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Transcript
Arvind Borde MATH 1: Week 7, Flat Geometry and Tilings A polygon is a flat two dimensional shape bounded by straight edges. NOTES: Examples: But not 1 NOTES: (1) One of these polygons is different from the other two. Which do you think? Polygons such as the first and last on the previous page are called convex. The middle one is called concave or non-convex. 2 NOTES: If a straight line enters a convex polygon, it can never re-enter after it leaves. For a concave polygon, reentry is possible: 3 A subclass of convex polygons are ones that are regular. For such polygons all internal angles are the same size and all edges are the same length. NOTES: 4 NOTES: (2) Which of these is regular? We’re going to study the interior angles and exterior angles of regular polygons. 5 NOTES: 6 The total exterior angle is the sum of all the exterior angles and is easy to calculate. One way is to walk around the polygon: By looking at the total rotation during the walk, we see that the total exterior angle of a convex polygon is 360◦ . NOTES: 7 NOTES: 8 NOTES: For a regular polygon all the exterior angles are equal. Knowing the total we can calculate each individual exterior angle. (3) If the polygon has 6 sides, what will each exterior angle be? 9 This leads to the formula: NOTES: Each exterior angle of a regular polygon of n sides is 360◦ /n. What is the exterior angle of a regular polygon with (4) 4 sides? (5) 10 sides? 10 We can now work out what each interior angle must be. Look at one vertex in the diagram on page 6: NOTES: (6) What is the interior angle in terms of the exterior angle? What are the interior angles of a regular polygon with (7) 9 sides? (8) 3 sides? 11 NOTES: 12 Int. and Ext. Angles of Regular Polygons Edges 3 4 5 6 7 8 9 10 Name Ext. What about a polygon with (9) 12 edges? (10) 3, 751 edges? (11) n edges? 6 Int. NOTES: 6 13 NOTES: 14 As you notice the internal angle gets larger as the number of sides increases. Visually this means the polygons “open out” and get more circular: 15 NOTES: Here’s what a 20-gon and a 40-gon look like: Tilings NOTES: 16 NOTES: A tiling of the plane is a set of shapes that can completely cover it. We will focus initially on tiling by regular polygons, starting with tilings by single tiles: only squares, or only equilateral triangles, for example. 17 NOTES: The key requirement for a tiling to work is that the tiles fit together without gaps. The next page shows how a tiling by squares fits together. 18 NOTES: (12) At every vertex, the interior angles fit together to exactly total what? 19 This is the key criterion for tilings by polygons (regular or not): If a set of polygons meet at a vertex, the interior angles at that vertex must total 360◦ . NOTES: If we are looking at tilings by a single regular polygon, then this is the same as saying that the interior angle must be a factor of 360◦ . Go back to the table on page 13, and check off which interior angles obey this, and how may times they divide into 360◦ . (For example, 90◦ divides into 360◦ 4 times.) This means that the only tilings with single regular polygons that are possible are with triangles, squares and hexagons. 20 NOTES: (13) Looking at the table, what tilings can you come up with that use two regular polygons? 21