a) besides the soccer ball sha

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.