τ (Tau) — Two-Digit Pair Analysis

Exploring the decimal expansion of τ = 2π (the true circle constant) through two-digit pairs (00–99)
using the first 100,000 digits. Visual proof that τ also behaves like a normal number.

Stacked Horse Race — How Each 10k-Digit Chunk Contributes

Final Analysis at 100,000 Digits

10×10 Transition Heatmap
(From Digit → To Digit)

Color intensity = occurrence count
Dark blue = lower • Bright red = higher

Statistical Summary

Chi-square (uniformity)
98.72
(99 df)
p ≈ 0.52 — excellent uniformity
Serial Correlation
−0.00187
Extremely close to zero → no detectable correlation
Total Pairs Analyzed
99,999
Expected per Pair
999.99
Minimum Count
931
Maximum Count
1,092

Just like π, τ shows essentially no correlation between consecutive digits and excellent uniformity.

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This page is part of an exploration of the deep mathematics hidden in a simple circle.

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