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The physical spaces within which organisms live affect their biology and in many cases can be considered part of their extended phenotype. The nests of social insect societies have a fundamental impact on their ability to function as complex superorganisms. Ants in many species excavate elaborate subterranean nests, but others inhabit relatively small pre-formed cavities within rock crevices and hollow seeds. Temnothorax ants, which often nest within acorns, have become a model system for studying collective decision making. While these ants have demonstrated remarkable degrees of rationality and consistent precision with regard to their nest choices, never before has the fine scale internal architecture and spatial organization of their nests been investigated. We used X-ray microtomography to record high-resolution three-dimensional (3D) scans of Temnothorax colonies within their acorns. These data were then quantified using image segmentation and surfacebased 3D visibility graph analysis, a new computational methodology for analysing spatial structures. The visibility graph analysis method integrates knowledge from the field of architecture with the empirical study of animalbuilt structures, thus providing the first methodological cross-disciplinary synergy of these two research areas. We found a surprisingly high surface area and degree of spatial heterogeneity within the acorn nests. Specific regions, such as those associated with the locations of queens and brood, were significantly more conducive to connectivity than others. From an architect's point of view, spatial analysis research has never focused on all-surface 3D movement, as we describe within ant nests. Therefore, we believe our approach will provide new methods for understanding both human design and the comparative biology of habitat spaces. [ABSTRACT FROM AUTHOR]