From the entropy of black holes to the complexity of plant leaves: an intriguing link

The complexity of biological forms has fascinated humanity over the years. Different plant species have different leaf shapes. Have you ever wondered why this is so? Why does this diversity of forms exist? Plants can change the shape of their leaves over time and space. But how?

Does the distinct shape of leaves play an important role in energy optimization? In fact, leaf shape has a lot to do with adaptation to their environment. What is the connection between the development of form and the evolutionary process of nature? These intriguing questions led us to focus on quantitative approaches to the complexity of plant leaves.

Quantifying leaf shapes using Euclidean shapes, such as circles, triangles, etc., is only suitable for a few plant species. Therefore, various quantitative measures of leaf shape have been developed with varying precision. But is the shape of an object really its true shape? The visual perception of a defined shape or geometry of physical objects is only an abstraction.

By digging deep into the sustenance of form, we reveal that the patterns or limitations we see are not perfect. The shape and boundaries of any physical object are only a perception rendered by human vision. A realistic boundary will change with magnification and can be seen as diffuse interfaces at the microscopic scale of finite thickness.

What is the connection between leaf geometry and black hole entropy?

In 1972, physicist Jacob Bekenstein developed a clever formula for calculating the entropy of a black hole. The entropy formulation is known as Bekenstein-Hawking entropy and is proportional to the area of ​​the black hole’s event horizon. This is one of the few prominent examples linking geometry to entropy.

Later, in 2008, the structure of the Bekenstein-Hawking entropy formula was formulated by scientist Georg J. Schmitz using geometric considerations of a geometric sphere based on a continuous 3D extension of the Heaviside function, which relies on the concept of a diffusion phase field. interfaces.

We followed multidisciplinary approaches to quantify leaf complexity. We derived the complexity of plant leaves as geometric entropy from an informational perspective by adopting the notion of the Bekenstein-Hawking formulation of black hole entropy by Georg J. Schmitz. Our results are published in the journal PLOS ONE.

While the geometry perceived at the sharp interface of an object (macro) creates a Euclidean illusion of real shape, the notion of diffuse interfaces (micro) allows us to understand the realistic shape of objects. We perceived the leaf boundary as a narrow diffuse interface between the leaf and the environment, which we view as analogous to the diffuse interface of phase field theory.

Using the concept of mereotopology, a lesser-known discipline in the scientific world that relates the static relationship between objects through logical expressions, true or false, we finally derived the geometric entropy of a geometric circle, which is then transformed for geometric entropy. entropy of plant leaves.

Our approach was purely theoretical and based on a continuous 2D extension of the Heaviside function and phase field functions on a narrow diffuse leaf-environment interface. The description of the shape of the diffuse leaf-environment interface was obtained by the statistical distribution of gradients in the diffuse interface. The expression for geometric entropy is proportional to the leaf perimeter and the square root of the leaf area and corresponds to the well-known leaf dissection index.

What are the potential applications of geometric entropy?

Geometric entropy is a measure of inherent complexity that outperforms other complex geometric morphometric measures. It does not require tedious preprocessing techniques and provides a prospective method to quantify the extent of variation in leaf shapes, such as deep lobism, dissections, serrations, and leaf perimeter.

Conventional geometric morphometric techniques primarily focus on homologous features sensitive to leaf size rather than leaf shape, which limits their reliable utility for discriminating leaf shapes at taxonomic levels. However, despite slight imperfections, geometric entropy offers a potential method for classifying leaf shapes at the genus level. We hope this could inspire plant biologists to explore its potential use in taxonomy.

Leaf morphology is a heritable trait in plants and influences light absorption, sap transport, and photosynthesis. Plants optimize leaf structure to increase the efficiency of energy exchange and maximize carbon assimilation, reproduction and resistance. We know that knowing the complex shapes of leaves has vast potential for understanding geometry and how it relates to energy capture.

Since complex leaves have greater adaptive stability in changing environments, we propose our geometric entropy as a derived plant trait to describe leaf complexity and adaptive stability. This will help in artificial leaf design studies to genetically engineer optimal leaf shapes in the future.

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Vishnu Muraleedharan holds a Ph.D. student at CV Raman Laboratory of Ecological Informatics, Indian Institute of Information Technology and Management – ​​Kerala, India. His research focuses on quantitative approaches to explore plant leaf morphological diversity as functional traits.

Sajeev C Rajan holds a Ph.D. student at CV Raman Laboratory of Ecological Informatics, Indian Institute of Information Technology and Management – ​​Kerala, India.

Jaishanker R is an ecological physicist and professor at the School of Ecology and Environmental Studies, Nalanda University, Bihar, India. Previously, Jaishanker worked as a professor at the School of Computer Science, Kerala Digital University.