Euler’s Formula
Attention! The Title May Mislead You.
There is no one who does not know or hear the name of Swiss mathematician Leonhard Euler. He is among the best mathematicians in the world for his contributions to mathematics. Mathematicians are more or less aware of his contributions, but he has a formula that has made him popular around the world.
If you type “Euler” into Google and look at the images, the first formula you will see is e^(iπ)+1=0. However, in this article, we will not talk about this formula, which has been spoken, written and videotaped many times. The name may be similar, but we will talk about another formula of Euler that is named after him. The formula V-E+F=2.
Now take paper and pencil and draw any shape similar to the figures below. The edges of the shapes you draw must be linear. Then in this way;
1-Number of vertices (V)
2-Number of edges (E)
3-Find the number of regions (F).
Subtract the number of sides in the number of vertices from the numbers we have found and add the number of regions. Probably your answer is 2. Is not it ? So how did we guess this correctly? Let’s try to explain with examples.
As you can see, no matter what shape we draw, if we subtract the number of edges in the number of vertices and add the number of regions to the result, the result is always 2. This is exactly what Euler’s formula tells us.
The result is a mathematical formula that a middle school child would understand. On the other hand, the examples we have mentioned so far are two-dimensional examples. He also mentions a fundamental property of three-dimensional solids, which we call polyhedra, which has fascinated mathematicians for over 4000 years. In fact, we can go even further and think that Euler’s formula tells us very profound things about shape and space.
On the other hand, the origin of the formula goes back to ancient times. Long before Euler, in 1537, Francesco Maurolico explained the same formula for the five Platonic solids. Another version of the formula belongs to Descartes in 1630, 100 years before Euler. Descartes explained it as the sum of the face angles of a polyhedra, equivalent to Euler’s formula, but later authors such as Lakatos, Malkevitch, and Polya disagreed, considering that the distinction between face angles and sides was too great for it. It was then (re)discovered by Euler; He wrote about it twice in 1750, and in 1752 published the result with an inductive erroneous proof for triangulated polyhedra, based on the removal of a vertex and the retriangulation of the hole formed by its removal. In other words, there were scientists working for the formula before Euler, and Euler handled this formula in a different way and revealed the Euler formula or Euler characteristic.
First of all, we need to know the polyhedra in order to better understand the Euler Formula in three dimensions and to examine the formula in depth.
Polyhedra
A polyhedron is what you get when you move up one dimension. It is a closed, solid object whose surface consists of a series of polygonal faces. We call the edges of these faces edges — two faces meet along each of these edges. We call the vertices of faces as vertices, so any vertex lies on at least three different faces. To illustrate this, two examples of well-known polyhedra are given here.
A polyhedron consists of only one part. For example, it cannot consist of two (or more) fundamentally separate parts joined only by an edge or vertex. This means that none of the following objects are a true polyhedron.
Also, Euler’s Formula is not valid for all polyhedra. For example, Euler’s formula might tell us that there is no simple polyhedron with exactly seven sides. On the other hand, using Euler’s formula similarly, we can discover that there is no simple polyhedron with ten faces and seventeen vertices. The prism shown below, with an octagonal base, has ten faces, but the number of vertices here is sixteen. A 9-sided pyramid also has ten faces but only ten vertices. But Euler’s formula tells us that no simple polyhedra has exactly ten faces and seventeen vertices.
Although polyhedra are like this, we can see Euler’s Formula below in Platonic solids, which are the most basic polyhedra.
Now you may wonder how many different Platonic Solids there are. Since the discovery of the cube and tetrahedron, mathematicians have been so impressed with the elegance and symmetry of the Platonic Bodies that they have looked for more and tried to list them all. This is where Euler’s formula comes into play. You can use this to find all the probabilities for the number of faces, sides, and vertices of a regular polyhedron. What you will discover is that there are actually only five different regular convex polyhedra! This is very surprising; After all, there is no limit to the number of distinct regular polygons, so why expect a limit here? The Five Platonic Solids are the tetrahedron, cube, octahedron, icosahedron, and dodecahedron shown above.
Euler’s Formula, which started in two dimensions, also appeared in Platonic solids, and if we think a little more, we can easily see that shapes evolve into spheres. Thus, the first question that comes to mind is is the Euler Formula valid as a sphere?
If we use the formula F + V — E = 2 for a sphere; We see that 4–6 + 4 = 2. We can change the pattern on the sphere . Yes, the formula is valid in a sphere, except for Platonic solids. In all the ways we’ve talked about so far, the Euler formula worked perfectly, always yielding 2.
Now that we’ve seen how the formula works, let’s explore how it doesn’t (i.e. when the result is not 2). So let’s examine other shapes whose result is not 2. Here are some cases where Euler’s formula doesn’t work (that is, the result of the formula is not equal to two).
We have seen that every shape may not be equal to 2 in Euler’s formula, but every shape can have Euler Characteristics (ie Euler’s formula). So the formula works perfectly, but the result may not always be two. We can examine other examples below.
Euler characteristic of these shapes; Torus:0, Double torus:-2, Triple torus:-4, Klein bottle:0, Möbius strip:0
Finally, we would like to point out that a coffee cup and a donut have the same characteristics, and we end the article with a GIF.
As a result, the formula, which starts with a middle school level question, is characterized as the Euler characteristic and this opens the doors of Topology, a branch of Mathematics.
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References:
1- https://en.wikipedia.org/wiki/Euler_characteristic
2- https://www.ics.uci.edu/~eppstein/junkyard/euler/
3- https://en.wikipedia.org/wiki/A_History_of_Folding_in_Mathematics
4- https://plus.maths.org/content/eulers-polyhedron-formula
5- https://www.mathsisfun.com/geometry/sphere.html
6- https://mathworld.wolfram.com/EulerCharacteristic.html
7- https://www.slideshare.net/shrekym/eulers-formula-155420282
