Question

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

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

Answer 1

A)

Euler’s Formula to form a sphere-like object  

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,

under the assumptions of convexity and regularity:

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

· If only squares, then .

· You cannot use only hexagons or higher.

B)

As one traverses the polygon the direction of movement must turn from 0 to 2π. Thus the turning angle at a vertex of a regular n-gon must be 2π/n. The interior angle then must be π−2π/n, which reduces to (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°) and for a pentagon 3π/5 (108°). The sum of the interior angle of the polygons impinging upon a vertex must be less than 2π (360°). There cannot be just two polygons coming together at a vertex. Therefore the only possibilities are: 1. Three pentagons 2. Two pentagons and one hexagon 3. One pentagon and two hexagons. In all three cases there are three edges terminating a vertex.