60 shows up in lots of places. It’s the smallest number divisible by 1 through 6, and perhaps because of that, it’s the basis of our timekeeping and angular measurements.
Of course the angles in an equilateral triangle are 60 degrees. But 60 also appears in solid geometry. There are four Archimedean solids (regular polyhedra made with any mixture of regular polygons) with 60 vertices. You can use Nat Alison’s beautiful polyhedra viewer to explore them: