Geometric reasoning has an inherent dissonance: its abstract axioms and propositions refer to perfect,
idealized entities, whereas its use in the physical world relies on dynamic perception of objects. How do
abstract Euclidean concepts, dynamics, and statistics come together to support our intuitive geometric
reasoning? Here, we address this question using a simple geometric task – planar triangle completion.
An analysis of the distribution of participants’ errors in localizing a fragmented triangle’s missing corner
reveals scale-dependent deviations from a deterministic Euclidean representation of planar triangles.
By considering the statistical physics of the process characterized via a correlated random walk with
a natural length scale, we explain these results and further predict participants’ estimates of the
missing angle, measured in a second task. Our model also predicts the results of a categorical reasoning
task about changes in the triangle size and shape even when such completion strategies need not be
invoked. Taken together, our fndings suggest a critical role for noisy physical processes in our reasoning
about elementary Euclidean geometry.