UNIT BALL: |x|^1.00 + |y|^1.00 = 1
Drag orange probe anywhere
Presets:
Intrinsic Half-Perimeter (π_p)
4.0000
Measured using L^p arc length
Euclidean Half-Perimeter
2.8284
2√2 ≈ 2.8284 (Grant's first tweet)
Draggable Probe (x, y)
(0.50, 0.50)
||(x,y)||_p = 1.000
Ball Containment
On Boundary
||v||_p ≤ 1 check
The Pi Day Tweet Mystery: Grant Sanderson noted in his tweet that for taxicab geometry (p=1), "half the perimeter will give other values... at p=1, 'π' = 2√2". That is the Euclidean arc length of the diamond! But as he corrected: in the taxicab metric itself, each side has length 2, so the whole perimeter is 8 and half-perimeter π_1 is 4.
π_p AS A FUNCTION OF NORM p (Gołąb's Bounds [3, 4])
Min: p ≈ 2.59
Gołąb’s Theorem: In any 2D normed plane (Minkowski space), the metric perimeter of its unit ball is always between 6 and 8 (so half-perimeter π_p satisfies 3 ≤ π_p ≤ 4). The Euclidean circle achieves π_2 = 3.14159..., while the minimum occurs at p ≈ 2.59 where π_p ≈ 3.129!