Question

Exercises for Chapter 9 1. Consider a convex polyhedron, all of whose faces are squares or regular pentagons. Say there are m squares and n pentagons. Assume that each vertex lies on exactly 3 edges. (a) Show that for this polyhedron, the following equations hold: 3V = 2E, 4m + 5n = 2E, m+n=F. (b) Using Euler's formula, deduce that 2m+n=12. (C) Find examples of such polyhedra for as many different values " as you can.

Answer 1

Salution :- . Given that, -) Actually there's no need for proof through induction : since, It says in the equation question, "each vertex lies on exactly 3-Edges". "so degree of each vertex is '3. so, by hand shakė lemma (i.e; sum of degrees=2€) we get; 3v = 20 5 Hence, There are 'm' squares (each square has 4-edges) and there are n- pentagons (each Pentagon has 5-edges). If we sum up all the edges we get (hem+ 5n). Bot each edge is counted twice.

Since, It belongs to Lem+ 5n = QE". exactly two shapes. so, we get There are 'm' squares shape is a face of "m+n= F" and the no pentagons and each Polyhedron. so, we get; 1. b) Here, from Euler's Formula; V- E+F = 2 using the equations in (a), this can be written as; - E+F = Q -> (after using the ist equation ) -) - Ep +F = 2 +) - (m + 1) Tona +F=2 (using after and equation) =) {tum + sn) ... +F a 6 (using after 3rd equation) -) -(4m+ sn) - tmtn = 2 -) (-4m-51 +6m +61) 6 =) amth = a am tn => am +1 = 12] "Hence Proved",