Leonard Euler: Mathematics Isn't Just About Numbers
The Euler Characteristic and Topology
Suggested level: middle school and up
Leonard Euler may have been the most productive mathematician of all time. Born in 1707, in his 76 years he produced a prodigious amount of original research and wrote textbooks in virtually every area of mathematics as well as in physics, astronomy, and mechanics. So far, 805 of his works have been translated by the Euler Archive, an online digital library dedicated to Euler. Besides his technical work, he wrote “Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess,” a popular account containing 200 letters on topics such as gravity, astronomy, why the sky is blue, why evil exists, and how sinners could be converted. The ‘Letters’ was his most widely-read book, becoming an international best seller. By the time he was in his early 50s, Euler had gone blind, but that didn’t slow him down. He still wrote about one paper per week as well as textbooks on algebra and calculus, continuing his amazing productivity up to his death in 1783.
In the 1700s, everyone thought that mathematics was purely a quantitative science. No less an authority than the Greek philosopher Aristotle had pronounced it so. Mathematics is concerned with measurement, numbers, and calculations. Euler thought that too. But because he was interested in so many areas, Euler sometimes took up questions that he himself thought weren’t necessarily mathematical. Euler amused himself by tackling the problem of how to classify polyhedra and in doing so he inadvertently kicked off new areas of mathematics—topology and graph theory—that are not strictly about measurement but still immensely important in mathematics today.
The Polyhedron
The polyhedron had been studied for thousands of years by the time Euler began to look at them. Polyhedron is a Greek word: poly means ‘many’ and hedron means ‘seats.’ So, a polyhedron is a 3-dimensional geometrical figure that has flat areas that would allow you to seat it on a table. The pyramid is an example of a polyhedron. You could seat it on a table on its square base, or you could seat it on one of its triangular sides.
You can build this pyramid out of polygons. Polygons are 2-dimensional shapes formed by connecting straight lines to each other. We can build the pyramid polyhedron by taking a square polygon and connecting each side of the square to the side of a triangle and then arranging the triangles so that they meet in a point.
There are an infinite number of polyhedra you could build from polygons. The Greeks noticed that there were some very special, beautiful polyhedra that could be made from polygons that all had the same shape and were symmetric: for each polygon, the length of each side of the was equal to every other side and the angles between all sides was the same.
The simplest polygon that fits this description is the square. The length of each side of a square is the same and all sides are at right angles to each other. The square creates the first special polyhedron, the cube. Nowadays, each side of a polyhedron is called a ‘face’ rather than a seat. A cube has six faces.
We can make three additional polyhedra from a triangle in which all sides of the triangle have the same length and all angles between each side are the same. The three polyhedra are the tetrahedron, the octahedron, and the icosohadron. The tetrahedron is constructed from four triangles. “Tetra’ means four in Greek. The octahedron is built from eight triangles. “Octa’ means eight in Greek—an octopus has eight legs. And the icosahedron is built from combining twenty triangles. ‘Icosa’ means twenty in Greek.
You can also make a polyhedron from a pentagram, which has five equal sides, with equal angles between each. This polyhedron that results is called the dodecahedron. Dodeca means twelve in Greek; a dodecahedron has twelve faces, each of which is a pentagram.
That’s five so far. How many more can we make? These polyhedra are special because they are only polyhedral that can be constructed from polygons that have equal sides and angles and with each face being the same polygon. That there are only five special polyhedra was discovered by the Greek mathematician Theaetetus, who lived at the same time as Plato. Plato was so impressed that he based his physical theories on the five special polygons in his dialogue “The Timaeus.” Today we call these special polyhedra the regular polyhedra.
The Greeks believed that there were four fundamental substances in the universe and that everything was made from them: earth, water, fire, and wind. Plato thought that the four substances must be made from the regular polyhedra, given their uniqueness. Plato decided that the earth was made from cubes, probably since dirt crumbled into pieces in his hands. Also, you can pack cubes together, making a hard material like rock or iron. Fire was made from the tetrahedron, since fire hurts like the prick you might feel from the tetrahedrons pointy edges. The icosahedron was associated with water: it’s nearly circular and so would readily flow. Plato assigned the octahedron to air, because it is more mobile than air but less sharp than fire. Plato thought the last polyhedron, the dodecahedron, arranged the constellations in the heavens.
