Euler's Relation Of Vertices, Faces and Edges

         As I have mentioned past posts, mathematical concepts that are not proven trouble me greatly.  To this end I have struggled to prove those that are yet to be explained to me.  One such principle is the relation given by Euler for components of solids:

                             V + F = E + 2

          Where, for a given solid, the number of vertices is given by V, F is the number of faces, and E gives the number of edges (between these Vertices).  At the beginning of my efforts, I tried to employ the principle that, beginning with a triangular base, one can choose between two operations: adding a point and joining a polygonal base to this point; or joining another polygonal base to an already defined point.  From this all solids can be formed.  Eventually, this needed to be expanded in order to reduce the cases that exist (which are infinite).

          My original method can be reestablished to define a given solid by the process of creating and translating points in a plane.  The rules for such a process are given by the following definitions.

1.          Creating a point from an already existing point, and subsequently translating this to some other locality, produces a line segment between the two points.

2.          When a point reaches an existing line segment or point, it encloses a surface, that is said to define a face, who's vertices are given by the collection of points so translated, and edges are those line segments that are produced by these vertices.  When this occurs, translation should halt.  To further the travel of the point, another must be created and subsequently the edge defined by this further translation will join to the vertice created by the intersecting edges of the surface.  In this discussion, any vertices and edges that do not define a surface will be disregarded, however they do cancel out in Euler's relation.

3.     When a set of surfaces are joined such that an infinitely small particle, given freedom to undergo any path of motion so long as it does not coincide with any vertice or any point lying on a face or edge of such surfaces, cannot reach a group of points other than those that constitute the surfaces themselves, the surfaces are said to enclose these points and such a set of surfaces is said to form a solid.
4.  When multiple such solids are formed, the total enclosure is said to be a composite solid, and such solids are the component solids of the composite solids.  Sometimes the term 'composite solids' will be used to describe those solids that shared either a set of vertices (and thus the number of common edges is 0 or greater), without sharing any faces, and vice versa.

5.          Edges on a surface may be removed, so long as doing so will allow a new continuous surface to exist, in other words, it must join two existing faces.

6.          When removing a vertice, one must consider any solids including such a point (it should only be removed if it belongs only to one surface (e.g. in the centre of a polygonal face, but not connected to any other solid).  This is the only case in which an edge may be removed without joining existing faces.

7.     Any composite solids require that its component solids share at least one face and one edge or vertice, or that there exists an edge between one or more vertice of each on the shared face.

8.          Therefore a vertice may only be removed, if in doing so, no composite solid is separated into component solids.

       To simplify the proof for the relation, I decided to employ the principle that any solid can be formed, beginning with a triangular base, by adding a number of (mostly irregular) triangular pyramids.  This corresponds to the creation process of points and their connection to other faces (which may be split into triangular bases if necessary) that was mentioned earlier. The problem is then reduced to proving a triangular base satisfies the formula, that then applying a triangular pyramid to any base, falling short by any polygon, given its attachment to two or fewer edges, or a vertice, will also satisfy the formula, and lastly that for this application over a number of bases will also suffice.

        I demonstrate this principal with aid of a few diagrams and whilst keeping the initial definitions and rules in mind.  Fig. 16, shows how different bases can be produced from an elemental triangular base, Fig. 17 shows how any pyramid can be produced from triangular pyrimid component figures, Figures 18 and 19 depict the construction of a triangular prism or 'wedge' from triangular pyramid base components and further, how these can be adapted to create any prism, lastly Fig. 20 demonstrates how a triangular pyramid can be applied to multiple polygonal faces.

