What cognitive systems underlie geometric competence in human children and nonhuman animals? Could those fundamental systems of geometric representation serve as a basis for human Euclidean understanding of space? The use of geometric properties of both large-scale environments and small-scale objects has been found across many distantly related species of animals and from very early stages of development. I will discuss the nature of these two systems of core spatial geometry and their potential role in the development of abstract Euclidean concepts. First, I will present evidence that the geometric analysis of the navigable environment is an evolutionarily ancient process that specifically computes relative positions and directions with respect to the 3D terrain. I will then present studies that suggest that the geometric analysis of the environment is independent from that of manipulable objects and that each system is limited in its geometric content. Finally, I will speculate that universal human Euclidean intuitions are neither innate nor acquired through general learning processes, but that they may be constructed from the foundations of core systems of geometry that we share with other animals.