The five regular polyhedra are known to this day as the Platonic solids. From ancient times through well past the middle ages, the Platonic solids were analogous to the modern concept of atoms, since they were the building blocks of everything that exists. At the end of the 16th century, the German astronomer Johannes Kepler was tried to explain the solar system using the Platonic solids, which he thought would reveal God’s geometrical design of the universe. By that time, astronomers knew there were six planets. Kepler took each Platonic solid and constructed a circle within it and outside of it, leading to six circles that he thought were the planets’ orbits. Kepler’s theory was bound to fail, of course, since it implied that there were only five planets. As flawed as Kepler’s theory was, however, the three laws of planetary motion that came out of theory ultimately led to Newton’s mathematical theory of gravity, a stunning success.
By the time Euler started to examine polyhedra, over a hundred years later, Platonic solids were no longer important in physics, but they continued to be immensely interesting to mathematicians. While looking for some general way to classify them, Euler stumbled on an observation that was so simple it’s hard to understand why no one else hadn’t seen it in the over the 2000 years before Euler. Everyone else had studied polyhedra geometrically, looking at measurements of sides and angles. Euler focused how the parts of a polyhedron were put together without any measurement of the parts.
No one before had ever looked at the basic parts of a polyhedron or had even named all of them. Euler saw that every polyhedron is built from faces, edges, and vertices. The faces, which the Greeks called the ‘seats,’ are polygons. The edges are the lines where the polygons meet. And the vertices are the points where the edges meet.
The pyramid, for example, has four faces that are triangles and one face that’s a square—five faces. It also has eight edges and five vertices. Euler noticed that the number of faces F, minus the number of edges E, plus the number of vertices V, equals 2. Let’s check for the pyramid. F = 5, E = 8, and V = 5. 5-8+5 = 2.
What about the cube? It has six faces, 12 edges, and 8 vertices. 6 – 12 + 8 = 2. Astonishingly, all polyhedra, as long as they don’t have unusual features such as holes, satisfy the equation that Euler found, Euler’s polyhedron formula:
Proof That There Are Only Five Platonic Solids
Euler’s polyhedron formula gives us a new way, without using geometry, to prove that there only five platonic solids. Euclid’s classic proof used the geometric features of the polygons such as the lengths of the sides and their angles. With the Euler method, we only need to think about how the polyhedra are broadly constructed, how the faces, edges, and vertices fit together.
For any polyhedra, we know that F – E + V = 2. What else can we say about the faces, edges, and vertices for the Platonic solids? Let’s call m the number of edges of each face. One observation is that the number of edges of each face multiplied by the number of faces is always twice the total number of edges. That’s because each edge is where two faces meet. So, one condition for being a Platonic solid is
This condition contains the assumption that all faces on the polyhedron are the same in the sense that they all have the same number of edges. But the condition does not say anything about the geometry.
Let’s call the n the number of faces that meet at each vertex. If we take the number of faces n that meet at each vertex and multiply it by the number of vertices, that’s also twice the number of edges. That’s because any edge has two vertices. So, a second condition for being a Platonic solid is
Let’s take a moment and verify these observations for the cube. In the first condition, m, the number of edges per face, is 4. If we take each of the 6 faces and multiply by four edges, we get 6 X 4 = 24, which is twice the 12 edges. mF = 2E for the cube.
Now let’s take the number of faces that meet at each vertex, n, which is three for the cube. If we take three and multiply it by the number of vertices, which is eight, we get 24, twice the cube’s twelve edges. The double counting of edges happens because every time you count two vertices you count the edge between them twice. nV = 2E for the cube.
When you look at the other Platonic solids, you see the same pattern. The number of edges per face multiplied by the number of faces is twice the number of edges. And the number of faces that meet at each vertex multiplied by the number of vertices is twice the number of edges. This means that we can say for all the Platonic solids that
Solving for F and V
Let’s substitute F and V into Euler’s polyhedron formula, which holds for all polyhedra, including the Platonic solids.
Re-arranging
And, dividing through by 2E,
The number of edges is positive, so 1/E is positive, which means ½ + 1/E > ½.
