# a) besides the soccer ball sha

a) besides the soccer ball shape (20 hexagons, 12 pentagons),are there other polyhedrons composed of just pentagons andhexagons? Using Euler’s formula of polyhedrons, prove what otherpolyhedrons are made of pentagons and hexagons.

b) Is this breakdown of edges 50/50? That is, must there be anequal number of edges adjacent one pentagon and one hexagon asthere are edges between two hexagons? If claim so, justify yourclaim and if you believe there is another ratio, explain why thisratio must be correct.

A)

Euler’s Formula to form a sphere-likeobject

Here, V: vertices, E : edges, F: faces.

Let other object be triangles , squares, pentagons.

Two faces meet at an edge so meets n faces.

So,

Now putting above in Euler’s formula,

underthe assumptions of convexity and regularity:

· If onlytriangles are used (as opposed to triangles and hexagons),then and .

· If onlysquares, then .

· You cannotuse only hexagons or higher.

B)

As one traverses the polygon the direction of movement must turnfrom 0 to 2π. Thus the turning angle at a vertex of a regular n-gonmust be 2π/n. The interior angle then must be π−2π/n, which reducesto (1−2/n)π. Thus for n=6 the interior angle is (1-2/6)π or 2π/3.For a pentagon the interior angle is (1−2/5)π or 3π/5.

Thus interior angle at a vertex of a hexagon is 2π/3 (120°) andfor a pentagon 3π/5 (108°). The sum of the interior angle of thepolygons impinging upon a vertex must be less than 2π (360°). Therecannot be just two polygons coming together at a vertex. Thereforethe only possibilities are: 1. Three pentagons 2. Two pentagons andone hexagon 3. One pentagon and two hexagons. In all three casesthere are three edges terminating a vertex.

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