Research
Tau Pattern Explorer
An interactive mathematical instrument to visualize and analyze the discrete structures and nested cycles of the Tau block pattern.
Tau Pattern Explorer
A dedicated interactive visualization tool for exploring the discrete structural properties of the transcendental constant τ = 2π. The explorer provides computational insights into the function mod(⌊n/τ⌋, 9), revealing deep self-similar patterns, repeating decimal fractions, and prime-generating functions at scale.
Core Capabilities
| Compute | Visualise | Analyze |
|---|---|---|
| Generate structural cycle maps spanning up to 102,240 computational steps. | Interactive exploration of the 113-block nested architecture of τ at different scale-gauges. | Pinpoint prime-generating regions and complex-exponential extensions in the dynamic block pattern. |
The Tau Block Architecture
The function mod(⌊n/τ⌋, 9) produces an invariant hierarchy of nested cycles that the Explorer makes visually accessible:
- Base Cycle: A 710-number cycle containing exactly 113 blocks with lengths drawn from {6, 7}.
- Symmetric Phase Breaks: A strictly repeating sequence
{7, 6, 6, 7, 6, 6, 6}interrupted predictably by structural phase shifts. - Super-Cycles: Complete value cycles preserved precisely at 6390, 25560, and 102240 scale markers.
Features
- Dynamic Navigation: Pan, zoom, and inspect computational blocks across the transcendental spectrum.
- Data Export: Extract structural blocks and numerical patterns for extended computational mathematics research.
- Quarter-Turn Dynamics: Visualize the complex-exponential quarter-turns of
e^(τ/4)underlying the integer progression.