Interesting questions (with answers) about concepts in topology for an amateur audience

I always liked the: hang a picture on two nails, such that if you remove one the picture falls down .

While the solution to this can be found with a bit luck and without the knowledge of fundamental groups the more complicated ones (hanging it on $n$ nails) is probably impossible without any mathematical advanced ideas.


Here is a list of suggestions:

  1. Take a disk of paper. Crumpled it and place the crumple paper over the place of the initial disk. One point of the crumpled paper will be at the vertical of its initial position in the disk. This is the fixed point theorem.

  2. Create a Möbius strip. An example of a surface with only one side.

  3. Describe the construction of the Peano curve. A curve filling a square.

  4. How to paint an infinite surface with a finite amount of painting. The painter paradox.

  5. The Schwarz lantern. Or how to approximate a surface of finite area with triangles whose areas are infinite.

You may find other examples on topology in my mathcounterexamples.net web site.


A nice (and classic) problem would be the Bridges of Königsberg (https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg).

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