Gather an orange and a kitchen knife. How can you cut the orange peel into four triangular pieces? (We’ll allow the sides of these triangles to be curved to make the cutting easier.) It turns out that you can do this by making six cuts and that there are many different possible cuttings that produce four triangular pieces. In order to better understand these different cuttings, I created this map. Each black dot represents a different division of the sphere (i.e. an orange) into triangles, like a dot representing a city on a map. The red lines connect divisions of the sphere which have most of the cuts in common, like highways connecting nearby cities. Visualizing the information in this way allowed me to prove a new property about divisions of spheres into triangles and produced this intriguing patterned image!