Recreational mathematician. Grew up a math nerd and consider the subject a major part of my identity. Fascinated by just about everything. Here's hoping this blog will open me up to some more creative outlets. That might be good for me at this juncture in my life :)
The sums of the reciprocals of the binomial coefficients over successive diagonals in Pascal’s triangle converge into beautiful patterns, apart from the first and second diagonal (which lead to the series 1 + 1 + 1 + 1 + … and the harmonic series, respectively).
A cool thing that you can construct from a plane graph is called its dual. If the graph’s Vertices, Edges, and Regions are represented G=(V,E,R), then the dual D=(R,E,V), where each vertex in G is a region in D.
They have the same set of edges. Basically, an edge separates two regions just as much as it connects two vertices. (Also, the dual needs to include self-loops for edges that are not part of a cycle, and multiple allowed edges between a pair of vertices to work for vertices of degree 2, but this is a good enough description to start out with)
Also, the dual of the dual is the same as the original graph.