Möbiusband: eine Fläche, eine Seite
About this app
Ein Papierstreifen klebt sich mit null bis drei Halbdrehungen zum Band. Eine Ameise malt: Beim Ring bleibt eine Seite frei, beim Möbiusband ist am Ende alles bemalt, und der Rand ist eine einzige Kurve. Die Schere schneidet in der Mitte oder beim Drittel, vorher stimmt die Klasse ab.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
What is a Möbius strip?
If you glue a paper strip into a ring, it has two sides and two edges. If you first give one end a half twist, you get the Möbius strip. An ant walking on it without climbing over the edge arrives on the back after one lap. So the strip has only one side, it is non-orientable, and its edge is a single curve as well. In general: with an even number of half twists there are two sides and two edges, with an odd number one side and one edge. Bending and twisting do not change this, so these are properties of topology.
What the app shows
A paper strip twists and bends into a band in 3D and is glued, with zero to three half twists. An ant with a paint roller walks two laps: on the ring one side stays blank, on the Möbius strip everything is painted in the end. Each edge curve lights up in its own colour. Scissors cut lengthwise down the middle or at one third of the width, then the pieces pull apart. Cut down the middle, the Möbius strip gives one single strip, twice as long, with two full twists; cut at one third, it gives a Möbius strip and a long strip, linked together.
Try it yourself
Choose the number of half twists and start the ant, or drag it over the band yourself. Before every cut the app pauses and asks “What happens next?”: the class votes by show of hands, then it is revealed. Try two half twists (two linked strips) and three half twists (one long, knotted strip). Under Practice short tasks to call out are waiting. The real experiment with paper strips, glue and scissors is worth doing alongside.
Frequently asked questions
- Why does the Möbius strip have only one side?
- The half twist joins the front of the strip to its back when gluing. Walking along the centre line, you arrive at the same spot on the other side of the paper after one lap without crossing the edge. After two laps the whole surface has been reached, so there is only one side.
- What happens if you cut a Möbius strip down the middle?
- It does not fall into two pieces. You get one single strip, twice as long, with four half twists, that is two full twists. This strip has two sides and two edges. The reason: the centre line closes after just one lap, and there is only a single edge next to it.
- What do you get when cutting at one third of the width?
- The scissors go round twice before they are back at the start. You get two strips that are linked together: in the middle a Möbius strip as long as the original, and around it a strip twice as long with two full twists.
- What can I try in the app?
- Choose zero to three half twists and watch the gluing, let the ant paint or drag it yourself, light up the edge, colour front and back, cut down the middle or at one third, pull the pieces apart, vote first and solve tasks to call out.
Subject: Mathe ohne Zahlen
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