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Structurally complex habitats tend to contain more species and higher total abundances than simple habitats. This ecological paradigm is grounded in first principles: species richness scales with area, and surface area and niche density increase with three-dimensional complexity. Here we present a geometric basis for surface habitats that unifies ecosystems and spatial scales.The theory is framed by fundamental geometric constraints among three structure descriptors-surface height, rugosity and fractal dimension-and explains 98% of surface variation in a structurally complex test system: coral reefs. We then show how coral biodiversity metrics (species richness, total abundance and probability of interspecific encounter) vary over the theoretical structure descriptor plane, demonstrating the value of the theory for predicting the consequences of natural and human modifications of surface structure. Main textMost habitats on the planet are surface habitats-from the abyssal trenches to the tops of mountains, from coral reefs to the tundra. These habitats exhibit a broad range of structural complexities, from relatively simple, planar surfaces to highly complex three-dimensional structures. Currently, human and natural disturbances are changing the complexity of habitats faster than at any time in history [1][2][3][4] . Therefore, understanding and predicting the effects of habitat complexity changes on biodiversity is of paramount importance 5 . However, empirical relationships between commonly-used descriptors of structural complexity and biodiversity are .