Geometric Factor: Boundary Efficiency in 2D and 3D for Natural and Artificial Systems

The article “Geometric Factor: Boundary Efficiency in 2D and 3D for Natural and Artificial Systems” presents the Geometric Factor (GF) as a mathematical ruler for quantifying the structural cost of boundaries. Based on the isoperimetric inequality, the GF compares a real shape with the geometrically most efficient configuration possible, yielding an adimensional and scale independent scalar. In natural systems, whether living or nonliving, such as cells, tissues, bubbles, physical interfaces and natural surfaces, there is no fixed ideal size. What is preserved instead is an efficient geometric regime. When a system loses energetic efficiency, this loss almost always manifests first in its geometry. The boundary becomes more irregular, more costly and less efficient. The GF makes this process measurable. It allows the identification of normal geometric regimes and the detection of structural deviations before functional failures or evident dimensional changes occur. The article demonstrates, through examples in two and three dimensions, how shape itself can be used as a diagnostic tool in natural systems. Case Study 1 In serial laboratory examinations, an individual may present red blood cells with unchanged mean area and volume over time. According to traditional metrics, the system remains within dimensional normality and does not indicate any relevant alteration. However, when the geometry of the cellular boundary is analyzed, a systematic increase in the Geometric Factor is observed, indicating a higher structural cost of the membrane. This alteration reflects a geometric signature of efficiency loss, even in the absence of evident volumetric changes. In this context, the GF acts as an early geometric indicator, capable of signaling structural deviations in natural biological systems before the manifestation of functional failures or dimensional alterations detectable by conventional methods.