Figure 3.

Three triangles illustrating nonpositively curved spaces. The center illustration represents three points (trees) in treespace, a, b, and c, with the geodesics running between them. Notice the paths are made of sequences of linear segments that sit in the Euclidean cubes of the cube complex, but together the geodesic path has an overall negative curvature (the triangles are thin compared to the Euclidean comparison triangle on the right). The left triangle depicts the extreme situation in which the space is so negatively curved as to be a tree.