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Description: Philosophers throughout history have debated the relations between the abstract geometry of formal mathematics and the physical geometry of the natural world. We provide evidence that abstract geometry reflects the geometry humans and non-human animals use for spatial navigation. Across two preregistered experiments, educated adults watched short videos of two points and two line segments forming an open figure on an otherwise blank screen. These simple visuals were described with sparse and minimally different language, creating different spatial contexts. After watching each video, participants were asked to click: anywhere (anywhere condition); to complete the triangle (triangle condition); where the next corner of an object would be (object condition); where the next stop on an agent’s path would be (navigation condition); or where the next point on an abstract plane would be (abstract condition). Participants produced strikingly different sets of geometric representations across spatial contexts; in particular, preserving distance and directional for open paths in the navigation condition but preserving length and angle for closed shapes in the object condition. In the navigation and abstract contexts, however, the elicited geometry was remarkably similar. Human language may thus effectively isolate phylogenetically ancient geometric representations used for navigating the physical world and recognizing the objects in it. Moreover, the cognitive origins of uniquely human abstract geometry may lie in representations used for navigating the physical world.