Visualizations

Module 1

Bijections Φ_m and Ψ_m

The map Φ_m sends {1,…,2^m} bijectively onto {0,1}^m via n ↦ (c₁,…,c_m). The map Ψ_m sends {1,…,6·2^m} bijectively onto the paths of length m in the graph G mod 6. Click any row to see the trajectory and the formula for T^m.

c_i = Tⁱ⁻¹(n) mod 2 — parity of the i-th iterate (0 = even, 1 = odd)

h_i = Tⁱ⁻¹(n) mod 6 — residue mod 6 of the i-th iterate

s = ∑ c_i — Hamming weight (number of odd iterates)

T(n) = n/2 if even · (3n+1)/2 if odd — accelerated Collatz map

Binary antisymmetry: n and n+2^m share c₁…c_m; bit c_m+1 is always the complement (0↔1)

Mod 6 antisymmetry: n and n+3·2^m share h₁…h_m; residue h_m+1 is always the opposite (r↔r+3 mod 6)

m = Antisymmetry +1 period

Module 2

Corrective residue R(C^m) and T^m(n)

Explore the formula T^m(n) = (3^s·n + R(C^m)) / 2^m: three methods to compute the corrective residue (recursive, explicit, matrix), the chain concatenation law and the power formula. Supports n ∈ ℤ.

Fundamental identity

T^m(n)= 3^s·n + R(C^m(n))2^m s = Σ c_i

R(C^m)= 2^m·T^m(n) − 3^s·n

c_i = Tⁱ⁻¹(n) mod 2 · r(i) = c₁+···+c_i · s = r(m) · n ∈ ℤ

Input

n = m =

Computing R

Application: T^m(n)

Concatenation law

Click a · point between c_i positions to add or remove a cut.

Chain powers

R(B^N) = R(B) · 2^pN − 3^r₀N 2^p − 3^r₀

B = base chain of length p B^N = concatenation of B with itself N times p = length(B), r₀ = weight(B) = Σ c_i

Repeat current chain N = times

Module 3

Graph G mod 6

Preamble — What does this graph show?

T: accelerated Collatz map — T(n) = n/2 (even), (3n+1)/2 (odd), T(0)=0. The trajectory is T⁰(n), T¹(n), T²(n), …

Graph G mod 6: 6 nodes = residues 0–5, 12 edges. Write n = 6k + r. Each node r has two outgoing edges: ●green (k even) and ●amber (k odd).

Periodicity: all n with the same residue mod 6 and the same parity sequence follow exactly the same path in G for m iterations — i.e., members of a congruence class modulo 6·2^m share their trajectory for m steps.

SCCs: V₁={0} fixed point · V₂={3} transient · V₃={1,2,4,5} closed attractor.

G' = G[V₃]: induced subgraph on the attractor. Stationary distribution: π(1)=π(4)=1/6, π(2)=π(5)=1/3.

n = m = 0/14 speed

Bijections Φm and Ψm

Corrective residue R(Cm) and Tm(n)

Graph G mod 6

Bijections Φ_m and Ψ_m

Corrective residue R(C^m) and T^m(n)