Topologie der Buchstaben: Welche sind gleich?
About this app
Großbuchstaben als dicke Drähte: Homöomorphe Buchstaben biegen sich weich ineinander (A→R, L→I), Enden, Verzweigungen und Löcher bleiben als Marker sichtbar. Vergleiche zeigen, warum O und Q verschieden sind; im Sortierspiel und mit dem eigenen Namen zerfällt das Alphabet in acht Klassen-Inseln.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
When are two letters homeomorphic?
If capital letters are seen as thin wires, topology only asks about their shape. Two letters are homeomorphic if one can be deformed continuously into the other: bending, stretching and squeezing are allowed, tearing and gluing are not. Three things stay the same: the number of holes, the number of ends and the branch points where three or more strokes meet. Corners do not count, an L can be straightened into an I. A and R both have one hole, two ends and two branch points and are homeomorphic. O and Q look alike, but Q has an end and a branch point.
What the app shows
Every letter is a thick wire with markers: rings at the ends, dots at the branch points, a filled area in each hole. Below it the profile appears as a row of symbols, without any numbers. When you tap a second letter, a card first asks “What happens next?” so that the class can vote. If both belong to the same class, the wire bends smoothly into the new letter and the markers move along. Otherwise both stand side by side with an inequality sign, and the differing markers blink. You can pull the wire like rubber.
Try it yourself
In Sort mode you drag all 26 letters onto eight soft islands whose emblem shows the basic form of the class, for example ring, lasso, tripod or figure eight. Check marks misplaced letters, Show solution lets every letter fly to its island. This makes visible that “is homeomorphic to” is an equivalence relation that splits the alphabet into classes, a partition. In Your name mode a typed word breaks apart into its classes. The challenge poses tasks to call out: find a partner, spot the odd one out among four letters, find the letter without a partner.
Frequently asked questions
- Which letters of the alphabet are homeomorphic?
- In a sans-serif typeface there are eight classes: C G I J L M N S U V W Z are a line, E F T Y a tripod, X a cross, H and K a double fork, D and O a ring, P and Q a ring with one tail, A and R a ring with two tails and B a figure eight. In other typefaces the classes can change, for example for K or Q.
- Why are O and Q not homeomorphic?
- Q has an end and a point where three strokes meet. O has neither. An end cannot appear by bending; you would have to tear the ring open. That is why O cannot be deformed into Q, although both have one hole.
- Why is homeomorphic an equivalence relation?
- Every figure is homeomorphic to itself, you deform nothing. If A can be bent into R, then R can be bent back into A. And whoever can bend A into R and R into a third letter can do both one after the other. Reflexive, symmetric and transitive give an equivalence relation whose classes form a partition.
- What can I try in the app?
- Bend letters into each other and vote first, compare two letters with their profiles, pull the wire, sort the whole alphabet onto class islands, let your own name break apart into classes and solve tasks to call out.
Subject: Mathe ohne Zahlen
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