One of the most interesting areas from mathematics is topology. It is quite complex and the theory is sometimes hard to digest, but it is so interesting and it has some of the most beautiful visual representations (when we can actually do them).
Topology (often described as the rubber-sheet or plasticene geometry - where distances have little meaning) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
Given the diversity of forms that a soap film or minimal surface can take on, mathematicians have developed schemes for putting surfaces into different categories, depending on their various features. One classification s based on topological type.
Create your own Möbius strip :) Making one is very easy:
take an A4, or similar, sheet of paper
cut a rectangle,
bring the two long ends together
twist one of the ends through 180°,
tape the two ends together.
Play with the Möbius strip:
A line drawn along the edge travels in a full circle to a point opposite the starting point. If continued, the line returns to the starting point, and is double the length of the original strip: this single continuous curve traverses the entire boundary.
Cutting a Möbius strip along the center line with a pair of scissors yields one long strip with two full twists in it, rather than two separate strips.
If the strip is cut along about a third in from the edge, it produces two linked strips. The center third is a thinner Möbius strip, the same length as the original strip.
Other topological ideas:
Picture frame puzzles
Mathematically correct breakfast
If you want to find out more about the above activtities and find out other great ones, you should check Maxine Elena Calle - Topology Fun and Games.