Euler's polyhedron formula | promovare-site.info
of this. 4. What is the relation between the number of vertices (V) of a polygon 3 Vertices, Edges and Faces of Polyhedra. Complete the. "Traditional polyhedra" consist of flat faces, straight edges, and vertices, but exactly .. Relationships between graph theory and other areas of mathematics. You can use it to find all the possibilities for the numbers of faces, edges and vertices of a regular promovare-site.info you will discover is that.
A regular polyhedron is a convex polyhedron whose faces are congruent regular polygons edges are all the same length and such that the same number of edges meet at the same vertex.
The five Platonic solids are the only regular polyhedra.
The only polygons meeting these requirements are the triangle 60square 90and pentagon Unfortunately some of Descartes papers were not discovered until many years later. A nice history of this can be found in E. Richeson — which is required reading for this course. A great book on this is I. Lakatos, Proofs and RefutationsCambridge — also required reading. A graph if a finite set of points vertices which are connected by line segments edges such that the graph is connected.
Here are a few examples of graphs: By this we mean that if there is a face that is not a triangle, we add some edges to it so that every face is a triangle. Here is an example of what we mean. Here is our original graph Notice how there are several faces which are not triangles. Now add in some edges in red to make all the faces triangles.
Vertices, Edges and Faces
Notice that every time you add in an edge to make a triangle the number of vertices remains unchanged, the number of edges obviously increases by one, and the number of faces also increases by one.
Who are the 10 best dressed men and 10 worst dressed women? One can also construct more grandiose lists. Who were the 10 best pitchers of all time or what were the 10 greatest movies? What appears on a list constructed by the same person can change dramatically with slight wording changes. Thus, the list of my 10 favorite movies might not coincide with my list of the 10 greatest movies ever made.
Does it make sense to construct lists related to mathematics? What about a list of the 10 greatest mathematicians? The 10 most influential theorems? The 10 niftiest theorems? Leonard Euler On the one hand constructing lists is perhaps silly. How can one make a list of the 10 greatest composers of classical music?
Must I leave Tchaikovsky out to include Mahler or Handel? Yet, from another perspective constructing lists of this kind makes one think about a wide variety of value-laden issues. What makes a composer great?
Should a composer of a few great pieces be put on a short list of greats while another composer who perhaps composed nothing that rose to the heights of the first person, yet composed times as many pieces at a very high level of inspiration, is omitted?
This being the first of my last two columns as solo editor of the Feature Column, perhaps readers will indulge me if I write two columns about a theorem which would make both my list of 10 favorite theorems and my list of 10 most influential theorems.
This theorem involves Euler's polyhedral formula sometimes called Euler's formula. Today we would state this result as: Aspects of this theorem illustrate many of the themes that I have tried to touch on in my columns. Basic ideas Polyhedra drew the attention of mathematicians and scientists even in ancient times.
The Egyptians built pyramids and the Greeks studied "regular polyhedra," today sometimes referred to as the Platonic Solids.
- Vertices, Edges and Faces
- Euler's polyhedron formula
What is a polyhedron? This question is remarkably hard to answer simply! Part of the problem is that as scholars have studied "polyhedral" objects in greater detail, the idea of what is a "legal" polyhedron has changed and evolved.
Which of the following objects below should be allowed to qualify as polyhedra? A cube with a triangular tunnel bored through it. The "faces" that lie in planes are not always polygons. The portion of the surface of three pairwise intersecting vertical planes e.
This surface does not have any vertices. The surface formed based on three rays which meet at a point as shown below. The "faces" are not polygons but unbounded portions of planes.
Almost certainly, in the early days of the study of polyhedra, the word referred to convex polyhedra. A set is convex if the line segment joining any two points in the set is also in the set. Among the nice properties of convex sets is the fact that the set of points in common to a collection of convex sets is convex.
