Ever seen a globe—that round thing showcasing every landmass in colorful outlines—perched on a classroom shelf or tucked into the dusty and abandoned corner of your room? If so, then you already have an inkling of what models are.
Take that little piece of ball in your hand, spin it, fiddle around with your fingers, and you will experience the gist of what models are: a re-creation of the actual object in a simplified and transformed form. Here we have used two seemingly innocuous words—simplified and transformed—but don't be misled by their benignity; these words, in a sense, contain the crux of mathematical modeling and deserve in-depth exploration. (More on that later.) For now, just keep in mind that a model is a sort of replica of some concrete physical object, and its purpose is to tell us something about its real-world counterpart, but not everything!
Now, what is a mathematical model, then? Again, it is a re-creation of an actual physical object, but this time in the form of mathematical equations. Note that our persistent use of the phrase physical object might not be suitable here, as it restricts the scope of things that can be represented by mathematical equations—but let's stick with it for now.
It's time for you to wonder: How on earth can I represent a physical object using a mathematical equation? Hmmm… how on Earth indeed!
Wait—Earth is a physical object, right? And we’ve already been introduced to its physical model, which purports to tell us something about it. But what exactly is the globe telling us about the Earth? Is it the color of the Earth? Maybe. Or is it telling us about the size of the Earth? Perhaps. Whatever it's telling us, we need to find a way to capture that information mathematically. What about its shape? You see, a model might convey all of the above information, some of it, or none of it, and we don't necessarily aim to capture the essence of the object in the sense of Aristotelian philosophy or exhaustively enumerate all of its conceivable properties; rather, we want to pluck and abstractize some of its relevant features. This is one aspect of what we had alluded to earlier by saying models are “simplified form.”
Without further ado, let's just pick one heck of the Earth's features already and model it: its shape. And right out of our bag of tricks, here is the mathematical representation of the shape of the Earth:
But how is this equation (1) telling us about the shape of the Earth, and more importantly, where did this equation come from? Let's score a double win by answering both questions in a single go. To be able to conjure up that equation, you must be familiar with at least two branches of mathematics: algebra and geometry. Algebra allows you to assign variables to physical quantities such as length, speed, temperature, volume, etc. You can just as easily say, Let the volume of water in the water tank be represented by \(X\)—no harm in saying that. And if you go on to take the extra effort of measuring that volume and it turns out to be, let's say, 10 liters, then you can simply write, \(X = 10\). This physical excursion of measuring something tangible and writing the corresponding equation has really paid off! You've managed to pull off the feat of writing a mathematical model—albeit a rudimentary one, but a model nevertheless. This model is stating a simple fact—and you know what that is!
Let's get back to our quest for the origin of Equation 1. As you can see in Figure 3, that sphere is embedded in a surrounding space, which is eerily empty. But that empty space is not there without a reason; it is there for one crucial purpose: to provide a stage for the sphere to merely exist—the same way our existence is anchored in the fabric of spacetime that surrounds and sustains us. But what does this mean for our figment of imagination sphere? Do we also need to curate a crucible for its sustenance? You bet your bottom dollar we do—no stage, no performance! Don't worry, it's a fairly simple but astronomically rewarding process. Let's walk through it step by step.
First, let's conjure up our "empty" stage as nothing more than a single horizontal line stretching unbroken from negative to positive infinity, containing all real numbers ($\mathcal{R}$). Next, imagine there lives a tiny-winy organism on that line, and he decides to go on a stroll to explore $\mathcal{R}$. However, there is not much for him to admire, as he can only scuttle left or right! That's why the realm he lives in has deservedly earned the name one-dimensional (1D) world; which consequently means that to pinpoint its whereabouts on the number line, one number suffices; let's call it $x$.
Next, make a twin of that line and rotate the copy by 90° so that it stands perpendicular to the original. In an instant, our little wanderer gains two new directions—up or down—alongside its old left-and-right routine; and the universe has blossomed into a two-dimensional plane. And what does this mean for our organism’s address? No longer can a single number do the job. We must now assign an ordered pair, $(x,y)$, to capture its position in this extended, 2D realm.
