Boundary (x, y)
0.500, 0.500
Tangent Vector
-1.000, 1.000
Infinitesimal ds_p
2.000 × dx
Euclidean ds_2
1.414 × dx
The Pi Day Correction: In the Taxicab metric (p=1), the “unit circle” is the diamond |x| + |y| = 1. Its Euclidean perimeter is 4√2 ≈ 5.6569 (half-perimeter = 2√2). But measured intrinsically in the L¹ metric, each side has length |Δx| + |Δy| = 1 + 1 = 2, making total circumference = 8 and intrinsic π¹ = 8 / (2 × 1) = 4.0000!
Intrinsic L^p Metric
4.0000
Circumference: 8.0000
Formula: ∮ (|dx|p + |dy|p)1/p
Diameter: 2.0000
Formula: ∮ (|dx|p + |dy|p)1/p
Diameter: 2.0000
Euclidean Measurement
2.8284
Perimeter: 5.6569
Formula: ∮ √(dx² + dy²)
At p=1: 4√2 ≈ 5.6569
Formula: ∮ √(dx² + dy²)
At p=1: 4√2 ≈ 5.6569
| Quadrant | Arc Range | Intrinsic L^p Length | Euclidean Length |
|---|---|---|---|
| Q1 (+, +) | (1,0) → (0,1) | 2.0000 | 1.4142 |
| Q2 (-, +) | (0,1) → (-1,0) | 2.0000 | 1.4142 |
| Q3 (-, -) | (-1,0) → (0,-1) | 2.0000 | 1.4142 |
| Q4 (+, -) | (0,-1) → (1,0) | 2.0000 | 1.4142 |