This article has been floating around in my head for quite some time, and now is a good time to let it loose. If you have been following my articles, then you know that I enjoy mathematics, physics equations, and formulas, and I really like to tell others about them. I have run across similar articles, but they tend to define the beauty of the equation (and sometimes they don’t even show you the equation) by what it describes in the physical world.
In this article, I am going to take a different approach. I want you to look at these equations as you would look at a work of art. I find them to be beautiful, and by the end, hopefully you will, too. I will cover their physical and mathematical meanings, as well; often with art, the beauty is hidden in the interpretation.
Newton’s Second Law of Motion
The beauty of this equation lies in its simplicity. It portrays a simple relation between three quantities: force, mass, and acceleration. Its deeper beauty lies in its versatility. The more complete form of this equation is:
If the big Greek letter (sigma) confuses you, don’t be alarmed; it is merely mathematical shorthand indicating that we are adding a bunch of forces together, and i is a index (1, 2, 3,…) identifying each of the forces. So, even if we have fifty different forces acting on our object, we can add them all together, and it has to equal the mass times the acceleration.
The versatility of this equation is that many other formulas are derived from it. For example, if you hang a mass on a spring attached from the ceiling, pull it down, and let it go, we can use this equation to derive the equation of motion of the object. There are two forces acting on the object, the first is the force of gravity – mg, where m is the mass and g is the acceleration due to gravity. The second is the force from stretching the spring. Here we can use Hooke’s law, which states that the force from the spring is proportional to the distance that it is stretched, – kx where k is the spring constant (recall that “proportional” means “multiplied by a constant”), and x is the distance that the spring is stretched. x is also the position of the object, and we will choose x = 0 to be the position where the mass is hanging on the spring but not moving. From our equation, we can write:
kx is negative because it is in the opposite direction of the pull of gravity. Now we just have to make the leap and realize that x is a function of time and apply a little Calculus (that scary word again; I promise that it’s just a little). Acceleration is the second derivative of the position function with respect to time. If you avoided Calculus or have worked hard to forget it don’t worry, a derivative is a operator that acts on functions and we have some pretty cool notations for it:
Einstein’s Equation of Mass – Energy Equivalence
This in another simple equation but its meaning changed the thinking of physicists all over the world (eventually). This equation states the equivalence of two factors that were generally considered to independent, mass and energy. Mr. Einstein turned the world on its head with this equation.
This equation shows the relationship between the five most fundamental constants of mathematics. Though they may seem benign to most people, the numbers 1 and 0 have deep significance in the mathematical world. In number theory, 1 is the starter; once you have that, everything else falls into place (look up the Peano axioms). And 0, it took a long time before people accepted that nothing is a valid number.
Hopefully you are familiar with π, the ratio of the circumference and diameter of a circle, but what is i? This is the one that violates everything that you learned in Algebra: it is the square root of -1. Before you start yelling about the violation of fundamental rules, you must understand that a great deal of progress in mathematics comes from taking a step back and wondering what would happen if we forget the old rules and poke around to see where things end up.
Many people have a difficult time accepting the existence of the square root of -1, but give me a little time to clarify some things about numbers in a way that you might not have considered (or been taught). We generally start to learn about numbers with what we mathematical types call natural numbers: – 1, 2, 3, etc. We first learn addition, which gives us no problems because adding two natural numbers always results in another natural number. When we learn subtraction, we run into some issues: 1 – 1 = 0 or 4 – 5 = -1. 0 and -1 are not natural numbers, so we extend our number system to include them and get the integers.
This suffices for our next lesson in multiplication (integers times integers yield integers), but division gives us the next headache. Dividing two integers does not always result in another integer, for example 5 divided by 2. At first they pass it off as a remainder, but eventually start teaching us fractions which gives us the next extension of the number system: rational numbers, which are ratios of integers. Next in the curriculum, we learn about exponents: square, cube, etc. Again, this doesn’t cause problems until we learn to undo the exponent and take a square root. It’s a fairly simple proof (left as an exercise) to show that the square root of 2 is not a rational number.
