The background of this site is a random graph, rebuilt every time you load the page or resize the window: nodes are scattered around the hero text, and each one connects to a few of its nearest neighbors. The ▲ = n in the corner isn’t decoration, it’s the number of triangles in whatever graph you happened to get.
Counting them is the classic warm-up: for each vertex \(i\), look at pairs of its neighbors and check whether they’re adjacent. With adjacency sets that’s \(O\!\left(\sum_i d_i^2\right)\), which is fine at this size. The linear-algebra version is prettier: if \(A\) is the adjacency matrix, then
\[t = \frac{\operatorname{tr}(A^3)}{6}\]since each closed walk of length 3 traverses a triangle, and each triangle is counted once per vertex and once per direction.
I’m planning to use this space to write about things I’m reading and thinking about, mostly algorithms and complexity, occasionally not. If a post exists, it’s because I wanted to work something out by explaining it.