A Mathematical View of Our World

1. A Mathematical View of Our World 1st ed. Parks, Musser, Trimpe, Maurer, and Maurer

2. Chapter 2 Shapes in Our Lives

3. Section 2.1Tilings • Goals • Study polygons • Vertex angles • Regular tilings • Semiregular tilings • Miscellaneous tilings • Study the Pythagorean theorem

4. 2.1 Initial Problem • A portion of a ceramic tile wall composed of two differently shaped tiles is shown. Why do these two types of tiles fit together without gaps or overlaps? • The solution will be given at the end of the section.

5. Tilings • Geometric patterns of tiles have been used for thousands of years all around the world. • Tilings, also called tessellations, usually involve geometric shapes called polygons.

6. Polygons • A polygon is a plane figure consisting of line segments that can be traced so that the starting and ending points are the same and the path never crosses itself.

7. Question: Choose the figure below that is NOT a polygon. a. c. b. d. all are polygons

8. Polygons, cont’d • The line segments forming a polygon are called its sides. • The endpoints of the sides are called its vertices. • The singular of vertices is vertex.

9. Polygons, cont’d • A polygon with n sides and n vertices is called an n-gon. • For small values of n, more familiar names are used.

10. Polygonal Regions • A polygonal region is a polygon together with the portion of the plan enclosed by the polygon.

11. Polygonal Regions, cont’d • A tiling is a special collection of polygonal regions. • An example of a tiling, made up of rectangles, is shown below.

12. Polygonal Regions, cont’d • Polygonal regions form a tiling if: • The entire plane is covered without gaps. • No two polygonal regions overlap.

13. Polygonal Regions, cont’d • Examples of tilings with polygonal regions are shown below.

14. Vertex Angles • A tiling of triangles illustrates the fact that the sum of the measures of the angles in a triangle is 180°.

15. Vertex Angles, cont’d • The angles in a polygon are called its vertex angles. • The symbol  indicates an angle. • Line segments that join nonadjacent vertices in a polygon are called diagonals of the polygon.

16. Example 1 • The vertex angles in the pentagon are called  V, W, X, Y, and Z. • Two diagonals shown are WZ and WY.

17. Vertex Angles, cont’d • Any polygon can be divided, using diagonals, into triangles. • A polygon with n sides can be divided into n – 2 triangles.

18. Vertex Angles, cont’d • The sum of the measures of the vertex angles in a polygon with n sides is equal to:

19. Example 2 • Find the sum of measures of the vertex angles of a hexagon. • Solution: • A hexagon has 6 sides, so n = 6. • The sum of the measures of the angles is found to be:

20. Regular Polygons • Regular polygons are polygons in which: • All sides have the same length. • All vertex angles have the same measure. • Polygons that are not regular are called irregular polygons.

21. Regular Polygons, cont’d

22. Regular Polygons, cont’d • A regular n-gon has n angles. • All vertex angles have the same measure. • The measure of each vertex angle must be

23. Example 3 • Find the measure of any vertex angle in a regular hexagon. • Solution: • A hexagon has 6 sides, so n = 6. • Each vertex angle in the regular hexagon has the measure:

24. Vertex Angles, cont’d

25. Regular Tilings • A regular tiling is a tiling composed of regular polygonal regions in which all the polygons are the same shape and size. • Tilings can be edge-to-edge, meaning the polygonal regions have entire sides in common. • Tilings can be not edge-to-edge, meaning the polygonal regions do not have entire sides in common.

26. Regular Tilings, cont’d • Examples of edge-to-edge regular tilings.

27. Regular Tilings, cont’d • Example of a regular tiling that is not edge-to-edge.

28. Regular Tilings, cont’d • Only regular edge-to-edge tilings are generally called regular tilings. • In every such tiling the vertex angles of the tiles meet at a point.

29. Regular Tilings, cont’d • What regular polygons will form tilings of the plane? • Whether or not a tiling is formed depends on the measure of the vertex angles. • The vertex angles that meet at a point must add up to exactly 360° so that no gap is left and no overlap occurs.

30. Example 4 • Equilateral Triangles (Regular 3-gons) • In a tiling of equilateral triangles, there are 6(60°) = 360° at each vertex point.

31. Example 5 • Squares (Regular 4-gons) • In a tiling of squares, there are 4(90°) = 360° at each vertex point.

32. Question: Will a regular pentagon tile the plane? a. yes b. no

33. Example 6 • Regular hexagons (Regular 6-gons) • In a tiling of regular hexagons, there are 3(120°) = 360° at each vertex point.

34. Regular Tilings, cont’d • Do any regular polygons, besides n = 3, 4, and 6, tile the plane? • Note: Every regular tiling with n > 6 must have: • At least three vertex angles at each point • Vertex angles measuring more than 120° • Angle measures at each vertex point that add to 360°

35. Regular Tilings, cont’d • In a previous question, you determined that a regular pentagon does not tile the plane. • Since 3(120°) = 360°, no polygon with vertex angles larger than 120° [i.e. n > 6] can form a regular tiling. • Conclusion: The only regular tilings are those for n = 3, n = 4, and n = 6.

36. Vertex Figures • A vertex figure of a tiling is the polygon formed when line segments join consecutive midpoints of the sides of the polygons sharing that vertex point.

37. Vertex Figures, cont’d • Vertex figures for the three regular tilings are shown below.

38. Semiregular Tilings • Semiregular tilings • Are edge-to-edge tilings. • Use two or more regular polygonal regions. • Vertex figures are the same shape and size no matter where in the tiling they are drawn.

39. Example 7 • Verify that the tiling shown is a semiregular tiling.

40. Example 7, cont’d • Solution: • The tiling is made of 3 regular polygons. • Every vertex figure is the same shape and size.

41. Example 8 • Verify that the tiling shown is not a semiregular tiling.

42. Example 8, cont’d • Solution: • The tiling is made of 3 regular polygons. • Every vertex figure is not the same shape and size.

43. Semiregular Tilings

44. Miscellaneous Tilings • Tilings can also be made of other types of shapes. • Tilings consisting of irregular polygons that are all the same size and shape will be considered.

45. Miscellaneous Tilings, cont’d • Any triangle will tile the plane. • An example is given below:

46. Miscellaneous Tilings, cont’d • Any quadrilateral (4-gon) will tile the plane. • An example is given below:

47. Miscellaneous Tilings, cont’d • Some irregular pentagons (5-gons) will tile the plane. • An example is given below:

48. Miscellaneous Tilings, cont’d • Some irregular hexagons (6-gons) will tile the plane. • An example is given below:

49. Miscellaneous Tilings, cont’d • A polygonal region is convex if, for any two points in the region, the line segment having the two points as endpoints also lies in the region. • A polygonal region that is not convex is called concave.

50. Miscellaneous Tilings, cont’d