Pattern formation is a ubiquitous phenomenon that describes the generation of orderly outcomes by self-organization. In both physical society and online social media, patterns formed by social interactions are mainly driven by information flow. Despite an increasing number of studies aiming to understand the spreads of information flow, little is known about the geometry of these spreading patterns and how they were formed during the spreading. In this paper, by exploring 432 million information flow patterns extracted from a large-scale online social media dataset, we uncover a wide range of complex geometric patterns characterized by a three-dimensional metric space. In contrast, the existing understanding of spreading patterns are limited to fanning-out or narrow tree-like geometries. We discover three key ingredients that govern the formation of complex geometric patterns of information flow. As a result, we propose a stochastic process model incorporating these ingredients, demonstrating that it successfully reproduces the diverse geometries discovered from the empirical spreading patterns. Our discoveries provide a theoretical foundation for the microscopic mechanisms of information flow, potentially leading to wide implications for prediction, control and policy decisions in social media.