygiene

same word, different denominator

Four quantities that must stay separate

shell population#Sₙ = φ(n)
annular normalizationDₙᵃⁿⁿ = φ(n)/π
circumferential densityDₙᶜⁱʳᶜ = φ(n)/(2π√n)
average disk densityΦ(R²)/(πR²) ~ 3R²/π³

The measure fence

The points lie on circles. The equal-area budget belongs to the assigned annuli. Therefore φ(n)/π is an annular normalization, not a literal pointwise density over the disk.

None of these quantities is the number-theoretic visible-lattice density 1/ζ(2).

Lock: before saying density, name the population, denominator, measure, and ambient space.

The TFG collapse

many cells can share one reduced polar address

(N,K)→ g = gcd(N,K)→ (s,t) = (N/g,K/g)→ t/s→ (√s, 2πt/s)

(12,8) collapses to 2/3

g=4 → (s,t)=(3,2) → (√3, 4π/3)

(6,4) lands on the same point

g=2 → (s,t)=(3,2) → (√3, 4π/3)
The square grid supplies the cells, reduction supplies the denominator shell, the polar map supplies the coordinates, and the totient supplies the population.

Where the zeta question changes

same metaphor · different mathematical object

Polar denominator shells

rₙ = √n   ·   n↑ ⇒ r↑

Reduced fractions expand outward by denominator. Shell n carries exactly φ(n) discrete points, and its assigned annulus has area π.

Local zeta-isolation shells

rₘ(ρ) ≈ [m|ζ′(ρ)|]⁻¹   ·   m↑ ⇒ r↓

For a known simple zero, local analytic level sets contract inward and support a finite-range diagnostic. They do not inherit the denominator-shell counting law.

Same shell metaphor. Different generator, geometry, and theorem family. A shared picture is not a transferred theorem.

Claims and boundaries

the fence travels with the figure

What the page claims

  • The stated polar map is exact.
  • Shell n has φ(n) points.
  • The angular positions are primitive n-th roots of unity.
  • Each assigned denominator annulus has area π.
  • The TFG denominator-collapse map is exact.

What the page does not claim

  • No new totient or cyclotomic theorem.
  • No constant-density filling of the disk.
  • No transfer of visible-lattice density into this geometry.
  • No shared shell law with the zeta annular-shell diagnostic.
  • No claim that absence of found prior art proves novelty.