Additionally, suppose that we are a little concerned that our little friend may go too far and we may lose him forever in his adventures; that's why we want to know how far away he currently is from the home, which is located at the origin. To do that, we draw a vertical line from the creature’s current point down to the \(x\)-axis, then we draw a horizontal line from the same point over to the \(y\)-axis. Then those two segments form the legs of a right triangle whose hypotenuse runs from the origin straight out to \((x, y)\); by the Pythagorean theorem, the creature’s distance from home is: \[ d=\sqrt{x^2 + y^2}\,. \]
Now, let’s take one more leap. Copy your entire 2D plane and rotate it so that it rises out of the page at a right angle—think of lifting it toward you as you keep the original \(X\!-\!Y\) sheet flat. Instantly, a third direction springs into being: forward or backward. Our tiny-winy creature, once confined to left-right and up-down motions, now roams a full three-dimensional (3D) world—the same way we do. To chart its position here, we can no longer rely on just two numbers. We must introduce a third coordinate—call it $z$—so that every location in space is specified by the ordered triple $(x, y, z)$. And how far away is he from home? A 3D version of Pythagoras’s theorem alerts us about his distance: \[ d \;=\; \sqrt{x^2 \;+\; y^2 \;+\; z^2}\,. \]
With those three values and the distance formula in hand, algebra and geometry together give us a precise “map” and “metric” of our little explorer’s universe—and that, in essence, is how we build the mathematical stage on which all of physical space unfolds.
But wait! Like our little wanderer, haven’t we, too, strayed from our original objective of tracing the origin of Equation 1? Quite the contrary, we are right on the cusp of our goal.
Suppose we tether our little 3D adventurer to home with a perfectly rigid “leash” of length $R$. He still enjoys a roam in three dimensions, but every step he takes must keep him exactly $R$ units from the origin. In other words, any point $(x,y,z)$ that he is allowed to reach must be at the same distance $R$ from his home $O$. Translating the last line in mathematical terms mean that we set the distance formulae (3) equal to $R$ $$ \sqrt{x^2 + y^2 + z^2} \;=\; R, $$ taking squre on both sides: $$ x^2 + y^2 + z^2 \;=\; R^2. $$ Doesn't this equation look familiar? It stands before us as a perfect replica of Equation (1), whose origin we have been fervently tracing out! So here is the recap: The 3D Euclidean space contains infinitely many points, each represented by an ordered triplet $(x,y,z)$, and we are interested in only those points that are at the same distance $R$ from the origin. Thus we express our interest as a constraint on the Euclidean space by using an equality: $d=R$. By doing so, we get our very first model of the Earth’s shape in all its simplified glory.
So far we have only modeled things that are entirely static; that is, there is nothing in these models that is changing. The volume of the water, which you, in a classic face-palm moment, measured with a stick, stays at the same level, even as our mathematical counterpart of the Earth is fixed in the infinite space, doomed to remain suspended till eternity. But the universe is nothing like that; it is filled with ceaseless change and flux: the swirling currents of Earth’s atmosphere and oceans, the spread of an epidemic through a community, and the random walk of stock prices and exchange rates are some examples of how dynamic our universe is.
However, in the classical viewpoint of science, nothing in the universe can budge from its place by its own. Things need an act of external agency or cause to set them in motion. Thus, to add an element of dynamism to our Earth, let's bring it out of the vast voidness and emptiness of the surrounding Euclidean universe and immerse it into something dynamic, engaging, and interactive. In its natural setting, the Earth is mostly affected by its cosmic neighbor: the Sun. One of the ways the sun affects the earth is by exerting a gravitational force on it. Simultaneously, the Sun also hurls away a bunch of other electromagnetic (EM) radiation towards us, some of which we experience as heat and light. But let's just focus on gravitational force—why, you ask? Because this time we want to build a model that tells us how the Earth moves in space as a result of the continuous action of the Sun's gravitational force. Apparently, those other EM radiations don't significantly impact the motion of the Earth, so we callously throw them out of the window to keep our model simple.
Remember how we said earlier that mathematical models are the re-creation of not just physical objects? Their scope is broader. Here, gravity is not a physical object in our day-to-day sense that we can see with our naked eye or grasp in our hand; it is an interaction. In our pursuit of understanding the physical universe, we have made two big categorizations: matter and interaction. Gravity falls into the second category. But the fact that gravity is not an object doesn't hamstring us from writing its mathematical equation. In fact, Sir Isaac Newton provided a clean definition for it; here is what he had to say on this subject: "Every particle of matter in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers."
Let's just mold what Newton said and interpret it using the conceptual narrative we have built so far. First, it prompts us to assume and visualize that there is nothing in the universe except a massive blob of matter that has zero dimension; next, build up the coordinate system (CS), as we did earlier, and place its origin so that it aligns with the blob. With CS as our reference, assign each point in Euclidean space a vector that is directly pointing towards the origin, with the vector being farther apart from the origin having a shorter length. To aid your visualization, look at the following animation—most of the heavy work has been done.
