This picture illustrates a number of the mathematical things I like to
think about:
- Discrete math / combinatorics: the dots
are definitely discrete
- Algorithms: some mysterious process is
going on in this series of pictures
- Polygons / polyhedra / polytopes: well,
there's
a triangle here (I like more complicated an higher dimensional ones
too, but they're harder to draw and explain, and there's lots of
interesting things going on even with triangles)
- Lattice points: namely, points in space
whose coordinates are all integers; all of the dots here are integer
points
- Lattice points in polytopes: combining
the last two things; that's exactly what the first and last pictures
are showing
- Generating functions: I didn't write them
down
here (I just wanted to show the pictures), but generating functions are
actually why these pictures exist in the first place.
Still
curious what's going on here? These pictures are from
a talk
I've given to undergraduates about these things. You can flip through
that talk to learn about these pictures and what exactly they have to
do with generating functions.