That gives us the condition for a polyhedron to be a Platonic solid: one over the number of edges per face plus one over the number of faces that meet at a vertex must be greater than one half.
To find which Platonic solids are possible, we need to find all values of m and n that satisfy the condition that 1/m +1/n >1/2. Which m and n work? First, notice that m and n must be three or greater. m is the number of edges per face. A face is a polygon so we need at least 3 edges. n is the number of faces that meet at each vertex. If the number of faces that meet at a vertex is not at least three, we can’t have a three-dimensional figure.
The table below shows that the only combinations of m and n that will work are (m = 3, n = 3), (m = 3, n = 4), (m = 3, n = 5), (m =4, n = 3), and (m = 5, n = 3). For any other combinations of m and n,1/m + 1/n <= ½. For example, if m = 3 and n = 6, 1/3 +1/6 = 2/6+1/6 = 3/6 = ½, but we need it to be greater than ½.
How does that compare to the five Platonic solids? It checks out! Each of the (m,n) combinations corresponds to a Platonic solid. We have proved that there are only five Platonic solids using Euler’s polyhedron formula.
You might be thinking, “so what?” We have derived a different proof that there are only five Platonic solids, but there are always tons of different proofs for the same mathematical fact. There are scores of different proofs of the Pythagorean theorem.
This proof is fundamentally different though. Notice that we didn’t talk about numbers or measurement at all. We didn’t use the particular shapes of the faces—whether they were triangular or square. We didn’t use the fact that the polygons that make up the faces had to all have equal sides and equal angles. In Euler’s new perspective, what defined the Platonic solids was not their geometry, but rather how the pieces of the polygon, the faces, edges, and vertices, had to be put together. We used the fact that the number of edges per face had to be twice the number of total edges in the polygon; and we also used the fact that the number of faces that meet in a vertex must also be twice the total number of edges. Those conditions define the Platonic solids, but they are not geometric conditions. When we combined those conditions with Euler’s polyhedron formula, which applies to all polyhedra, we could prove that there are only five Platonic solids. Euler’s polyhedral formula sparked the beginning of a new branch of mathematics—topology.
Why is Euler’s Polyhedral Formula True?
Geometry comes from ‘ge,’ Greek for earth and ‘metria,’ Greek for measure. Geometry literally means measurement of the earth. Geometry is all about measuring the lengths, areas, volumes, and angles of geometric objects such as polyhedra. Topology comes from ‘topos’ which is place or location in Greek and ‘logos,’ which means study. The faces, edges, and vertices are like the places on the polyhedral. The topological property Euler discovered is that the number of faces minus the number of edges plus the number of vertices is always 2, no matter what the polyhedron looks like geometrically. That fundamental relationship between the places doesn’t change even if the polyhedra look very different.
Another way to think of topology is that it is the study of properties of geometric objects that don’t change when you stretch, deform, or bend them, as long as you don’t tear them: the ‘places’ of the geometric object keep their relationships with each other. We can see the topological nature of Euler’s polyhedron formula by working through why it must be true. Euler provided a proof, but that proof turned out to be wrong. Later on, many mathematicians proposed correct proofs. We’ll use a proof developed by 19th century French mathematician Augustin-Louis Cauchy, since it illustrates the topological nature of Euler’s polyhedron formula.
The idea behind the proof is to simplify the problem by projecting the three-dimensional polyhedron into a two-dimensionsal polygon and then reducing the polygon to a triangle using a series of changes that don’t change the value of F – E + V for the polygon. The topological nature of the proof comes about because when we flatten the polyhedron into a polygon, we will have to stretch the edges and change the angles between them.
We reduce the polygon to a triangle because a triangle is simple. In two dimensions, Euler’s formula says that F-E+V = 1 for a triangle. That’s because every triangle has one face, three edges, and three vertices, so 1-3+3 = 1. If we can always reduce a polygon to a triangle without changing the value of F – E + V, then every polygon projected from a polyhedron must have F – E + V = 1, since a triangle does. Then, when we take the polyhedron back to three dimensions, we’ll add an extra face to get the third dimension, so that F-E+V = 2.
