Are numbers beautiful?
Category: Physics Published: November 6, 2026
By: Christopher S. Baird, author of The Top 50 Science Questions with Surprising Answers and Associate Professor of Physics at West Texas A&M University
Beauty is in the eye of the beholder, so it's a matter of personal opinion. If you are asking for my personal opinion, then my answer is yes, I find numbers to be beautiful. Let me explain.
First of all, what is the most meaningful interpretation of this question? This question could be referring to how we write numbers, such as drawing an oval to represent zero, which can certainly be beautiful. However, that's not unique to numbers. You can also write letters of the alphabet beautifully. Therefore, mentioning numbers specifically implies that we aren't talking about how they are written.
Or, this question could be referring to the numbers in a computer that store information about the physical world and artistic creations. For instance, a photo of a rose on a computer screen can certainly be beautiful, and the information that makes up that photo is indeed stored as numbers. However, that's also not unique to numbers. Old-fashioned photography does not store image information as numbers; it stores it as shapes on photographic film. The exact same beautiful photo of a rose can be displayed on a computer screen by using numbers or can be displayed by placing a film in an old-fashioned projector without using numbers. This indicates that the beauty of the photo of the rose is not really arising from the beauty of the numbers, but from the beauty of the physical world, which can be stored using numbers, but does not have to be.
With all of this in mind, the question of whether numbers are beautiful or not most meaningfully refers to the mathematical beauty inherently present in numbers, independent of data from the physical world, independent of anything created by an artist, and independent of how you write the numbers. So, are numbers beautiful in the pure, intrinsic, mathematical sense? In my opinion, the answer is yes. Let's look at a bunch of examples.
One of the simplest relationships between numbers is multiples. For instance, the multiples of two are the numbers that result when you multiply the number two by integers, which includes 2, 4, 6, 8, 10 and so forth. Let's form a grid of numbers by counting up from one, writing each number down as we go across the page, and then start a new row after every twelve numbers. Now, if we take this grid of numbers and highlight all of the multiples of two, we end up with evenly spaced vertical lines. Similarly, if we highlight all of the multiples of three, we end up with a different set of evenly spaced vertical lines. The same goes for multiples of four, six, and twelve. For other numbers, the multiples form sets of slanted lines. All of this is shown in the animation below.
These beautifully simple results arise from mathematical patterns that are intrinsically present in the numbers. Because we put twelve numbers in each row, for every number that is a factor of twelve, its multiples form vertical lines. Note that these patterns are present in the mathematical properties of the numbers themselves, independent of how we write the numbers. In other words, if instead of writing "1" for the number one, we put a dot, and instead of writing "2", we put two dots, and so forth, then the colored lines in the animation above would still be exactly the same.
Certain numbers are especially beautiful because they show up all over the place in mathematics, even in unrelated fields, and have all sorts of interesting properties. This includes the number pi, which is π = 3.14159265… The number π shows up in various formulas for the circumference, area, and volume of shapes that involve circles; and in trigonometric identities related to triangles; and in statistics; and in vector calculus; and in other fields. The beauty of the number π is so much admired that it even has its own holiday called "Pi Day," which occurs on March 14. (When this date is written out numerically as month/day, it is 3/14, which are the first three digits of the number π).
Another especially beautiful number is e = 2.7182818…, which shows up in equations involving exponentials and logarithms, in finances, in calculus, and in other areas. The number e has the special property that when you raise it to the power of x, then the derivative of this function is itself.
Another especially beautiful number is the number i = √-1, which is the basic unit that gives meaning to all imaginary numbers. Simpler cases of especially beautiful numbers are the numbers zero and one, which have all sorts of interesting properties. All of the special numbers that I have mentioned show up in one particularly simple, but true, equation called Euler's identity:
Euler's identity is often called one of the most beautiful equations in mathematics because it includes five of the most beautiful numbers (0, 1, π, e, and i), because it includes the three most important mathematical operations (addition, multiplication, and exponentiation), and because it is simple and true. Understanding Euler's identity requires advanced training in complex-number mathematics, so I won't try to explain it.
There are all sorts of beautiful patterns that are intrinsically present in numbers. These patterns can be revealed using a simple algorithm that repetitively does certain calculations. For instance, look at the programming code below that I wrote.
