Euler’s laws

For a convex polyhedron, topologically like a sphere, with F faces, V vertices and E edges, Euler s law states that V—E+F = 2. A sphere has genus zero and if another more complex polyhedron can be deformed to take the shape of a sphere with N handles, then it has the genusN. This is a useful, but not a complete, characterization... 

However, all microstructures satisfy a topological relationship represented by Euler s law which relates various dimensions in a geometrical figure. Let C (corner), E (edge), P (polygon) and B (body) be the numbers of... 

We can describe the characteristics of a two-dimensional microstructure using Euler s law. In Figure 3.7, a typical two-dimensional microstructure, three polygons meet at a point. Therefore >C=TinP + Ec, where P is the number of polygons with n edges and Ec the number of partially shared corners in the periphery of the figure where only two polygons form a corner. 

Consistent topology satisfies Euler s law. In Figure 3-16, the consistency of three topological structures is checked by Euler s law. The central structure consists of three faces, three edges and two... 

In the case of shapes having inner loops on faces, separate bodies, and through holes, the Euler-Poincare law can be applied. It is an extended version of Euler s law ... 

The simplest deltahedron is the equilateral triangle. When considered as a dihedron (two triangular faces when viewed from above and below), it obeys Euler s law for convex polyhedra... 

Euler s law is also applicable to dimeric polyhedra sharing common comers, edges or faces. Box 5 gives a few examples. 

Fullerenes (Cgo) are the third allotropic form of carbon. The Nobel Prize for their discovery was awarded in 1996 to CurL Smalley, and Kroto. Soccer-ball-structured Cgo, with a surface filled with hexagons and pentagons, satisfies Euler s law. Euler s law states that no sheet of hexagons will close. Pentagons have to be introduced for... 

In order to examine Newton s second law In terms of Euler s laws of mechanics, we consider a particle acted upon... 

At this point we are in a position to understand why physicists resolutely adhere to Newton s three laws of mechanics while engineers always adopt some form of Euler s two laws. The physicist, with an overriding Interest in the motion of particles, finds it convenient to tacitly accept the central force law in the discussion of non-relativlstic mechanics since this idea is easily altered when relativistic problems are encountered. If the physicist were to adopt Euler s two axioms of mechanics, the second axiom would require alteration when relativistic problems arise. Engineers, on the other hand, are immersed in the study of continua and Euler s laws for linear and angular momentum are perfectly suited to their purposes which rarely include relativistic effects. 

On the microscale, the DPM has been used to describe the particle dynamics in the apparatus (Deen et al., 2007). In the DPM, the motion of individual particles is predicted using Newton s and Euler s laws ... 

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