## Notes to The Epistemology of Visual Thinking in Mathematics

1. A geodesic space is a metric space in which for any two points there is a geodesic segment from one to the other. Let $$b$$ and $$c$$ be points in metric space $$(X, d)$$ such that $$d(b, c) = r$$. Then a geodesic segment from $$b$$ to $$c$$ is the image of an isometric embedding $$g$$ of real interval $$[0, r]$$ into $$X$$ with $$g(0) = b$$ and $$g(r) = c$$. The embedding $$g$$ is isometric when $$d(g(x), g(y)) = y - x$$ for all $$0 \le x \le y \le r$$.

2. Figure 13a was created by Wikipedia user Ylebru and released into the public domain; the original can be found at commons.wikimedia.org/wiki/File:Triangolo_iperbolico.svg. Figure 13b was created by Wikipedia user Stomatapol and licensed as Creative Commons Atttribution-Share Alike 3.0; the original can be found at commons.wikimedia.org/wiki/File:Delta_thin_triangle_condition.svg