Fig. 16 Polygonal bases

         The last figure in the above sequence represents how any polygonal base can be formed by a sequence of triangular faces whose edges are produced from a central point to each vertice of the required polygon.  For the original triangle there are three vertices and edges, and two faces (one either side).  When put into the formula we get (3) + (2) = (3) + 2.  So the formula holds for one triangle, and further development need only fulfill the requirement that any change to the number of edges be equal to any changes in the sum of the number of vertices and faces (i.e. ΔE = ΔV + ΔF).  Adding another, sharing an edge will add two edges, one vertice and one face.  In the new formula this is 2 = 1 + 1, which is true.  This can continue until any polygon is formed, the last triangle of which will share two edges and hence will add only one edge and a face, in the formula, 1 = 1 + 0, and the formula holds.  To then merge these triangles into a single face, we remove an edge one by one, for each edge thus removed, two faces become one and so ΔF = ΔE and ΔV = 0, so the formula holds.  For the last edge however, we cannot remove this as normal due to both the initial rules stated and the fact that this would not produce a new face, i.e. ΔF 

≠

 ΔE, and so the formula would not hold.   To overcome this, we remove the vertice from the centre and achieve ΔV = ΔE and ΔF = 0, thus the process of producing any base satisfies the formula.

Fig. 17 Composite triangular pyramids

          As the diagram demonstrates, by splitting a base into its component triangular faces (as previously shown), one can form a pyramid from a collection of triangular pyramids on those bases.  The correctness of the formula for these solids will be dealt with later more generally when we have discussed prisms and further composite solids.

Fig. 18 Triangular prism as a composite solid

          As in the diagram (shown in three stages), a triangular prism can be created using a combination of three different irregular triangular pyramids.  From this can be produced any solid prism.  Again, a discussion of Euler's relation for this situation will be reserved for later.

Fig. 19 Prism as a composite solid

          As we have seen, any prism can be produced by a series of smaller triangular prisms, which in turn can be produced from a variety of pyramids.  Thus all prisms can be created using only triangular pyramids.  To establish our discussion of the application of a solid to the face of another, I will demonstrate how a solid can be applied to a general polygonal face, and thus show how this is made to fit Euler's formula.  First I should note that a triangular pyramid is composed of four faces, four vertices and six edges.   Thus (4) + (4) = (6) + 2.

Fig. 20 A pyramid applied to a polygon

          Beginning with a polygonal pyramid, when a triangular pyramid is applied, it falls short by another given polygon.  Assuming that this first solid satisfies the formula, we can observe that we remove two edges and three vertices of the polygon, and add the vertices, edges and faces of the triangular pyramid, minus a face.  We can denote this by ΔE = E_Tri - 2, ΔV = V_Tri - 3, and ΔF = F_Tri – 1 (where ‘Tri’ stands for Triangular Pyramid).  Since F_Tri + V_Tri - E_Tri = 2 (by a rearrangement of the original relation), we have:

          ΔV + ΔF - ΔE = V_Tri - 3 + F_Tri - 1 - E_Tri + 2

                                   = (V_Tri + F_Tri - E_Tri) - 3 - 1 + 2

                                   = 2 - 2

                                   = 0

          Thus satisfying the second formula, and in turn the first.  To prove the formula for the case where the base shares only one edge with the face it is applied to, we note that transforming the original case, we add one edge and vertice, that cancel each other out.  For the case where one vertice is shared (and no edges), transform the previous transformation, to add an adge and vertice again.  The last case (where the base does not share anything with the face applied) will be discussed later.

          So now, assuming the last case is proved, we can conclude that applying to a triangular face is equivalent to an application to any other polygonal face.  Now all that is left is to show is that the formula holds when the pyramid shares multiple faces with the solid that it is to be applied to.  There are only two other possibilities: the situation where two faces are shared and that for when three are shared.  For the former, there are added the vertices, faces and edges of the pyramid, excepting those of the two shared faces (as in the diagram, Fig. 21).  These are: five edges, two faces (obviously), and four vertices.  Additionally, the lowest edge (most indented ridge) is lost as it is covered by the new faces.  This can be represented as:

          ΔV + ΔF - ΔE = V_Tri - 4 + F_Tri - 2 - ((E_Tri - 5) + 1)

                                  = (V_Tri + F_Tri - E_Tri) - 4 - 2 + 4

                                  = 2 - 2

                                  = 0

         Thus satisfying the formula.  This information is represented in the following figure.

