This research examines how TikTok users conceptualize and engage with personalized algorithms on the TikTok platform. Using qualitative methods, we analyzed 24 interviews with TikTok users to explore how algorithmic personalization processes inform people's understanding of their identities as well as shape their orientation to others. Building on insights from our qualitative data and previous scholarship on algorithms and identity, we propose a novel conceptual model to understand how people think about and interact with personalized algorithmic systems. Drawing on the metaphor of crystals and their properties, the algorithmic crystal framework is an analytic frame that captures user understandings of how personalized algorithms (1) interact with user identity by reflecting user self-concepts that are both multifaceted and dynamic and (2) shape perspectives on others encountered through the algorithm, by orienting users to recognize parts of themselves refracted in other users and to experience ephemeral, diffracted connections with groups of similar others. We describe how the algorithmic crystal framework can extend theory and inform new lines of research around the implications of algorithms in self-concept development and social life.