Complexity and entropy of natural patterns

Complexity and entropy play crucial roles in understanding dynamic systems across various disciplines. Many intuitively perceive them as distinct measures and assume that they have a concave-down relationship. In everyday life, there is a common consensus that while entropy never decreases, complexity does decrease after an initial increase during the process of blending coffee and milk. However, this consensus is primarily conceptual and lacks empirical evidence. Here, we provide comprehensive evidence that challenges this prevailing consensus. We demonstrate that this consensus is, in fact, an illusion resulting from the choice of system characterization (dimension) and the unit of observation (resolution). By employing a complexity measure designed for natural patterns, we find that the complexity of a coffee-milk system never decreases if the system is appropriately characterized in terms of dimension and resolution. Also, this complexity aligns experimentally and theoretically with entropy, suggesting that it does not represent a measure of so-called effective complexity. These findings rectify the prevailing conceptual consensus and reshape our understanding of the relationship between complexity and entropy. It is therefore crucial to exercise caution and pay close attention to accurately and precisely characterize dynamic systems before delving into their underlying mechanisms, despite the maturity of characterization research in various fields dealing with natural patterns such as geography and ecology. The characterization/observation (dimension and resolution) of a system fundamentally determines the assessment of complexity and entropy using existing measures and our understanding.