Fig. 21 Adding a pyramid sharing two faces

          For the case where there are three shared faces between the pyramid and solid to be added to, we can view the solid as having a pyramidical 'hole' to be filled (as shown in Fig. 22).  To achieve this,  there will be added 1 face, and lost the number of other components in the pyramid, apart from three verices, three edges and a face. Therefore:

          ΔV + ΔF - ΔE = -(V_Tri - 3) - (F_Tri - 1) + 1 - (-E_Tri + 3)

                                  = -(V_Tri + F_Tri - E_Tri) + 3 + 1 + 1 - 3

                                  = -2 + 2

                                  = 0

Fig. 22 Pyramidical 'hole'

         Therefore adding a triangular pyramid anywhere achieves the same result if done using the initial rules stated.  To add two component solids together, simply reconstruct one on the face of the other, using the method of adding triangular pyramids.  We have already shown this will achieve the formula, unless, as you might recall, the base does not share an edge or vertice with the face of the other solid.  By the initial rules, this would be achieved by translating a point to one vertice of the component solid yet to be constructed and creating it from this basis.  What this means is that the new solid will be constructed with one less face, however the overall composite solid will gain an edge (joined beteen the base first constructed vertices).  Thus, for the overall solid:

         F + V = F_first + (F_second - 1) + V_first + V_second

                E = E_first + E_second + 1

   F + V - E = F_first + (F_second - 1) V_first + V_second - (E_first + E_second + 1)

                   = F_first + V_first - E_first + F_second + V_second - E_second - 1 - 1

      But we assume F + V - E = 2 for both of the component solids, so we have:

   F + V - E = 2 + 2 - 2

                   = 2

         F + V = E + 2

        Therefore all composite solids satisfy the relation!  We may now additionally say that each solid that is applied to a face without sharing any edges or vertices will add an additional edge to the solid.  So we can instead remove these edges to obtain a new result, where n is the number of 'unconnected' component solids, sharing only a face with the original:

                  F + V = (E + n - 1) + 2

                            = E + (n + 1)

        A singular solid is now seen as a special case where n = 1.  If we were to accept solids that do not share a face as composite, we could say for the number of edges shared, there are shared the same number increased by one of vertices.  Thus for two edges, there are three shared vertices, for one, two and for zero, one.  This can be represented by the formula, where E_s is the number of shared edges, and doubling to represent the two solids:

                    F + V – E = F_first + F_second + V_first + (V_second – (E_s + 1)) – (E_first + (E_second - E_s))

                                    = F_first + V_first – E_first + F_second + V_second – E_second – E_s – 1 + E_s

                                    = 2 + 2 – 1

                                    = 3

                          F + V = E + 3

          We could arrive at this intuitively by noting that, the relation of the final solid will have one less edge to add than vertices, thus to balance the formula out we must add one to the right hand side.  Therefore we arrive at a general formula for all solids including those composite solids whose component solids do not share edges, those that do not share faces, and those that have no common vertices – this case defines solids that are completely separate, in which case two is added to the constant term for each such solid under investigation.  Using notation similar to previous formulas we have, where beginning with a pyramid, n_faces is the number of solids added subsequently, that share only a face with the component solid that it is applied to, n_edges is the number of such solids sharing only edges or a single vertice with the solid it is applied to, and lastly n_vert represents the number of completely separate solids, having no common elements with any previous component solids:

                    F + V = E + 2 + n_faces + n_edges + 2(n_vert)

          In this formula those sharing faces does not include the original, thus the constant term is increased to two.  Now we can treat the original relation for non-composite solids (where composite mean that component solids do not share at least one face and one vertice) as a special case where both n_faces, n_edges, and n_vert are equal to 0.