Circle inside polygon

Circle Inside a Regular Polygon. Several expressions have been developed for this system (Fig. 3.6). Two relationships that give results to within a fraction of 1 percent are given. The first is [52]... 

Figure 27. Seven possible cases of the polygonal surface representation in a single pyramid. The Euler characteristic is calculated as a sum of the number of faces and the number of vertices minus the number of edges of the polygons. The black and white circles represent points with higher and lower values relative to the threshold one. The gray area is the schematic representation of the surface inside a pyramid [225].

The qualitative ordering and, indeed, the numerical values of the energies of the 7T molecular orbitals for a cyclic system of N p orbitals can be derived in a very simple way. It is necessary only to inscribe a regular polygon with N sides inside a circle of radius 2/3 with a comer down. For example, for N — 5 we get the following ... 

Step [1] Draw the polygon In question inside a circle with its vertices touching the circle and one of the vertices pointing down. Mark the points at which the polygon intersects the circle. 

Draw a circle of radius 2/3 around a midpoint corresponding to the a value and place inside it the regular polygon which corresponds to the desired cyclic it system so that one apex is always at the lowest point. The projection of the apices perpendicular to the connecting line from the midpoint of the circle to the lowest apex gives the numerical values of the orbital energies in units of /3. 

The polygon is placed inside the circle so that one edge is always in the lowest position. 

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