Topology is the mathematical study of 3-D surfaces, shapes and spaces; specifically it is the branch of geometry that deals with distorted shapes. Topologists look at the properties of shapes that are preserved even when deformed by stretching, twisting, and bending. The way they see it, if an object can be distorted without being torn, cut, or glued, it maintains the same topology even though it may look completely different. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped into a coffee mug (see animation). Topology also deals with visual conundrums such as knot theory (endless loops) and non-orientable surfaces (on which you can move from outside to inside without crossing an edge). Examples include: Mobius Strip, Trefoil Knot, and Klein Bottle. Learn more about topology at the following links:
http://www.mychildmalaysia.com/topic/777/Topology+for+Kids – Topology for Kids
http://2000clicks.com/MathHelp/BasicSetTopologyKidsintro.aspx – Kids Intro to Topology
http://www.dadcando.com/default_DOING.asp?project=MobiusStrip&catagory=Experiments&lhs=Experiments – Mind Bending Adventures in Topology with a simple-to-make Mobius Strip.
http://toomai.wordpress.com/2008/05/25/more-experimental-topology-and-experiments-in-topology/ – Math With My Kids: More Experimental Topology and Experiments in Topology.
http://britton.disted.camosun.bc.ca/jbrubbergeom.htm – Rubber Geometry
http://www.math.wayne.edu/~rrb/topology.html – What is Topology?
http://www.math.osu.edu/~fiedorowicz.1/math655/ – Math 655 is an introduction to the history of topology and the basic concepts of modern topology.
Finally, for an interesting take on some topological topics, check out “Math That Makes You Go Wow,” an innovative multi-disciplinary project by a group of Yale undergraduates: http://www.carliner-remes.com/jacob/math/project/