To see how this would work, let’s start with one of the simplest cases, the cube. Imagine that it is empty box. We take the top square off and then flatten it down into two dimensions, like laying out pizza dough on a table. As we are flattening the cube, we are allowed to stretch or contract the edges as needed and also change any angles to get it flat. We just can’t tear it.
Now we can work with the flattened version. In two dimensions, it has five faces, 12 edges, and 8 vertices: 5 – 12 + 8 = 1 as expected, but we want to develop a general method to reduce any two-dimensional polygon to a triangle so that we can prove that any flattened polygon has F – E + V = 1.
The first step is to change each face into a pair of triangles by drawing a diagonal across it. That operation does not change the value of F – E + V, since when we add the diagonal we have added one edge, but then we’ve also added another face to compensate.
Let’s add one diagonal, in red, to see what’s going on. When we add the diagonal to the top polygon of the flattened cube, we don’t change the number of vertices, but we do add an additional face and an additional edge. Since the additional edge is subtracted from the additional face in Euler’s formula, F – E + V doesn’t change. So, we can convert every face into a pair of triangles in the flattened cube without changing the value of F – E + V.
The first step then is to draw a diagonal across every face, shown in red. Once we’ve created two triangles in every face, the next step is to start removing triangles without changing the value of F – E + V. If we have any triangles pointing out so that only one side touches the polygon, we should remove them first. Removing them doesn’t change the value of F – E + V, since we remove 2 edges, 1 face, and 1 vertex. The face and the vertex is 2 and then we subtract two edges, producing no change.
Looking at the diagram, we don’t have any triangles sticking out. So, we then should remove as many triangles as we need that have one external side. That process also doesn’t change F – E + V. The reason is that we remove one edge and one face and they cancel each other in F – E + V. Our goal ios to remove two triangles to leave one outward pointing triangle, which we then remove.
To proceed, we remove two triangles with one edge on the outside of the polygon. Each triangle we remove is marked with an ‘x’ before we remove it.
Performing one more round:
And then we perform the final round:
We end up with a triangle, which we know has one face, two edges, and three vertices, F – E + V = 1 – 3 + 3 = 1.
Now, here is the crucial step to get back to three dimensions. We took one face off the top of the cube before we compressed it to two dimensions. Now imagine that we reverse the process and pull it back up to three dimensions. Nothing has changed. But now let’s put the top face back on. That adds one face. Therefore, for three dimensions
Of course, we did a simple case, the cube, in which we already know that F – E + V = 2. What we learned, however, is the general method for how to transform into a triangle any polyhedron compressed into a polygon, without changing the value of F – E +V. Once we put the polygon back into three dimensions and add back the top face, we will therefore always get F – E + V = 2.
The key pitfall in doing the proof is that you have to be careful about the order in which you remove the triangles. You should always remove any triangles jutting out before you do anything else. If you can’t remove any triangles jutting out, you must first remove enough triangles with one external face to get at least one triangle that is sticking out, which you then remove. If you get the order of triangle removal wrong, you can end up disconnecting the triangles and the proof will fail.
To see how this would work in a general case, let’s look at a another polygon in two dimensions. As we can see, the faces are much less regular. If we put one diagonal in the top face, we won’t get triangles. We’ll need to put several diagonals in many of the faces. Once we have triangles everywhere, we remove the triangles that are pointing out, labeled with an x, first. Then we remove triangles with one side on the exterior of the polygon until we get new triangles pointing out to remove. We keep going until we end with a single triangle.
Working through Cauchy’s proof gives us a feeling for the topological nature of Euler’s polyhedron formula. The formula shows that if we can identify the relationship between the ‘topos,’ the places on the shape, which are the faces, edges, and vertices, then those relationships are the same even if the shape looks very different geometrically.
Euler’s formula started as a way for Euler to classify polygons, but it’s much more general than that. We saw for example how it also applied to polygons, where F – E + V had a different value, one. Euler’s formula is now used as a method for classifying surfaces. All surfaces have an Euler characteristic, which equals F – E + V. For polygons, the Euler characteristic is one. For polyhedrons, the value of the Euler characteristic is 2.
Euler characteristics apply to other shapes too. The Euler characteristic of a sphere is also two, implying that topologically all polyhedrons are the same as spheres. You can see that intuitively if you can imagine deforming a polyhedron into a sphere by stretching and bending it without tearing it.