Exploration is a central component of animal behaviour studied extensively in rodents. Previous tests of free exploration limited vertical movement to rearing and jumping. Here, we attach a wire mesh to the arena wall, allowing vertical exploration. This provides an opportunity to study the morphogenesis of behaviour along the vertical dimension, and examine the context in which it is performed. In the current set-up, the mice first use the doorway as a point reference for establishing a borderline linear path along the circumference of the arena floor, and then use this path as a linear reference for performing horizontal forays towards the centre (incursions) and vertical forays on the wire mesh (ascents). Vertical movement starts with rearing on the wall, and commences with straight vertical ascents that increase in extent and complexity. The mice first reach the top of the wall, then mill about within circumscribed horizontal sections, and then progress horizontally for increasingly longer distances on the upper edge of the wire mesh. Examination of the sequence of borderline segments, incursions and ascents reveals dimensional modularity: an initial series (bout) of borderline segments precedes alternating bouts of incursions and bouts of ascents, thus exhibiting sustained attention to each dimension separately. The exhibited separate growth in extent and in complexity of movement and the sustained attention to each of the three dimensions disclose the mice’s modular perception of this environment and validate all three as natural kinds.