  This paper investigates the geometrical properties of real world games (e.g.
Tic-Tac-Toe, Go, StarCraft II). We hypothesise that their geometrical structure
resemble a spinning top, with the upright axis representing transitive
strength, and the radial axis, which corresponds to the number of cycles that
exist at a particular transitive strength, representing the non-transitive
dimension. We prove the existence of this geometry for a wide class of real
world games, exposing their temporal nature. Additionally, we show that this
unique structure also has consequences for learning - it clarifies why
populations of strategies are necessary for training of agents, and how
population size relates to the structure of the game. Finally, we empirically
validate these claims by using a selection of nine real world two-player
zero-sum symmetric games, showing 1) the spinning top structure is revealed and
can be easily re-constructed by using a new method of Nash clustering to
measure the interaction between transitive and cyclical strategy behaviour, and
2) the effect that population size has on the convergence in these games.
