Möbius Strip Unrolled: Fundamental-Polygon & Twist Diagram (2D)

This 2D companion shows the Möbius strip the way topology textbooks actually define it — as a flat rectangle with one pair of edges glued together — instead of rendering the twisted surface in 3D space. An ant walks a chosen offset line while the rectangle's right edge wraps back to its left edge under the identification (2π,s)~(0,(-1)^k·s); a rotating twist dial tracks the same θ(t)=k·t/2 cross-section angle the 3D engine uses internally, and a scrolling face strip shows the ant's side flipping every lap when k is odd and never flipping when k is even, making the one-sided, one-edge topology visible from the flat diagram alone.