Findings synthesized across five empirical laboratory- and classroom-based studies of high school and college students engaged in geometric reasoning and proof production during single- and multi-session investigations (346 participants overall) are presented. The findings converge on several design principles for computer technologies to support embodied mathematical thinking and learning: (a) the importance of integrating linguistic and body-based processes, especially via "dynamic gestures," which enable simulated action of mathematical transformations for hypothesizing, testing, and generalizing universal truths about space and shape; (b) the "cognitive relevance" of directed actions to the concepts under consideration so that action-cognition transduction elicits the bodily interactions needed to influence mathematical reasoning in the intended manner; (c) "collaborative contexts" that foster common ground and shared meaning construction, production of socially mediated gestures, and the emergence of extended and distributed knowledge and representations; (d) conditions that encourage "transfer" of sensorimotor experiences across contexts in order to predict and make sense of the world; and (e) the need to assess learners' knowledge and learning that is expressed nonverbally. "The Hidden Village" is presented as an instance of an embodied learning environment that uses the narrative context of a visual novel to instantiate these design principles and investigate their influences on mathematical thinking and learning. These design principles are placed in terms of larger efforts to articulate a broader theoretical framework of grounded and embodied learning. Together, the interplay of theoretical, empirical and design considerations of grounded and embodied learning reframes learning, transfer, and assessment, offering promising new pathways for an emerging class of learning environments that may propel educational practice and learning theory. [This paper was published in: "Proceedings of the 29th International Conference on Computers in Education," edited by M. M. T. Rodrigo et al., Asia-Pacific Society for Computers in Education, 2021, pp. 348-58.]