HW5

Problem 1

• 1. • k = 4: tetrahedron
     • k = 5: F = 5, E = 7.5 (15 / 2), no dot exist
     • k = 6: two tetrahedron merge one facet with each other
• 2. • F = 4, E = 10, V = 8, not possible, total V must less than 7 (3 * 7 = 21 > 20)
     • F = 5, E = 10, V = 7, same reason
     • Yes, cube
• 3. • consider four side faces as a whole 5 by 3 face, and the top/down/inside faces as 1 by 1 faces
     • F = 4 + 21 * 2 + 4 * 8 + 12 + 8 + 4 = 102
     • E = 4 + 5 * 6 * 2 * 2 + 4 * 8 + 4 + 4 * 8 + 12 + 2 * 4 * 2 = 220
     • V = 6 * 6 * 2 + 4 * 8 = 104
     • Euler characteristic = 104 + 102 - 220 = -14
     • genus = (2 - (-14)) / 2 = 8 (-14 = 2 - 2g)

Problem 2

• 1. 
     • generated by Ipe
     • Delaunay edges: 18
     • Voronoi edges: 18
     • |Q| = 1: 10
     • |Q| = 2: 18
     • |Q| = 3: 7
     • |Q| = 4: 2
     • |Q| = 5: 0
     • |Q| > 5: 0
• 2.