As the most prolific mathematician in history (seemingly without peer), Leonhard Euler’s research spanned almost all fields of mathematics, including number theory, graph theory, and calculus. He was also a physicist; the calculus of variations, which he pioneered alongside Lagrange, brought the study of classical mechanics to a new height. Euler possessed extraordinary computational ability and mathematical intuition, which greatly aided his research. Today, in many fields, we can see numerous formulas and theorems named after him. Euler was extremely productive and reached a vast number of correct conclusions, but many of these conclusions originated solely from his mathematical intuition (creative thinking) and analogical reasoning. This was not because Euler did not seek rigor, but rather because the mathematical knowledge of the time was limited and difficult to formalize. Furthermore, the order of research is often: find the answer first, then prove it!
Moreover, creative thinking is often breathtaking and can further stimulate our cognitive abilities. Over-focusing on rigor and technical details often hinders us from reaching the correct answer. As stated in The Art and Craft of Problem Solving: Rough and inspired thoughts may lead to rigorous proofs; and sometimes, a rigorous proof can completely dilute the essence of the argument. Therefore, we need not worry about the lack of rigor in Euler’s proofs; instead, they offer a perfect visual and intellectual feast. For this reason, some ingenious, non-rigorous, and (to some extent) incorrect mathematical arguments that nonetheless yield correct results are referred to as “Eulerian Mathematics.” In fact, anyone and any research must pass through this non-rigorous early stage of “Eulerian Mathematics.”
Below is a formula regarding the faces, vertices, and edges of convex polyhedra. It belongs to the field of topology and is known as “Euler’s Formula.” (Of course, the formula is Euler’s; the argument provided here is merely a rough sketch by the author).
Euler discovered that for any convex polyhedron, the number of vertices (v), the number of edges (e), and the number of faces (f) satisfy the following relationship: v - e + f = 2
Why does such a relationship exist? (1) First, let us examine the case of a single point. As shown in the figure, n rays can be drawn from a single point, forming n convex faces. In this case, v - e + f = 1 - n + n = 1 = \text{constant}. This suggests that there might be a similar invariant relationship between v, e, f for any polyhedron.
Now let us examine an arbitrary tetrahedron. As shown in the figure, it is easy to count that for a tetrahedron, v - e + f = 2. In a plane, a triangle is the “building block” of all polygons (meaning every polygon can be divided into several non-overlapping triangles, which is the premise for deriving the formula for the sum of interior angles of a polygon). Similarly, in space, the tetrahedron is the “building block” of all polyhedra (at least convex ones). Therefore, for any n-faced polyhedron, we can divide it into (n-3) tetrahedra.
Now consider what happens when we assemble these tetrahedral “fragments” back into the original n-faced polyhedron. Consider only the case of joining two tetrahedra together: for two independent tetrahedra, we must have v - e + f = 4. At the same time, these two tetrahedra must have one congruent face. By “docking” these two faces, we undoubtedly lose 2 faces, 3 points, and 3 edges. Thus, we actually have v - e + f = 4 - 2 + 3 - 3 = 2, and the relationship remains unchanged. Of course, it is possible that two side faces merge into a single face; in this case, we will “lose” another 1 face and 1 edge, and v - e + f = 2 still holds!
Therefore, as the assembly continues, the relationship v - e + f = 2 remains invariant throughout the process. Thus, Euler’s formula holds for any polyhedron!
The above is a rough discussion of Euler’s formula. Of course, it is merely an exercise for the mind rather than a rigorous proof. But as stated at the beginning of this article, we should be more enthusiastic about creative exploration than about dull details. Creativity is a form of beauty, a touch of the soul that deeply moves our hearts to love and explore mathematics. This is precisely the charm of “Eulerian Mathematics”!
When reposting, please include the original link: https://kexue.fm/archives/1496
For more details on reposting, please refer to: Scientific Space FAQ