What is Geometry?

Geometry: The Rhetoric of Mathematics

“Let no one ignorant of geometry enter,” Plato once inscribed above his academy, hinting that geometry is more than a mere field of study—it’s a prerequisite for higher thought. Like rhetoric, geometry structures our understanding and shapes how we interpret the world.

But what exactly is geometry? It’s nearly impossible to define without it dissolving into algebra, topology, or analysis. When we talk about metrics, we often mean analysis, while the “geometry” of graphs is actually combinatorics, using terms like diameter or bridges for rhetorical flair. Even the concept of shape often implies a topological, rather than geometric, essence.

Geometry, in its truest form, evokes a sense of contextualization and intertextuality. Terms like distance, curvature, or surface suggest an act of looking at or envisaging something, rather than purely calculating. Geometry frames problems, giving them form and clarity. It brings abstract concepts into visual existence and offers a way to think about their relations and transformations. When we describe objects as having a certain “shape” or when we speak of spaces being “curved,” we’re not just borrowing terms—we’re invoking the imagery and context that give meaning to these abstractions.

Geometry is also both physical and computational. When our eyes are open, we see geometry everywhere; when closed, we imagine it still. Geometry’s transcendence lies in its ability to provide a vantage point—an eternal recurrence, as Nietzsche might put it—a lens through which reality is not just observed but deeply understood.