Don't worry about trying to understand what each line of this code does. I'm just showing it to you to demonstrate how small this tiny chunk of programming code really is and that it only carries out multiplication, addition, the storing of the results of its calculations, and the looking up of the stored results. If you are curious, I wrote this code in the programming language Julia. This code does not use any of the more advanced mathematical operations like square root, cosine, sine, logarithm, or factorial. Furthermore, this code does not load any data from an external source. It simply starts with a few zeros and ones and then does a series of simple mathematical calculations. And yet, this short, simple chunk of code directly produced this beautiful series of images:
Let me repeat that my code, which is what produced this beautiful set of images, did not load any external data. Therefore, the beauty of these images is directly contained in the mathematical properties of numbers themselves. The fact that I used programming code is not what created the beauty in these images. Using computer code simply enabled me to do a large number of mathematical calculations quickly. If you had enough patience and basic mathematical skills, you could carry out on paper the exact same series of basic mathematical calculations in my code and end up with the same images, without using any programming code.
My code is not deceptively trying to look simpler than it really is. It does not call on external code packages to do calculations. The only external code package that my code uses is the Julia Plots package, and all that this code does is present the final array of calculated numbers as an array of colored pixels, where each color represents the value of the number at that location. The Plots package does not do any calculations with the numbers or change them. It just presents them on the screen as colors. If you did the calculations by hand, using just paper and markers, and colored in a series of boxes according to the resulting numbers that you calculated, you would end up with the same colorful images shown above without ever needing to look at or follow the code in the Plots package. My point is that the beautiful series of images shown above arise directly from the mathematical patterns that are intrinsically present in numbers, and not from anything clever that the Plots code package is doing.
Also note that my programming code never uses a random number generator and therefore contains no element of randomness. This is an amazing thing to realize because the images are so complex that they seem to contain randomness. However, if you rerun my code, you will get the exact same series of images every time.
If you are curious, the set of calculations that my code shown above carries out is called a 1D cellular automaton. If you do a google image search with the phrase "1D cellular automaton," you will find some of the same types of images as in my animation above, because other people have also carried out these same calculations.
If you look closely at my programming code shown above, you see a list of numbers on the fifth line. You may be tempted to say that this list of numbers represents external data that I manually typed into the code, like the data that you get from taking a photo of a rose, and therefore the resulting images are not purely mathematical. However, these numbers are not data. Rather, they encode various rule sets. In other words, each number on the fifth line is a compact way of telling the code exactly what calculations to do. Each image in the animation above was produced by the set of rules encoded by one of the numbers on the fifth line of the code. In general, my code could have marched through every possible rule set by starting with "s" equal to one, then equal to two, and so on, up to 256. However, to keep the animation short, I picked out the rule sets that give the most interesting images, and that is all that the numbers on the fifth line represent.
Below is a tiny chunk of programming code that I wrote that is even shorter and still produces beautiful sets of numbers. This code calculates the Mandelbrot set of numbers. When you run the code below, it directly produces the beautiful image shown after the code, which is zoomed in on part of the Mandelbrot set. If you were to slightly tweak the starting numbers in this code (the x range and the y range), you would produce other beautiful images that are zoomed in on other parts of the Mandelbrot set of numbers.
My Mandelbrot code does not load any external data, such as numbers containing the information captured in a photo of a rose or numbers containing the information produced by an artist. Rather, the code directly reveals mathematical patterns that are intrinsically present in numbers themselves. Numbers can indeed be beautiful!
There are other purely mathematical sets of calculations that also reveal beautiful patterns that are present in numbers. I wrote the tiny chunk of code below, which directly produced the beautiful image shown after the code. In this case, the Plots package simply takes the final array of numbers produced by my code's calculations and plots them as a surface, where the value of each number in the array is represented as the height of the surface at that point.
This mountain-like surface is so complex that it appears to contain an element of randomness. However, my code that produced it never uses any random numbers. If you rerun my code, you get the exact same result every time, just as 2 × 2 always gives you 4. In my code above, the exact manner in which the calculations proceed depends on three numeric parameters: "pow," "mm," and "rag." By slightly tweaking the value of these numeric parameters, you get different resulting surfaces, but in a deterministic, orderly, mathematical way.
The parameter that I called "pow" simply specifies the resolution of the surface, i.e. how far the code should go in producing details, as a power of two. By smoothly varying the value of "pow" and recording the series of resulting images, my code produced this animation:
The parameter that I called "rag" specifies the raggedness of the surface by specifying how widely different the high points on the surface are from the low points. By smoothly varying the value of "rag" and recording the series of resulting images, my code produced this animation:
Lastly, the parameter that I called "mm" specifies how high the code should count before starting over again at one, where each number that is counted through determines the height of each subsequent point on the surface. By smoothly varying the value of "mm" and recording the series of resulting images, my code produced this animation:
My point in showing you these animations is to make clear that the surfaces produced by this algorithm are not random, but arise in a logical, predictable way from the patterns present in numbers. If you want to learn more, this type of algorithm is called a "fractal landscape generator." (Note, however, that many fractal landscape generators make use of random numbers, unlike my code.)
