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- The Ring Geometry
https://wessengetachew.github.io/MODZ/
For each modulus M ≥ 1, define the coprime residue set
R(M) = { r ∈ {1, …, M−1} : gcd(r, M) = 1 } |R(M)| = φ(M)
Each element r is placed on a unit circle at angle
θ(r, M) = 2π · r/M
In the concentric arrangement, ring M sits at radius proportional to M (scaled to fit the canvas). The result: nested circles, each carrying φ(M) dots. As M grows, the dot density per ring trends toward the average coprime density 6/π² ≈ 60.79%.
Global Rotation
A global rotation angle α is applied to every point:
θ_displayed(r, M) = 2πr/M + α + (M−1)·δ
where δ is the per-ring rotation increment (ring rot slider). Default: α = π/2 (90°, entered as 1/4 × 360°). Labels optionally stay fixed at their unrotated positions regardless of α.
- The Lift Condition
A residue r on ring M lifts to ring M+1 when
gcd(r, M+1) = 1
Every coprime residue satisfies gcd(r,M)=1 by definition. The lift condition adds the requirement for the next modulus. Lift lines: green when it lifts, red when blocked.
Chain-n Survival
Require r to lift through n consecutive rings:
gcd(r, M+j) = 1 for j = 1, 2, …, n
The chain slider restricts visible lift lines to residues satisfying all n conditions simultaneously. As n increases, fewer points qualify and the canvas thins.
- Live Counters
Three quantities update in the status bar on every render:
φ / total
Σ φ(M) / Σ (M−1)
→ 6/π² ≈ 0.6079
lift / φ = C(N)
Σ T(M) / Σ φ(M)
→ C ≈ 0.530712
M range
M_min – M_max
ring count, point count
where T(M) = |{r ∈ R(M) : gcd(r, M+1) = 1}| is the count of lifting residues on ring M.
The Lift Survival Constant C
C = lim_{N→∞} Σ_{M=2}^{N} T(M) / Σ_{M=2}^{N} φ(M) = ∏_p (p²−2)/(p²−1) = ζ(2) · ∏_p(1−2/p²) = ζ(2) · d_FT ≈ 0.530711806246…
where d_FT = ∏_p(1−2/p²) ≈ 0.3226 is the Feller–Tornier constant (OEIS A065469). The status bar shows the empirical C(N) for the current M range, converging toward 0.530712 as M_max grows.
Coprime Density
Σ_{M=2}^{N} φ(M) / Σ_{M=2}^{N} (M−1) → 6/π² ≈ 0.607927
This is the density of coprime pairs among all integer pairs — the fraction of the full grid occupied by points on the canvas.
- Color Modes
16 color modes control how every point is colored. Applied per-point at render time based on (r, M, θ).
- Display Overlays
Prime Spiral
For a fixed prime p, the residue r=p appears on every ring M where gcd(p,M)=1 — all M not divisible by p. Its angular position θ=2πp/M sweeps as M grows, tracing a spiral. Three geometric features emerge:
Equator gap
At M=2p: gcd(p,2p)=p≠1. The spiral always skips ring 2p. The gap is visible as a break in the colored path.
Upper path r=(M+1)/2 — always red
gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2. This residue never lifts to M+1. Always shown blocked.Lower path r=(M−1)/2 — alternating
gcd((M−1)/2, M+1) = gcd((M−1)/2, 2) = 1 iff M≡3(mod 4). At a prime q this is the condition for q to be inert in ℤ[i] — the primes not expressible as a sum of two squares.
Lift Lines
Green segment from (M,r) to (M+1,r) when gcd(r,M+1)=1. Red when blocked. Opacity and line width are adjustable. The chain slider restricts to n-consecutive-lift survivors.
N-gon Polygons
Connect the φ(M) coprime points on ring M in angular order — you get the coprime polygon, a geometric representation of (ℤ/Mℤ)×. Three modes:
Mode Vertices Example M=6
Coprime only φ(M) vertices at coprime r Triangle: r=1,5 (+ closure)
Full M-gon All M points Hexagon: all r=0…5
Both Both overlaid Triangle inside hexagon
Gap Chords
For a chosen gap value k, connect residues r and r+k on the same ring when both are coprime. k=2 shows twin-prime pairs geometrically; k=6 shows sexy pairs.
Non-Coprime Points
Points where gcd(r,M)>1 — the zero divisors of ℤ/Mℤ. Colored by their gcd value (hue = gcd×47 mod 360). Hoverable when inspect is on.
- The Inspect System
With Inspect ON, clicking any point opens a panel showing:
Field Value / Formula
r / M Residue and modulus
r/M decimal Fractional position on circle
θ angle 2πr/M in degrees
Farey sector n ⌊M/r⌋ — sector containing r/M
Half r/M > ½ (top) or r/M ≤ ½ (bottom)
Lift to M+1 gcd(r, M+1) = 1 ✓ or ✗
gcd(r,M) Should be 1 for coprime points
gcd(r,M+1) 1 = lifts, >1 = blocked
φ(M) Number of coprime residues on this ring
M prime Whether the modulus is prime
Mirror M−r gcd(M−r, M+1) shown
Appearances How many rings r appears on in [M_min, M_max]
Connect-same-r: when a point is inspected, gold dashed arrows connect all rings where r appears as a coprime residue, with arrowheads at midpoints and M= labels.