Gummiwelt: Die Tasse ist ein Donut
About this app
Ein glänzender Knetkörper verformt sich stetig: Donut zur Tasse, Kugel zum Würfel und Teller, Doppeltorus zur Brille, Dreifachtorus zur Brezel. An jedem Henkel hängt ein Ring; Reißen und Kleben zeigen mit roter Naht, wie die Form die Klasse wechselt. Mit Abstimm-Herausforderungen und Sortierspiel auf Klassen-Inseln.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Homeomorphic: equal in rubber-sheet geometry
In topology two shapes count as equal if one can be deformed continuously into the other: stretching, squeezing and bending are allowed, tearing and gluing are not. Such shapes are called homeomorphic. Lengths, angles and corners do not matter, so sphere, cube and plate are the same. For closed surfaces in space only the number of handles counts, the genus: sphere zero, doughnut one, double torus two, pretzel three. “Is homeomorphic to” is an equivalence relation, and its classes form a partition of all shapes.
What the app shows
A glossy clay body deforms smoothly between shapes with the same number of handles: sphere, cube and plate, doughnut and mug, double torus and glasses, triple torus and pretzel. The shapes are built so that during allowed kneading no hole closes and none is added. A glowing ring hangs on every handle, and a counter at the top shows one ring per handle. The hollow of the mug gets no ring, because it is only a dent. Tearing and gluing are shown with a red seam, after which the number of rings changes.
Try it yourself
Drag the body directly or use the slider and knead the doughnut into a mug. Under Forbidden you tear through a handle or glue on a new one and see the shape change its class. Under Challenge the class votes first, for example whether a plate can become a doughnut without tearing. In the sorting game you drag beaker, mug, straw, scissors, glasses, key, trousers, plate and pretzel onto islands with the same number of handles.
Frequently asked questions
- Why is a mug topologically a doughnut?
- Both have exactly one handle. You can shrink the body of the mug continuously into the handle without tearing or gluing; what remains is a ring. The hollow of the mug is only a dent, not a hole that goes through.
- What does homeomorphic mean?
- Two shapes are homeomorphic if there is a continuous, invertible map between them whose inverse is continuous as well. Intuitively: you can deform one rubber shape into the other without tearing and without gluing points together.
- How many holes does a pretzel have?
- A classic pretzel has three holes; topologically it is a triple torus of genus three. Glasses or scissors have two handles, a straw has one, because its hole goes right through once.
- What can I try in the app?
- Knead shapes continuously into each other, count handle rings, watch forbidden tearing and gluing with a red seam, solve challenges with a class vote and sort everyday objects onto class islands.
Subject: Mathe ohne Zahlen
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- Königsberger Brücken
- Möbiusband: eine Fläche, eine Seite
- Ordnungen und Hasse-Diagramme
- Relationen-Labor
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