Euler’s relation

In analysing polyhedra and the relationships between them, a useful formula is Euler s relation... 

Now imagine a net lying in a smooth surface. This net also facets the surface into curved faces, curved edges and vertices. If each node on the net has z edges (so that the net is z-connected), and the ring size of each ring in the net is n, Euler s relation for the net is ... 

Euler s relation (described in section 1.8) allows a relation to be drawn between the Euler characteristic (-x/N) per vertex to the average ring size (n) of rings on the surface, and the connectivity (z) of the atomic net on the surface. (A ring is defined to be the shortest circuit about a vertex along edges in the framework.) This relation is ... 

The Greeks had a considerable knowledge of polyhedra, but only during the past two hundred years or so has a systematic study been made of their properties, following the publication in 1758 of Euler s Elementa doctrinae solidorum. From Euler s relation between the numbers of vertices (A o)> edges (A i), and faces N ) of a simple convex polyhedron... 

Polyhedra related to the pentagonal dodecahedron and icosahedron In equation (1) for 3-connected polyhedra (p. 62) the coefficient of is zero, suggesting that polyhedra might be formed from simpler 3-connected polyhedra by adding any arbitrary number of 6-gon faces. Although such polyhedra would be consistent with equation (1) it does not follow that it is possible to construct them. The fact that a set of faces is consistent with one of the equations derived from Euler s relation does not necessarily mean that the corresponding convex polyhedron can be made. Three of the Archimedean solids are related in this way to three of the regular solids ... 

Other equations may be derived from Euler s relation which are relevant to polyhedra with, for example, a specified number of vertices or faces. The former are required in discussions of the coordination polyhedra possible for a particular number of neighbours the latter are of interest in space-filling by polyhedra. For 8-coordination we need polyhedra with eight vertices. These must satisfy the equation S( - 2)f = 12, that is. 

From the fact that the smallest kind of interstice between spheres in contact is a tetrahedral hole it follows that we should expect to find coordination polyhedra with only triangular faces, in contrast to those in, for example, cubic closest packing which have square in addition to triangular faces. Moreover, it seems likely that the preferred coordination polyhedra will be those in which five or six triangular faces (and hence five or six edges) meet at each vertex, since the faces are then most nearly equilateral. It follows from Euler s relation (p. 61) that for such a polyhedron, 1)5 -h Oug = 12, where 1)5 and are the numbers of vertices at which five or six edges meet, so that starting from the icosahedron (vg = 12) we may add 6-fold vertices to form polyhedra with more than twelve vertices. 

Since E is a path-independent state function, it satisfies its pertinent Euler s relation... 

Euler s relation connecting the zeta function with prime numbers, which we proved in Section 1.12 ... 

The spherical harmonics are often split into real and imaginary parts. This is straightforward by exploiting Euler s relation for the exponential exp(im ) of Eq. (4.121). For the example m = 1, we can express the exponential in terms of Cartesian coordinates... 

A less cumbersome representation can be developed by using Euler s relations ... 

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