As another example, the tiny chunk of programming code that I wrote that is shown below directly produces the animation shown after it. This code can produce beautiful tree-like images despite being such a tiny chunk of programming code, and despite not loading any external data, because it is accessing the beautiful patterns already present in numbers. If you want to learn more about it, this type of algorithm is called a "fractal tree generator."
As the last example, I wrote the tiny chunk of programming code that is shown below and the direct result is the animation shown after it. This code does not load any external data, does not use any random numbers, and does not use any external programming code except the Plots package, which is only used to present the numbers as a grid of colored boxes. If you want to learn more about it, this type of algorithm is called a "random walk," except that in my code I do not use any random numbers.
Although modern mathematicians typically consider all numbers to be beautiful, certain numbers have historically been considered ugly and illegitimate. (My historical account below is greatly oversimplified for the sake of brevity.)
The first numbers used by humans were the counting numbers, which are the numbers 1, 2, 3,… These numbers are directly used in counting whole objects. The counting numbers are the solutions to simple equations like x2 + 3 = 4x, which has the solutions x = 1 and x = 3.
Long ago, zero was considered too ugly and absurd to be acknowledged as a legitimate number. After all, you can't count zero apples in a basket, because there is nothing to count. However, if you take the equation 4 + x2 + 1 = 5 and systematically apply the rules of math in the usual way, you end up with the solution x = 0. Ancient mathematicians who refused to acknowledge zero as a legitimate number simply declared that equations like this don't have a valid solution. However, declaring that the number that results when you correctly apply the rules of math to a certain equation is not a legitimate number because you don't like it is not a rational way to do math. Eventually, zero was accepted as a number, so that the accepted numbers were now the whole numbers, which are: 0, 1, 2, 3,…
In turn, negative numbers were discovered but considered too ugly and absurd to be acknowledged as legitimate numbers. After all, you can't eat -3 apples. However, if you take the equation x2 + 6x + 8 = 0 and systematically apply the rules of math in the usual way, you end up with the solutions x = -2 and x = -4. Again, people who considered it too radical and absurd to admit that negative numbers exist simply declared that equations like this have no valid solution. However, declaring an equation and its solution invalid because you personally don't like it is not a good way to do math. Eventually, negative numbers became accepted as numbers, so that the accepted numbers were now the integers, which are: …, -3, -2, -1, 0, 1, 2, 3,…
For a time, people considered fractions to not be legitimate numbers. After all, if you have an untouched apple and a partially eaten apple in a basket, you still have two objects in the basket. However, the equation 6x2 + 2 = 7x has the solutions x = 1/2 and x = 2/3. Some people who stubbornly resisted the advance of math and could not bring themselves to accept fractions as real numbers declared that 7/10 isn't really a single number called a fraction, but is a compact way of comparing whole numbers, such as representing 7 red apples in a basket that contains a total of 10 apples. Eventually, fractions became accepted as numbers, so that the accepted numbers were now the rational numbers, which includes integers and fractions that are expressed as simple ratios of integers.
In turn, people considered irrational numbers to not be legitimate numbers. An irrational number is a number that cannot be expressed as an integer or as a simple ratio of two integers. When expressed in decimal form, an irrational number has an infinite, non-trivial, sequence of digits. The very name "irrational" suggested that anyone who believes them to be real is crazy or illogical.
However, the equation x2 - 2 = 0 has the solutions x = √2 and x = -√2, where √2 is the irrational number 1.41421356… Also, the equation ln(x) - 1 = 0 has the solution x = e, where e is the irrational number 2.71828… Also, the equation cos(x) + 1 = 0 has the solution x = π, where π is the irrational number 3.14159265… Eventually, irrational numbers became accepted as numbers, so that the accepted numbers were now the rational and irrational real numbers. Irrational numbers show up all over the place in physics.
But what about the equation 2x2 + 8 = 0 ? In solving this equation, you end up with x = 2√-1. But the square root of negative one seemed to be nonsense, and therefore this equation was declared to have no valid solution. But we've heard this story before! Declaring that a solution to a particular equation does not count as a number simply because you don't like it is bad reasoning. In time, mathematicians realized that √-1 is a legitimate number, labeled as i, so that the solution to the previous equation is the number x = 2i. Any number that is a real-valued number times i is called an imaginary number. The label "imaginary numbers" suggests that anyone who believes that they are legitimate numbers is crazy or illogical. However, like the irrational numbers, it's just a name. In time, the so-called imaginary numbers were understood to be quite logical, and were accepted as numbers. Imaginary numbers are used extensively in physics in meaningful and logical ways.
Although imaginary numbers, irrational numbers, and other seemingly absurd numbers have been passionately rejected throughout history, I personally find them to be beautiful numbers.