In this section, you will learn Euler's formula and apply it to convex polyhedrons including C60.


In the section of Carbon allotropes, we have learned the structure of C60 molecule, which is different from the common forms of diamond or graphite. C60 are made up of 60 carbon atoms bonded together in an array of hexagons and pentagons, like a football.

In the following activity, we will learn a mathematical formula known as Euler's formula for convex polyhedrons. We will then explore the geometric aspects of the C60 molecule with this formula.


1.

Look at the following 3-dimensional figures. Fill out the number of faces (F), edges (E) and vertices (corners) (V) of each figure in the table.

Fig. 1 Tetrahedron
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Fig. 2 Cube
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Fig. 3 Octahedron
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No. of faces

No. of edges

No. of vertices

Tetrahedron (Fig. 1)

 

 

 

Cube (Fig. 2)

 

 

 

Octahedron (Fig. 3)

 

 

 

 

2.

Now we shall look at the values for the Tetrahedron (Fig. 1), and try to find a relationship between F, E and V. Calculate V íV E + F. What value do you get?

3.

Do the same for the cube and octahedron (Fig. 2, 3) and see what you get.

4.

What is the value of V íV E + F for these three polyhedrons?

5.

Does this relationship work on other geometric figures you know? Try more complicated structures such as the Dodecahedron and Icosahedron (Fig. 4, 5).

Fig. 4 Dodecahedron
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Fig. 5 Icosahedron
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6.

If the formula applies to all the structures, then you have just found a formula which relates the number of faces, number of edges and number of vertices for the polyhedrons. In fact, this is known to mathematicians as the Euler's formula, which is named after Leonard Euler, a Swiss mathematician.

Fig. 6 C60
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7.

Now let us apply Euler's formula to C60 (Fig. 6). If we consider each atom in the molecule to be a vertex, we will have V = 60. Then, we need to solve for E and F. We can easily determine the number of edges E if we realize that each carbon atom in C60 is connected to 3 other atoms. Determine the number of edges E.

8.

Now we need to determine F. Calculate the number of faces F in C60 by determining the number of pentagons and hexagons.

9.

Try to apply Euler's formula. Does V íV E + F give the value you expect?


You may find more about Euler's formula on the following webpage:
http://en.wikipedia.org/wiki/Euler_characteristic

 
 
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