Modern geometry deals with shapes and spaces of arbitrarily high dimensions. While it is hard to physically construct or directly depict these objects, the language of mathematics lets us describe and study them. There are many different kinds of algebraic varieties. My favorites are Calabi–Yau varieties in the broad sense—K3 surfaces, higher-dimensional holomorphic symplectic manifolds, and Fano varieties of K3 type. These varieties possess rich symmetries and have been studied from different viewpoints—through sporadic finite simple groups such as the Mathieu and Conway groups, through dynamical systems via Salem numbers, and through various techniques in geometric group theory. My principal tool is the derived category of coherent sheaves. This categorical viewpoint reveals hidden symmetries and dualities. The Fourier–Mukai transform, for example, relates different varieties and clarifies the symmetries of an individual variety. Mirror symmetry, originally discovered in superstring theory as a duality between algebraic and symplectic geometry, can likewise be phrased in this categorical language. I hope to sense how much lies within a mathematical structure.