Problem Sheet 4 MT1100 From Euclid to Mandelbrot Hand in your solutions, with this coversheet, for feedback on Thursday 8 November 2018 YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER: 1
1. [This question shows that the trisection 'method' from the notes does not work when 0 = T/2.] In the diagram below, the angle QPX is T/2, |PQ| = |PX| = 1, and S trisects the line segment QX (so |QS| = |QX|/3). Q S P X (i) Find |QX|, |QS| and the angle PQS. (ii) Using the cosine rule, show that |PS| = V5/3. Now use the sine rule to find that sin Q = 1/v5. Use a calculator to find the approximate value of o in radians, and the approximate value of (T/2)/3. Does the line PS trisect the angle QPX? 2. Suppose we have a polyhedron with f faces, e edges, and v vertices. Suppose our polyhedron is regular: each face has n sides, and q faces meet at each vertex. (i) Draw arrows from the middle of each edge to the centre of each face adjacent to that edge. By counting the number of these arrows in two ways (see the great rhombicuboctahedron example in the lectures), show that fn = 2e. (ii) Now draw arrows from each vertex to the centre of each face adjacent to that vertex. By counting the number of these arrows in two ways, show that fn = qv. 3. A deltahedron is a polyhedron with N faces, all of which are triangles. Show that a deltahedron without tunnels must have - (N + 4) vertices. Why are there no deltahedra with 1001 faces? 4. In a truncated icosidodecahedron, every vertex is adjacent to one square, one regular hexagon and one regular decagon (10 sides). Suppose there are x4 square faces, x6 hexagonal faces and x10 decagonal faces. (i) Why are the following equations true? f=x4+ 6 + @10, 1 e == (4x4+6x6 + 10x10), 1 v = 3 (4x4+6x6 + 10x10), 4x4=6x6 = 10x10 = V. (ii) Using Euler's equation and the equations above, show that the truncated icosdodecahedron has 62 faces. 2