20.1 Tilings with Regular Polygons

1. Determine the measure of an exterior angle and of an interior angle of a regular dodecagon (12 sides).

2. Determine the measure of an exterior angle and of an interior angle of a regular decagon (10 sides).

3. Specify a formula for the measure of an interior angle of a regular -gon.

4. Using the formula from Exercise 3 and a calculator, make a table of the interior-angle measures of regular polygons with 3, 4, …, 12 sides.

5. Use the table of interior-angle measures from Exercise 4 to determine all the possible vertex types of regular polygons (with at most 12 sides) surrounding a point.

6. Which of the vertex types of Exercise 5 do not occur in a semiregular tiling?

7. In addition to the vertex types of Exercise 5, Xîp** exactly five others are possible, each involving one polygon with more than 12 sides. None of these vertex types leads to a semiregular tiling. The five many-sided polygons involved in these five vertex types have 15, 18, 20, 24, and 42 sides. Determine the other polygons in each of these five vertex types.

8. Use the center of the tiling to determine the measures of the angles of the isosceles triangle tile.

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9. Every vertex except the center vertex has the same vertex type, in terms of the measures of the angles surrounding the vertex. What is that vertex type?

20.2 Tilings with Irregular Polygons

10. For each tile below, find a tiling of the plane. (Adapted from Branko Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987, p. 25.)

11. You know that a regular pentagon cannot tile the plane. Suppose that you cut one in half into two mirror- symmetric pieces. Can this new shape tile the plane? (See Figure 19.4 on page 787 for a regular pentagon that you can trace.)

12. Design a nonconvex hexagon that can tile the plane.

13. If you “unbulge” an ordinary soccer ball so that each of its sewn pieces is flat, you get a polyhedron, but it is not a regular polyhedron. It is a truncated icosahedron, one of the semiregular polyhedra. Some of the faces are regular hexagons, and some are regular pentagons. Explain why not all of its faces can be regular hexagons. All the vertices have the same vertex type. What is it? How many pentagons are there and how many hexagons? How many vertices?

14. A soccer ball should be as round as possible, which suggests using many polygons, each of which is short across. Of course, making a ball with a large number of small pieces requires more sewing. A ball with 92 faces was used for some European soccer competitions: 12 pentagons, 20 hexagons, and 60 triangles (see the accompanying figure), all of them regular polygons. What are the possible vertex types for a polyhedron made up of pentagons, hexagons, and triangles? Explain why the polyhedron underlying this soccer ball is not semiregular.

20.3 Using Only Translations

15. For each tile (a) through (c), determine whether it can be used to tile the plane by translations. (From Frederick Barber et al., Tiling the Plane, COMAP, Lexington, MA, 1989, pp. p. 1 p. 8, 9.)

16. Repeat Exercise 15, but for tiles (d) through (g).

17. Start from a par-hexagon of your choice and modify it to tile the plane by translations. (You will probably find it useful to do your work on graph paper. If you choose a regular hexagon, special graph paper is available, ruled into regular hexagons, that would be particularly good to use.) Can you draw a design on the tile so as to make an Escher-like pattern?

18. Start from a parallelogram of your choice and modify it to tile the plane by translations. (You will probably find it useful to do your work on graph paper.) Can you draw a design on the tile so as to make an Escher-like pattern?

19. Is the L-tromino convex? Does the result about which hexagons can tile the plane (on page 834) give any information about whether the L-tromino can tile the plane or not?

20. Find a tiling of the plane using just the L-tromino and translations of it. Is there more than one way to do the tiling?

21. Show how alternative 2 of the Translation Criterion can be applied to the L-tromino.

22. Explain why alternative 1 of the Translation Criterion cannot be applied to the L-tromino. (Hint: Label each of the eight corners of the component squares of the tromino with the letters . Let these be our candidates for the points , , , and of the criterion.) Each of the sides of the tromino that is 2 units long has nowhere to go under a translation. Any application of the criterion must divide each side into two pieces, so their midpoints must be two of the points , , , and . Make a similar argument about two corners of the tromino. Thus, we have four points, which can be labeled consecutively , , , and , starting at any one of them. Show that none of the four possibilities “works.” (This argument can be generalized to show that trying , , , and at points other than the corners of the squares won’t work either.)

23. Explain why there are no other tetrominos (each made of four squares joined at edges) than the five shown below—plus differing mirror images of two of them (which ones?). In the order shown, they are called the square, straight, T-, L-, and skew tetrominos.

