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 Graph Theory Addison-Wesley. Peter Cameron Polynomial aspects of codes, matroids and permutation groups Welsh, DJA 1976 Matroid Theory Academic Press, London. Bollobás, B 1998 Modern
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 but with open questions. References: Wikipedia Braid group Epple, M 1998 Orbits of Asteroids, a Braid, and the First Link Invariant Mathematical Intelligencer, volume 20,
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 simple, but have remarkably surprising and deep mathematical properties, as well as giving beautiful tessellations. References: Austin, D Penrose tiles Penrose, R 1989 Tilings and
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 which angles are preserved but not lengths. References: Feeman, TG 2002 Portraits of the Earth: A Mathematician Looks at Maps American Mathematical Society. McCleary, J
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 group acts on this sphere, giving a fascinating set of transformations, crucially important in many branches of mathematics. A natural development is to consider multi-valued
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 is an important hint that complex geometry is somehow fundamental. One can see more of this in the projective geometry of twistor space, which is
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 sphere, interpreted in twistor theory as a Riemann sphere, or in other words a complex projective line. These complex projective lines lie inside the three