**Taming The Infinite/ Chapter 12/ Impossible Triangles.**

What are impossible triangles? Euclid is considered the father of Geometry, and in most ways he is. I say most ways because there is such a thing as non-Euclidean Geometry, meaning a geometry distinctive from that contained alone within Euclid. This different kind of Geometry exists for a reason, and it helps us explain things that Euclid had no account for. The surge of this non-Euclidean Geometry, the book points out, is that moment that came the struggle of navigation, when people needed tools and instruments that described and articulated properly the elements of a three dimensional space. These three dimensions couldn't be described by mere Euclidean geometry.

From this came trigonometry, and also came the explanation of curvature. Kemeny makes a distinction between Euclidean and non-Euclidian gemetry more accurately, he even gives us a chart.

Another interesting to consider that came from Euclidean impossibilities, and his eery postulate 5, is the notion of the shape of the universe; a truly complex idea. Is it a sphere? Perhaps a flat surface? Or something in between? .. The illustration below represents exactly those three possibilities.