Folks, We Have the Best π And some novel ideas for bicycle wheel design. By lcamtuf, Sep 15, 2025 --- Overview This article explores an intriguing mathematical discovery involving different metrics used to measure distance, leading to unexpected variations in π (pi) depending on the space's metric. The focus is on how π changes in various non-Euclidean geometries constructed by adjusting the metric exponent \( n \) in the formula: \[ dn = \sqrt[n]{|x|^n + |y|^n} \] where \( dn \) is the distance, and \( x, y \) are coordinates. --- Key Concepts Topology and Metrics Topologists study shapes based on continuity without regard for exact distances. Metrics define how distance is measured in a space. Euclidean metric (\( n=2 \)) uses the formula: \[ d = \sqrt{x^2 + y^2} \] Taxicab or Manhattan metric (\( n=1 \)) sums absolute values: \[ d\text{taxi} = |x| + |y| \] Chebyshev metric (\( n \to \infty \)) uses the maximum: \[ d\infty = \max(|x|, |y|) \] --- Circles in Different Metrics A circle is defined as all points at distance \( r \) from a center. The shape changes drastically based on \( n \): \( n=1 \) (taxicab): Square rotated by 45°, diamond shape. \( n=2 \) (Euclidean): Perfect circle. \( n \to \infty \) (Chebyshev): Square aligned with axes (max norm). Intermediate \( n \) yield shapes between these. Examples of n-circles for \( n=1.5, 2, 3, 5 \) show gradual morphing shapes from diamond to circle to square. --- The Value of π in These Metrics π is traditionally the ratio of a circle's circumference to its diameter. Non-Euclidean metrics change the measurement of circumference. For taxicab metric, common intuitive guess: \[ \pi1 = \frac{4 \times \sqrt{2}}{2} \approx 2.828 \] This is incorrect because the taxicab metric measures distance differently. Correct calculation uses taxicab distances: \[ \pi1 = \frac{4 \times 2}{2} = 4 \] Chebyshev metric also yields \( \pi\infty = 4 \), from perimeter of a square. Euclidean π (\( \pi2 \)) approximated numerically converges to 3.14159. Other values summarized: | n | π Approximate | |-------|---------------| | 1 | 4 (exact) | | 1.5 | 3.26 | | 2 | 3.14 | | 3 | 3.26 | | 5 | 3.50 | | ∞ | 4 (exact) | A plot shows π varies smoothly between 3.14 and 4 as \( n \) changes. --- Values for \( n < 1 \) Defining distance with exponents less than 1 produces concave "circles". Examples: | n | π Approximate | |------|---------------| | 0.8 | 4.7 | | 0.5 | 7.2 | | 0.3 | 11.9 | At \( n=0 \), the formula breaks down. --- Mathematical Insight Among all such metric spaces parameterized by \( n \), Euclidean π is apparently minimal. This is surprising since it places Euclidean geometry at a special position — a "π-dip". This insight derives from a 2000 paper by Charles Adler and James Tanton, providing formal proofs. --- Summary Different geometries defined by varying the metric exponent \( n \) yield a family of distance measures and corresponding "circles". The constant π varies across these metrics, ranging from 3.14 (Euclidean) to 4 (taxicab and Chebyshev). For \( n<1 \), circles become concave and π can be even larger. The usual π is thus minimal among these metric-derived constants. This offers a fresh perspective on geometry and metric spaces. --- Additional Notes The article also touches lightly on recreational math and encourages exploring deeper mathematical concepts through accessible discussions. Illustrations are provided for metric circles, approximate calculations of π for various \( n \), and related plots. Readers interested in formal proofs are referred to the linked academic paper by Adler and Tanton. --- *If you enjoyed this content, subscribing to lcamtuf's newsletter is suggested for more insightful mathematical