A strip with a single half-twist has only one side and one edge. Send a bug crawling along it and it returns mirror-flipped — standing where it began, but upside-down. Cut it down the middle and you don't get two loops; you get one longer, double-twisted loop. Drag the twist slider, run the crawler, and run the scissors to feel why "which side" can be a meaningless question.
Building the strip. Take the centre circle and, at each angle u around it, attach a short line segment of width 2h. As you walk once around, rotate that segment about the centre-line by a total angle of m·π, where m is the number of half-twists. With m=0 you get a plain cylindrical band; with m=1, the Möbius strip.
Carrying a frame. Put a tiny arrow (a surface normal) on the bug and slide it around the centre-line. The segment it rides has rotated by m·π by the time the bug is home. If m is even, that's a whole number of half-turns — the arrow comes back the way it left. If m is odd, it comes back reversed. A surface on which some loop reverses your frame is exactly what non-orientable means.
One side, one edge. "Two sides" would require painting the front without ever reaching the back. On the Möbius strip the flipped-frame loop connects front to back, so a single stroke of paint covers everything — one side. Trace the boundary and the same thing happens to the rim: it is one loop of double length.
The scissors trick. Cutting a band down its centre-line removes the centre circle and leaves a band of half the width whose edges are the two old halves. For a Möbius strip those two halves were already joined into one boundary loop — so the cut yields one connected band, twice as long, and now carrying m = 4 half-twists (two full twists), which makes it orientable.
For an even band the two halves are separate boundaries, so cutting gives two bands of the original length, linked around each other m/2 times. The applet computes and animates each of these outcomes from the twist you set — nothing is scripted per-case beyond the true topology.
Where it goes. Glue the Möbius strip's single edge to itself and you can't do it in 3D without self-intersection — the result is the Klein bottle. Two Möbius strips glued along their edges make the real projective plane. Orientability is the hinge on which the whole classification of surfaces turns.
Read on.
• Möbius & Listing (1858, independently) — the original discovery of the one-sided surface.
• Armstrong, Basic Topology — surfaces, orientability, and the classification theorem in a gentle, picture-first treatment.
• Munkres, Topology — orientation via the fundamental group and covering spaces; the orientation double cover of \(M\) is the cylinder.
• Francis, A Topological Picturebook — how to draw the cut, the Klein bottle, and \(\mathbb{RP}^2\).
• Classic experiment: cut a paper Möbius strip along the line one-third of the way from an edge — you get two interlocked loops of different lengths. Try predicting it from the same rule this applet uses.
Sibling in this repo: examples/knot.html — the same Three.js + bloom pipeline, one dimension of knotting up.