Introduced by Cole and Davie in 1969, Euclid is a combinatorial game based on the operations of the Euclidean algorithm. Using geometry, I'll prove Cole and Davie's result that describes which player should win under optimal play and how this depends on the Golden Ratio. The play of the game can be described as moving along the branches of the Calkin-Wilf tree. I will review how the Calkin-Wilf tree provides an enumeration of the positive rational numbers. I will explain how Euclid is played and its relationship to the Calkin-Wilf tree. Finally, I will prove that the arrangement of the losing positions in the tree form a fractal. This work is joint with Michael Ivanitskiy and Brittany Shelton.