📐 Generalized Metric Geometry

L^p Norm Pi Explorer

Investigate how “π” varies when defined as half the perimeter of the unit ball |x|^p + |y|^p = 1 in ℝ². Grounded in Grant Sanderson’s (@3blue1brown) Pi Day puzzle and taxicab metric correction.

Unit Circle Contour & Metric Caliper Drag Caliper Nodes (P₁, P₂)
Axes: [-1.4, 1.4] Caliper: Euclidean dist = 1.414, Intrinsic dist = 2.000
Norm Exponent (p) p = 1.00
Source Presets
Euclidean π_E(p)
2.8284
Half-perimeter via Euclidean ruler ∫√(dx²+dy²)
Intrinsic π_p(p)
4.0000
Half-perimeter measured in L^p metric itself
3blue1brown Pi Day Insight In his original Pi Day tweet, Grant noted that for taxicab geometry (p=1), “π” = 2√2 ≈ 2.8284 (half-perimeter of diamond using standard Euclidean distance). He soon posted an erratum: within the taxicab metric itself, each side is 2 units long, making the intrinsic “π₁” exactly 4.0!
Export:
Spectrum of “π” across 0.5 ≤ p ≤ 10 Live cursor marks current p
Euclidean Half-Perimeter π_E(p) [Min: 2√2 ≈ 2.8284 at p=1, Max: 4 at p=∞]
Intrinsic Metric Half-Perimeter π_p(p) [4 at p=1, π ≈ 3.14159 at p=2, 4 at p=∞]

The Pi Day Taxicab Puzzle

On March 14, Grant Sanderson tweeted: “Happy Pi Day! In a certain sense, π is not a constant, but a variable... applying other L^p norms on ℝ², half the unit circle's perimeter will give other values. For instance, at p=1 (taxicab geometry), “π” = 2√2.”

Shortly after, he followed up: “Ah! Correction, as many have pointed out, it should be 4 in the taxicab metric.”

Why the discrepancy? The unit circle for p=1 is the square diamond connecting (1,0), (0,1), (-1,0), and (0,-1). Each edge has Euclidean length √(1² + 1²) = √2. Four edges equal 4√2, so half-perimeter is 2√2 ≈ 2.8284.

Measuring With the Metric Itself

When geometry is defined by the taxicab norm ‖(x,y)‖₁ = |x| + |y|, distance is not measured by diagonal Euclidean rulers. Stepping along a segment from (1,0) to (0,1) incurs |Δx| + |Δy| = 1 + 1 = 2 units of distance!

Thus, each of the four sides of the diamond has intrinsic length 2, giving a total perimeter of 8. Therefore, the intrinsic half-perimeter is exactly 8 / 2 = 4!

In general Minkowski spaces (normed linear spaces), the Busemann and Schäffer definitions of metric circumference show that for any symmetric convex unit ball in ℝ², the intrinsic perimeter is always between 6 and 8, meaning intrinsic “π” lies in [3, 4].