
Atomic world > Euler's formula > Activities 

In this section, you will learn Euler's formula and apply it to convex polyhedrons including C_{60}.
In the section of Carbon allotropes, we have learned the structure of C_{60} molecule, which is different from the common forms of diamond or graphite. C_{60} 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 C_{60} molecule with this formula.

1. 
Look at the following 3dimensional figures. Fill out the number of faces (F), edges (E) and vertices (corners) (V) of each figure in the table. 


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 ¡V 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 ¡V 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). 


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. 


7. 
Now let us apply Euler's formula to C_{60} (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 C_{60} 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 C_{60} by determining the number of pentagons and hexagons. 
9. 
Try to apply Euler's formula. Does V ¡V 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

