The increasing reliance on digital platforms shapes how individuals understand the world, as recommendation systems direct users toward content "similar" to their existing preferences. While this process simplifies information retrieval, there is concern that it may foster insular communities, so-called echo chambers, reinforcing existing viewpoints and limiting exposure to alternatives. To investigate whether such polarization emerges from fundamental principles of recommendation systems, we propose a minimal model that represents users and content as points in a continuous space. Users iteratively move toward the median of locally recommended items, chosen by nearest-neighbor criteria, and we show mathematically that they naturally coalesce into distinct, stable clusters without any explicit ideological bias. Computational simulations confirm these findings and explore how population size, adaptation rates, content production probabilities, and noise levels modulate clustering speed and intensity. Our results suggest that similarity-based retrieval, even in simplified scenarios, drives fragmentation. While we do not claim all systems inevitably cause polarization, we highlight that such retrieval is not neutral. Recognizing the geometric underpinnings of recommendation spaces may inform interventions, policies, and critiques that address unintended cultural and ideological divisions.

翻译：随着对数字平台的日益依赖，推荐系统引导用户接触与其现有偏好"相似"的内容，这塑造了个体理解世界的方式。虽然这一过程简化了信息检索，但人们担心它可能催生封闭的社区，即所谓的回音室，从而强化现有观点并限制对替代观点的接触。为探究这种极化现象是否源于推荐系统的基本原理，我们提出了一个最小化模型，将用户和内容表示为连续空间中的点。用户根据最近邻准则选择局部推荐项，并迭代地向这些推荐项的中位数移动；我们从数学上证明，即使没有任何显式的意识形态偏见，用户也会自然地聚合成不同的稳定簇。计算模拟验证了这些发现，并探讨了群体规模、适应速率、内容生成概率及噪声水平如何调节聚类速度与强度。我们的结果表明，即使在简化场景中，基于相似性的检索也会驱动碎片化。虽然我们并非断言所有系统都必然导致极化，但强调此类检索并非中立。认识推荐空间的几何基础，可为针对非预期的文化与意识形态分化的干预措施、政策制定及批判性分析提供参考。