# 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