And now we are getting into the realm of what are known as the real numbers. At this point in our education, we are beginning to learn algebra, perhaps trigonometry, and maybe even calculus. Even in basic algebra, we run across basic equations that have no solution if we restrict ourselves to the real numbers. For example, everyone’s favorite, the quadratic equation:
Whose solutions are given by:
We generally learn algebra in the real number system, so we are told that if:
…then there are no solutions. Mathematicians don’t really like it when there are no solutions, so we extended the real number system to include the square root of -1. We call these numbers “complex”, they consist of a real number and an imaginary number (a factor of i). Even though it may seem to be an artificial implant, there is a funny thing about theoretical advancements like this: they eventually find their way into physics and become real. For example, the solutions to that last equation in the section on Newton’s second law reduces to a quadratic equation with imaginary solutions. In that example, if we don’t accept i, then the spring won’t bounce. We know it bounces, so it must be real (as in real life, not as in “real number”).
So that leaves e; if you haven’t studied calculus, then this one will be a complete mystery to you. This is often called Euler’s number or Napier’s number. Like π, it is an irrational number; its decimal expansion goes on forever and never repeats. Without throwing out some scary calculus stuff, there is no easy way to describe this number. Unless you are willing to take the time to study calculus and physics, you’ll just have to trust me that e is extremely important. I’ll just leave you with this:
Isn’t it interesting that two of the most important constants in mathematics are approximately 3?
Isn’t that a beauty? I won’t bother to try and explain all of the symbols in there, but jump right into the interpretation. This is the fundamental equation of Quantum mechanics. This equation is actually very similar to Newton’s second law; it describes the motion of subatomic particles (which aren’t really particles).
I put this one directly after Schrödinger’s equation, because they are very closely related. In this equation, y is an unknown function of the independent variable x; and p, q, and w are known functions of x. The parameter λ is an unknown value. This particular equation is unique, because along with finding the unknown function y, we also must identify values of λ for which the solution exists. In the math and physics world, λ is called an “eigenvalue” or a “characteristic value.”
In the quantum mechanics world, the Schrödinger equation can be reformulated to fit into the Sturm-Liouville form, where λ is the energy of the particle we are studying.
From this, we find that the energy of sub-atomic particles can only have discrete values. What this means is that, for example, the energy of an electron in a hydrogen atom can only be one of a set of specific values. It cannot ever be a fraction of one of those values. In other words, an electron can only be on a step on a ladder, never in between steps. This is where the term “Quantum leap” comes from. When an electron increases or decreases in energy, it has to jump up or down to the next rung, it will never be anywhere in between.
Heisenberg’s Uncertainty Principle
This one drifts slightly from the parameters of this article since it not an equation but an inequality. The symbol Δ indicates an uncertainty in the measurement of the variables p, which is the momentum or mass times velocity, and, x which is the position. This inequality basically states that the product of the uncertainties of these two variables is always greater than a constant value. In other words, if you decrease the uncertainty in the momentum (or speed), then the uncertainty in the position has to increase. You can never be exactly sure of either of these values; you can’t be totally sure of where you are or where you are going!
And the other, equivalent form:
I’ll let you decide which form is more beautiful; I like them both equally. These equations describe the interactions of the electric (E) and magnetic (B) fields. Though these equations are generally attributed to James Clerk Maxwell, the first two were discovered by Carl Friedrich Gauss, the third by Michael Faraday, and the fourth by André-Marie Ampère. Maxwell tied them all together into one cohesive group. The first equation describes how the electric field permeates a closed surface. The second equation says that the positive and negative parts of the magnetic field passing through a closed surface always cancel each other out; there cannot be an isolated positive or negative magnetic element (no monopoles). The last two equations describe the intertwining of the electric and magnetic fields. As one of them varies in time, it creates the other.
Einstein’s Gravitational Field Equation
This one is just plain beautiful. This is actually a collection of equations in four dimensions; μ and ν are indices that vary over the integers 1 to 4. This collection of equations describe how matter interacts with space itself. is called the Ricci curvature tensor, which depicts the deviations in the curvature in space. is called the stress-energy tensor and describes the gravitational field itself. \If you are unfamiliar with the mathematical term “tensor”, suffice to say that , , and are each a collection of sixteen numbers (four times four). Besides the wondrous results that come from this equation (black holes, worm holes, gravitational lenses…), the simplicity of its notation lends a special aesthetic quality.