Loosely speaking, non-convex sets in two dimensions have either notches or holes. The diagram below shows a non-convex polygon and a convex polygon. The planar set below is not convex, but note that it does not satisfy the usual definition of a polygon, even through it is bounded by sections of straight lines. The faces of a convex polyhedron consist of convex polygons.
However, this approach to defining polyhedra rules out a "polyhedron" which goes off to infinity, such as the surface below: This polyhedron has three rays which, if extended, should meet at a point and three line segments as edges of the polyhedron, rather than having edges which are line segments.
One can also have examples where there are only rays see earlier diagram. Traditionally, what is essential for a polyhedron is that it consist of pieces of flat surfaces, but as time has gone on, the definition of "legal" polyhedra has been broadened.
Typically this has resulted in a dramatic improvement in the insights that have been obtained concerning the original "narrower" class of polyhedra and the newer class of polyhedra under the broader definition. For example, in Euclid's Elements there is a "proof" that there are 5 regular polyhedra, based on the implicit assumption that the faces of the polyhedra are convex polygons.
A regular polyhedron is one in which all faces are congruent regular convex polygons and all vertices are "alike.
Using this definition one finds there are 5 regular polyhedra. If one has a supply of regular pentagonal polygons such as the one below, one can assemble 12 of them, three at each vertex, to form the solid known as the regular dodecahedron. However, the Greeks knew about the polygon that today is called the pentagram: This polygon has a good claim to be called a regular polygon because all its sides have equal length and the angle between two consecutive sides of the polygon is always the same.
However, this polygon, when drawn in the plane, does not define a convex set and the sides of the polygon intersect each other. Nonetheless, when mathematicians considered the idea of "non-convex" regular polyhedra where such "star polygons" were permitted, they discovered some new examples of "regular polyhedra," now known as the Kepler-Poinsot polyhedra. Euler mentioned his result in a letter to Christian Goldbach of Goldbach's Conjecture fame in He later published two papers in which he described what he had done in more detail and attempted to give a proof of his new discovery.
It is sometimes claimed that Descartes discovered Euler's polyhedral formula earlier than Euler. Though Descartes did discover facts about 3-dimensional polyhedra that would have enabled him to deduce Euler's formula, he did not take this extra step.
With hindsight it is often difficult to see how a talented mathematician of an earlier era did not make a step forward that with today's insights seems natural, however, it often happens. If one is wandering through a maze of tunnels in a cave, perhaps one comes across a large hall of impressive formations with many tunnels. You may explore some of these without successfully discovering any impressive new formations and go back home contented.
However, it might have turned out that one of the many tunnels you did not have time to explore would have led you to an even more spectacular chamber. Descartes' Theorem is a very lovely result in its own right, and in 3 dimensions it is equivalent to Euler's polyhedral formula. It is tempting to speculate about why all the able mathematicians, artists, and scholars who investigated polyhedra in the years before Euler did not notice the polyhedral formula.
There certainly are results in Euclid's Elements and in the work of later Greek geometers that appear more complex than Euler's polyhedral formula. Presumably a major factor, in addition to the lack of attention paid to counting problems in general up to relatively recent times, was that people who thought about polyhedra did not see them as structures with vertices, edges, and faces.
It appears that Euler did not view them this way either! He seems to have adopted a fairly conventional view of polyhedra. In his view the "vertex" of a polyhedron is a solid angle or a part of a "polyhedral cone" that starts at the vertex.
However, he does not appear to have thought of polyhedra as graphs, a step not exploited until rather later by Cauchy Because he did not look at the polyhedra he was studying as graphs, Euler attempted to give a proof of the formula based on decomposing a polyhedron into simpler pieces.
This attempt does not meet modern standards for a proof. His argument was not correct. However, results proven later make it possible to use Euler's technique to prove the polyhedral formula.
Although Euler did not give the first correct proof of his formula, one can not prove conjectures that have not been made. It appears to have been the French mathematician Adrian Marie Legendre who gave the first proof, though he did not use combinatorial methods.