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If we replace the massive blob with the sun, then the above animation tells us that right from the sun’s immediate vicinity to the far reaches of infinity, there is a force field—a bunch of directed arrows—and the vector of the arrow (i.e., its magnitude and direction) tells us how much and in which direction another unit mass will feel the force if placed at that point in space. In other words, what Newton says is that around any given point particle that carries a mass, there is a corresponding gravitational field that envelops the whole universe around it.
All of this mumbo-jumbo about gravity, force fields, vectors, yada yada can be shoved into one liner that does it all: \[ \mathbf{F}(\mathbf r) \;=\; -\,G\,\frac{M\,m}{r^3}\,\mathbf r, \] where \(G\) is the gravitational constant, \(M\) the mass of the Sun, \(m\) the mass of the other body, and \(\mathbf r\) the position vector from the Sun to that body (with \(r = \|\mathbf r\|\)). This captures both magnitude (\(G M m / r^2\)) and direction (towards the Sun via the minus sign). For the gravitational field (force per unit mass), set \(m=1\).
It’s all the more remarkable that equation (6) and the last animation are saying the same thing in two entirely different languages—like two sides of the same coin. It is this bridge between two realms of language that you will have to traverse time and again, teasing its edges from both ends. What we have done, in essence, is set up an agency—courtesy of Newton—in mathematical form (6); this agency, in turn, will be utilized to catapult the Earth into motion. I think, at this point, it’s safe to assume you don’t need to fret any further about (6), and we can readily tread toward building the model with visual intuition alone.
A Note on Your Modeling Journey
Remember, we didn’t derive the mathematical model of gravitational interaction from scratch. Rather, we borrowed it from Newton. It is a recurrent motif in mathematical modeling that we don’t reinvent the wheel every time but take little bits and bytes of different components—developed by other people—and fit them together as in the missing piece of a puzzle.
However, (6) alone can't describe the motion of the earth as affected by the sun's gravitational interaction—it merely describes a pull. We are still missing a key piece here: how an object, matter, or particle reacts when put under the effect of external agency? To answer this, we are indebted once again to Newton and his Second Law of Motion: \[ F = m \cdot a \]
Translating (7) into our mundane human language, it simply means that when an external agency imparts the force on some particle, it will move in the direction parallel to the direction of the force--simple as that. If you’ve been all ears (and maybe a bit earwigged) to the mantra of ‘simplification,’ then you should effortlessly see how Newton shrewdly stripped an object of all its fundamental properties, i.e., shape, size, color, and temperature, until the only thing left standing was its mass! That’s by design: Newton’s sole mission here was to explain and quantify motion, so he tossed out every non-essential trait and zeroed in on mass.
By now, we have all the ingredients to propel the Earth into orbital motion: the object (Earth), the agency (gravitational force), and the mathematical description of how the object moves under the duress of force. But how do we combine all these into a perfect recipe? To do that, we must take a leap of faith—a giant intellectual leap: we equate the two expressions for \( F \): \[ m \cdot a = G \frac{M_s \cdot M_e}{r^2} \]
By doing so, we connect the cause (gravity) to the effect (acceleration), creating a model that can predict the Earth’s orbital path. However, note that equation (8) is an incomplete model for fully describing orbital motion, as it contains variables whose values require another equation to determine. But don’t break a sweat, as the purpose of this article is to give you a bird’s-eye view of the philosophical paradigm of what the process of mathematical modeling entails.
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And there you have it; Archimedes said, “Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.” Unlike Archimedes, who was pining for a long enough lever, you don’t need to go chasing after one. With just the right understanding of the world and a knack for translating that understanding into math, you can, with a flick of thought, move the world!
While we duly noted that the picture remains incomplete, you should nevertheless take a moment to appreciate the intellectual voyage we’ve just completed. We began with nothing but a harmless globe in our hands and spun it into the abstract yet powerful realm of mathematics. And don’t forget those two seemingly innocuous words from the beginning—simplified and transformed. We have now come full circle to see them sprung into vigorous life: we simplified the Earth to a mere point defined only by its mass, callously tossing aside its vibrant colors, swirling oceans, and complex atmosphere. We then transformed the invisible pull of gravity and the very concept of motion into precise, symbolic statements. And finally, we constructed a mathematical stage—our 3D coordinate system—placed our actors upon it, and handed them a script, written in the language of Newton’s laws, dictating how they must interact and evolve.
This, in a nutshell, is the art and science of mathematical modeling. It is a creative process of asking a specific question, stripping reality down to its essential and bare components, and translating those components into a language that allows for prediction and understanding. It’s a tool that lets us hold a universe of complexity in the palm of our hand, not as a colorful globe, but as an elegant equation.