This is another equation with aesthetic qualities but extremely deep mathematical and physical meaning. It stems from this:
…where we desire that J have a minimum or maximum value. If you’ve only been through basic calculus, then this one is a bit of a stretch of the mind. The symbol f is called a “functional”; it is a function whose arguments are other functions, in this case y and y’ are functions of t (y’ is the derivative of y with respect to t; in most cases, it is the velocity of an object).
If all of that is gibberish to, you don’t be alarmed; this equation allows us to prove some deep mathematical and physical questions, like “what is the shortest distance between two points?” We all know that it’s a straight line, but without the Euler-Lagrange equation, it is quite difficult to prove that fact.
Mandelbrot Set Generator
This is another simple equation with profound implications. This equation defines a sequence of numbers, the parameter n runs through the integers: n = 1, 2, 3… We start with an initial value of z, and c is a constant value specified at the beginning. What we are looking for are starting values of z, for which the sequence of numbers generated by the equation remains bounded; they never go above or below some particular values. For example, if we start with z = -1 and c = 0, then the values will always be 1 or -1, so the sequence is bounded; -1 is included in our set of solutions.
If we restrict the starting values of z and c to the real number system (you might need to reread the section on Euler’s identity), then the set of values that satisfy the equation is not all that interesting. If we extend the possible values into the complex numbers, then things get really interesting.
We can plot complex numbers in a plane like we would plot x and y coordinates. The y axis, we generally think of a complex number as z = x + iy, where y is the imaginary part, so plotting the complex number z is simply a point (x, y) in the plane. When we do this for the numbers that satisfy the conditions (don’t worry, we mathematicians have a big bag full of tricks to use to determine if a sequence of numbers remain bounded without having to sit there forever and generate all of the values), we get a picture like this:
In this picture, the black indicates numbers in the set, blue are numbers not in the set, and white is the boundary. The first thing that you’ll probably notice is that the boundary looks fuzzy, and that’s because it is. If you zoom in on the boundary anywhere in the picture, you will see the same pattern repeated (though rotated) again and again no matter how far you zoom in. This was the first example of what is called a fractal set; actually, the boundary falls into this category. A fractal set has the basic property that I just mentioned, there is a non-trivial (not a straight line) pattern that is repeated at every scale of focus.
This leads to some deep mathematical issues such as how to measure the size of a fractal set. Fractals are also closely related to what is commonly termed Chaos Theory. Chaos Theory is a subset of a larger branch of mathematics known as Dynamical Systems. Dynamical Systems give us a set of tools that allows us to determine the general behavior of solutions to systems of equations without actually solving the equations.
Hopefully, I haven’t lost everyone at this point. It might not be obvious after all this, but the one point I’m trying to get at here is that a great deal of what I see as beauty in Mathematics is in the notation that mathematicians create to simplify very complex ideas into symbols that can be manipulated by a set of rules. The true beauty of mathematics is that it has its own language. It differs from other languages of this world in its precision. Unlike other languages, the definition of a mathematical term is exact. There are no synonyms, there are no multiple meanings. When I say “a function is continuous”, every mathematician in the world knows exactly what I mean, and every textbook has the same definition. Unfortunately, we tend to use words that you are already familiar with in your own language.
I’ve heard it said that mathematics is mankind’s most perfect creation, but is it really? A mathematical system begins with a set of axioms. Axioms are statements that are accepted as true, without any doubt, they do not have to be proven. Next come theorems that are statements that are proven by applying the axioms and other theorems that have been previously proven true. This seems valid, but there is a flaw. Back in 1931, Kurt Gödel presented his incompleteness theorems. Without going into painful detail (though the genius of his proofs is incredibly beautiful), the gist of his findings is that no finite set of axioms can result in a infinite set of valid theorems. Basically, you will eventually encounter statements that can neither be proven true nor false. Since the statement cannot be proven or dis-proven, you now have a choice: either the statement or its negation is an axiom, but which do you choose? Other than examining the consequences of either choice and deciding which one you like the best, there is no absolute guidance. Either system is valid mathematically, but is it valid to chose one over the other simply because it leads to the results that I want?
I’ll stop here to avoid going completely off the deep end. Let’s consider what we’ve learned here:
- Blaise has a weird sense of beauty.
- Mathematical formulas are cool even if you don’t understand them.
- Mathematics is a foreign language, no wonder I don’t get it.
- Precision does not lead to perfection.