2. Combinatorics of Schlfli symbol. Consider the regular polytope {p, q, r}. Suppose it has V vertices, E edges, F faces, and C cells. Suppose that its vertex figures have V, vertices, E, edges, F, faces and that its cells have Ve vertices, Ec edges, Fc faces (a) Use what we learned in Euler's proof that there are only five possible regular polyhedra to find expressions for V, Ev, Fy, Vc, Ec, Fc in terms of p,q,r. (b) Prove that VV, = 2E. (c) Each face belong to two cells. What is the relationship between F and C? (d) Each edge of {p, q, r} belong to r cells, and each cell has Ec edges. Knowing that, write down the relationship between C and E. (e) Using all this information and Euler's formula in dimension 4, find out formulae for V, E, F in terms of p, q, r, C. (Hint: if you are wise, you will double check these formulae against actual values for the regular polytopes. That might help you find mistakes in your previous answers.) 2. Combinatorics of Schlfli symbol. Consider the regular polytope {p, q, r}. Suppose it has V vertices, E edges, F faces, and C cells. Suppose that its vertex figures have V, vertices, E, edges, F, faces and that its cells have Ve vertices, Ec edges, Fc faces (a) Use what we learned in Euler's proof that there are only five possible regular polyhedra to find expressions for V, Ev, Fy, Vc, Ec, Fc in terms of p,q,r. (b) Prove that VV, = 2E. (c) Each face belong to two cells. What is the relationship between F and C? (d) Each edge of {p, q, r} belong to r cells, and each cell has Ec edges. Knowing that, write down the relationship between C and E. (e) Using all this information and Euler's formula in dimension 4, find out formulae for V, E, F in terms of p, q, r, C. (Hint: if you are wise, you will double check these formulae against actual values for the regular polytopes. That might help you find mistakes in your previous answers.)