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Why is this a Möbius band?
Take a closed curve γ(t) with parameter 0 ≤ t ≤ 1. An unordered pair of points on
the curve is given by parameters (x, y) with 0 ≤ x ≤ y ≤ 1 — a triangle. On its two legs, the pair
(0, y) describes the same pair as (y, 1), so those edges are glued together with a flip.
A triangle with two edges glued that way is exactly a Möbius band.
The construction
Each pair {A, B} of curve points is mapped to the point in space directly above the
midpoint of A and B, at height equal to the distance |A − B|.
Pairs with A = B land on the curve itself at height 0 — so the boundary of the Möbius band is the
original curve.
To render it, the parameter triangle 0 ≤ x ≤ y ≤ 1 is triangulated on a regular grid and each grid
node (x, y) is mapped to ((γ(x)+γ(y))/2, |γ(x) − γ(y)|). Because γ(0) = γ(1), the glued edges match
up automatically and the mesh closes into the band.
Inscribed rectangles
This is the surface from Vaughan's proof of the inscribed rectangle theorem. A
Möbius band cannot be embedded in the half-space above the plane with its boundary on the plane
without intersecting itself. A self-intersection means two different pairs of curve points
share the same midpoint and the same distance — and four such points always form a
rectangle. Hence every closed curve in the plane has an inscribed rectangle!
Watch for the self-intersections of the surface as it rotates.
This vibe is inspired by 3Blue1Brown's video
“This open
problem taught me what topology is” — watch it for the full story.
Controls
- Drag to rotate, scroll to zoom, right-drag
to pan.
- Detail: number of samples along the curve (mesh resolution).
- Height scale: visually exaggerate or flatten the band (1 = exact distances).