BUILDING BUCKYBALLS FROM SCRATCH Introduction to Euler's…

           BUILDING BUCKYBALLS FROM SCRATCH
Introduction to Euler's rule

         Leonhard Euler was a mathematician born in Basel, Switzerland in the early 1700's. One
of the simplest relationships he discovered was the relationship between the faces, vertices
(corners), and shared edges of a closed polyhedron. Using the two figures shown below, a
pyramid and a cube, fill in the accompanying table and determine a consistent mathematical
relationship between the edges of each face and the total number of faces, corners, and edges of
a closed polyhedron.




                    figure 1                                figure 2

figure      face edges (n)           faces (F)            corners (C)         total edges (E)
   1

   2

Relationship between F and C from the table above:




Relationship between C and E from the table above:




Relationship between n, F, and E from the table above:
Building C60

   1. Buckminsterfullerene is another example of a closed polyhedron and is subject to all of
      the same relationships that you created above. Using these mathematical equations,
      determine the number of faces and edges found in one molecule of a buckyball assuming
      that you will be starting out with 60 vertices (the carbon atoms).

         figure                  faces (F)             corners (C)              edges (E)
  buckminsterfullerene                                      60

   2. Using the number of F, C, and E's found in step (1) for one molecule of
      buckminsterfullerene, is it possible to create this structure using only hexagons? Only
      pentagons? Explain your answer and show all work.




   3. Hopefully you have now proven that buckminsterfullerene cannot be created from only
      one type of polygon. Buckyballs actually contain a mixture of both hexagons and
      pentagons in a fixed ratio. Calculate the number of hexagons and pentagons that would
      be needed to create one buckyball. Show all calculations. (Hint: the number of hexagons
      and pentagons must equal the total number of faces)




   4. You are now ready to build your buckyball model.
      Start by making a pentagonal carbon ring using
      the trigonal planar carbon atoms. To each side of
      the pentagon, add a hexagon so that they share
      only one common edge as seen in the picture.
      Construct one more of these structures so that
      you have a set of two and put one of them aside.
      You have just created the "top" and "bottom" of
      your buckyball and have already accounted for
      two pentagons and ten hexagons, don't lose
      count.

   5. You will now finish your model of the buckyball as you connect the top and bottom using
      one simple rule: no two pentagons may share a corner or edge (no touching!).
      Remember to keep track of the number of hexagons and pentagons you are creating and
      stay within the limits you set for yourself in step (3). You should now use this model
      along with your representations of diamond, graphite, and CNTs while investigating the
      physical properties of each allotrope in the next section and answering the attached
      questions at the end of the lab.