The first time I heard the word model was in the context of “The Liril Girl,” a video promotion for a soap meant for movie theaters. The concept of using a model dressed in a bikini enjoying her time in a cascading waterfall was way ahead of its time for Indian society. The brilliant production made the soap, and its model became famous forever. I used to look forward to the Liril soap promotion just as much as the movie.
Strictly speaking, the word model means a proxy. Historically, builders made a scaled-down model, a miniature replica of a proposed construction, primarily to understand the geometry of the planned building. Nowadays, engineers build scaled-down models of an airplane to test their design in a scaled-down model of the sky in the form of a wind tunnel. Medical scientists use small animals such as rabbits and rats to understand many things about humans. Though this may seem strange, humans share a significant percentage of their DNA with many mammals, and that is sufficient to understand much about humans by studying these small mammals. One can imagine that many of these experiments lead to the death of these animals. Every medicine we use results from the sacrifice of many such mammals. Over the last few decades, humans have become so sensitive to the rights of animals that such experiments are considered highly barbaric. Fortunately, there are (partial) alternatives to animal models. This write-up is all about such models.
Humans have been using models throughout their (civilized) history. Architects built miniature replicas to plan a strategy for building their structures. Military commanders used more dynamic models to plan their battle strategies. It is important to note that a model is never identical to what it is supposed to represent. The model is chosen so that it is helpful to study a particular feature of the actual scenario. In a model for battle, a horse may be represented by a small pebble to understand and plan its optimal placement to win the battle. It is unimportant whether the object representing a horse looks exactly like a horse.
Every culture in every part of the world has its own model for our world and the universe. Every culture had its own model for the genesis of the world and its constituents. Most of the old cultures imagined that our world appeared in an instant because of a superpower. They also imagined that the same superpower was responsible for the running of this world. The majority of people believe that our world was created in seven days. The ancient Indians believed that every world is created and destroyed in a cyclical manner. They imagined the world was shaped like a giant mountain supported by eight elephants with skies above and the seas below.
The intellectuals among these constantly updated their models based on their observations. They would try to create models (either in their minds or in reality) consistent with their observations. A series of such brilliant observations concluded that our planet is curved and shaped close to a sphere. Ironically, a section of the population, calling themselves flat-earthers, still believes that the Earth is flat.
Before the invention of electricity, the world was mostly dark for half of the day. The curious lot spent most of its time observing the night (and day) skies. They observed the light coming from the sun, the moon, and the stars. They were fascinated by the repetitive and predictable sun's rising and setting and the moon's waxing and waning. They also observed that a few star-like objects wander much more than others. They called these wandering objects planets and were given far more attention than the more stationary stars. As they developed calendars based on the sun’s and the moon’s motion (relative to the Earth), they perceived the notion of time and eventually developed methods to measure time. Interestingly, this same fascination with light and time led to a deep understanding of the farthest (both in time and space) parts of our universe.
Astronomy, the study of the motion of all the heavenly bodies, was one of the earliest sciences that resulted from a meticulous study based on accurate observations over millennia. Astrology, its cousin, was the applied counterpart meant to understand the cause between the motion of stars and one’s own destiny. Astronomy was mostly out of curiosity, and astrology was mostly out of propensity for profit.
The mundane skills of counting and understanding shapes led to the concepts of numbers and geometry. Astronomers and astrologers refined these mathematical concepts to predict the motion of planets and constellations. They imagined models of the universe with the Earth at its center and everything else revolving around it. These models helped them understand the relative motion of these bodies and made the predictions simpler. The essence of any model is to provide means to predict the future state of the object in question accurately.
The stubborn few astronomers spent their lives improving these models to make them more consistent with the observations and used them for better predictions. All this was accomplished without a telescope or an accurate instrument to measure distances and time. To glimpse into such a wonderful world of ancient astronomical models and the human drama around them, one must watch a movie called Agora. Agora is loosely based on the life of a woman astronomer and mathematician named Hypatia and how her contemporaries were intolerant of her ideas. The movie maker imagined that Hypatia came close to inventing the more modern and simpler heliocentric (sun-centered) models for our solar system instead of the more commonly accepted (of her time) complex geocentric (earth-centered) model.
It took more than a thousand years since Hypatia to culminate in a satisfactory model of the universe. During these thousand years, the world of Hypatia fell into the dark ages, and the other parts discovered and invented many different models of the universe. Religions came and went. Some grew powerful and, unfortunately, intolerant of any ideas threatening their beliefs. But science and its scientists continued to propose models of many kinds to explain everything they see in our nature and world. One such young scientist was apparently sitting under an apple tree and was hit by a falling apple. This made him realize that a body, whether a small apple or a heavenly body, attracts another body through a phenomenon called gravity. He used observations of all his predecessors to represent the gravitational relationship of two bodies using a simple mathematical formula. His was no longer a complicated geometric model that Hypatia used but an abstract model in the language of mathematics. Newton’s was certainly not the first mathematical model to explain the physics of our world. Still, his was and is one of the most used mathematical models to understand our universe.
Mathematical models, too, are a proxy for a real system. However, they are not visual in nature, such as a model for a building or a battlefield. Mathematical models use symbolism and abstract concepts to represent various quantities describing a system. Today, we use computer simulations and their graphical displays to provide a visual representation of mathematical models. The latest among these tools involves computer simulations of mathematical models displayed in a “virtual” reality environment. It is a make-believe visual environment where mathematical models come alive through computer simulations.
The earliest mathematical models were all static; that is, they were mostly representing a static relationship between different physical quantities. As Newton was interested in understanding the motion of planets relative to time, he also needed to understand how motions change instantaneously. With a single brilliant idea called derivative, he revolutionized mathematics overnight. He also related this idea to the already existing idea of integration, laying the foundation for “calculus.” He at once became highly revered among the curious and the most hated among struggling high school students. Newton provided many mathematical models, including gravitational force between two bodies, how fluids flow, and how light travels.
All the interesting mathematical models since Newton are expressed in terms of equations involving derivatives and are called differential equations. My area of expertise lies in studying the qualitative behavior of quantities described by differential equations and, to a lesser extent, the applications of these studies. My colleagues and I used these concepts to study the wild behavior of combustion in a jet engine, the way a drug is distributed among different body organs or the uncertain behavior of the price of a stock in the trade market. These studies were further useful in determining control strategies to control the behavior of various quantities of a given system. We invented algorithms to optimally inject anesthesia in an operating patient to ensure that the patient is sedated at a desired consciousness level. Though I dabbled a bit in understanding models from different applications, my primary interest and expertise lie in filling pages of theorems and proofs for understanding the behavior of different classes of differential equations. The abstract models and their analysis allow me to understand various social, physical, and psychological behaviors.
Like Newton’s gravity model, Einstein’s mass-energy equivalence is probably considered one of the most beautiful mathematical models representing physics. The other famous models include Schrodinger’s equation, Maxwell’s electromagnetic equations, Clausius inequality for entropy in the subject of thermodynamics, the wave equation applicable for fluids, strings, quantum phenomenon, and many more, and the Heisenberg’s mind-boggling uncertainty principle. It is interesting to note that other than Newton’s models, all the other models were discovered within a span of sixty years. They resulted from some brilliant theoretical insights of some of the most brilliant scientists ever lived based on some brilliant experimental observations. They are genuinely supermodels in the world of mathematical physics.