The object-responsive cortex of the visual system has a highly systematic topography, with a macro-scale organization related to animacy and the real-world size of objects, and embedded meso-scale regions with strong selectivity for a handful of object categories. Here, we use self-organizing principles to learn a topographic representation of the data manifold of a deep neural network representational space. We find that a smooth mapping of this representational space showed many brain-like motifs, with (i) large-scale organization of animate vs. inanimate and big vs. small response preferences, supported by (ii) feature tuning related to textural and coarse form information, with (iii) naturally emerging face- and scene-selective regions embedded in this larger-scale organization. While some theories of the object-selective cortex posit that these differently tuned regions of the brain reflect a collection of distinctly specified functional modules, the present work provides computational support for an alternate hypothesis that the tuning and topography of the object-selective cortex reflects a smooth mapping of a unified representational space.