There are some simple and amazing polynomial invariants of graphs, such as the chromatic polynomial, which are special cases of the Tutte-Grothendieck polynomial invariants of matroids. References: Harary, F 1969
There are many intriguing invariants of knots, some only discovered recently. Some of these come from studying the related braids, which lead to extremely interesting groups, relatively simple to define
It was discovered only fairly recently that there is a pair of tiles with which the entire plane can only be tessellated in a non-periodic way. These tiles are very
Cartographers use a wide variety of projections from the sphere to the plane, which have a very rich structure studied in differential geometry. Particularly interesting examples are conformal maps, in
Instead of thinking of complex numbers as lying on a plane, it is much better use stereographic projection to put them on a sphere, called the Riemann sphere. The Möbius
The spacetime described in Einstein’s special theory of relativity is a very beautiful geometry. In particular, there is a deep relationship between the Lorentz group and the Möbius group, which
In twistor theory, the points of Minkowski space-time are picked out by the families of null geodesics through them. In relativity, each such family has the conformal structure of a