24. Show how alternative 2 of the Translation Criterion applies to the T-tetromino.

25. Apply alternative 1 of the Translation Criterion to the skew-tetromino and show how it can tile.

26. Show how alternative 2 of the Translation Criterion applies to the L-tetromino.

27. Show how alternative 2 of the Translation Criterion can be applied to the skew-tetromino and show how it can tile.

28. In Exercises 25 and 27, we indulged in what would appear to be “overkill,” demonstrating the same result in two different ways. Doing so can be useful in mathematics for giving greater understanding about why and how something occurs or is true. Here, those exercises should give you the idea that alternative 2 of the Translation Criterion can reduce to (and hence is more general than) alternative 1 if some points are allowed to coincide. For such a reduction, which pairs of points must coincide? (You are allowed to relabel the remaining four distinct points.)

20.4 Using Translations Plus Half-Turns

29. Explain how the final shape of Figure 20.18 on page 843 satisfies the Conway Criterion by identifying relevant points , , , , , and F.

30. Do the same as in Exercise 29, but for the final shape of Figure 20.19 on page 843.

31. For each tile (a) through (c), determine whether it can be used to tile the plane by translations and half-turns.

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32. Repeat Exercise 31 for tiles (d) through (g).

33. Show how an arbitrary pentagon with two parallel sides, such as the one shown below, can tile the plane.

34. The pentagon below is a pentagonal tile of type 13, which was discovered by Marjorie Rice. Show how it can tile the plane. (Hint: Carefully trace and cut out a dozen or so copies and try fitting them together.)

The parts of this pentagon satisfy the following relations:

[Adapted from Doris Schattschneider, “In Praise of Amateurs,” in David A. Klarner (ed.), The Mathematical Gardner, Wadsworth, Belmont, CA, 1981, p. 162.]

35. Start from a triangle of your choice and modify it to tile the plane by translations and half-turns. (You will probably find it useful to do your work on graph paper.) Can you draw a design on the tile so as to make an Escher-like pattern of a plant, animal, or other object?

36. Start from a quadrilateral of your choice and modify it to tile the plane by translations and half-turns. (You will probably find it useful to do your work on graph paper.) Can you draw a design on the tile so as to make an Escher-like pattern of a plant, animal, or other object?

37. By experimenting, determine which of the pentominos can tile the plane by translations. (Hint: There are nine.)

38. Apply the Conway Criterion to the f-pentomino and show how it can tile by translations and halfturns.

39. Apply the Conway Criterion to the U-pentomino and show how it can tile by translations and halfturns.

40. Apply the Conway Criterion to the T-pentomino and show how it can tile by translations and half-turns.

41. (Thanks to Doris Schattschneider, Moravian College.) In the text, we discuss some criteria and methods for generating Escher-like patterns that involve just translations or translations and half-turns. A slight variation on one such method allows construction of tilings that feature a tile and its mirror image. We modify a parallelogram so that two reflected tiles are joined and the joined pair tiles by translation. You get to design the shape of the tile and put whatever art you like on it.

Begin with an isosceles triangle and mark the midpoint of each of its sides (it’s best to use graph paper or a dynamic geometry program). Half-turn the triangle about the midpoint of one of its two equal sides to produce a parallelogram.

Modify half of the (odd) third side in any manner—here is where you get to be creative!— joining a vertex to the midpoint of that side. Then reflect that modification in the side and translate it so that it joins the midpoint to the other vertex of that side.

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Now translate the lower half of the modified side to the upper half of the opposite side of the parallelogram, and translate the upper half of the modified side to the lower half of the opposite side of the parallelogram.

Now modify one of the remaining sides of the parallelogram in any way you wish, and then translate that modification to the opposite side of the parallelogram.

Join a midpoint of the diagonal of the parallelogram to a midpoint of the side you have just modified (this line segment will be parallel to the other sides of the parallelogram). Reflect the side just modified in this segment, then translate the reflected modification so that it replaces the diagonal of the parallelogram. Your original parallelogram is now replaced by two tiles that are reflected images of each other. The outline of the joined pair of tiles is a modified par-hexagon that tiles by translations.

42. Does the Conway Criterion apply to the tilings created by the method of Exercise 41?

20.5 Nonperiodic Tilings

43. (a) Describe the monohedral tilings of the line. (b)Describe the periodic tilings of the line that use two tiles of different lengths.

44. A line can be tiled quasiperiodically with a pair of tiles but only if their lengths are in the right proportion, so that the pattern can be scaled up. We determine what that proportion must be. Let the two lengths be (for “short”) and (for “long”), with and , where the scaling factor is . The key requirement for the scaling factor is that scaling up a tile of length produces a total length of , that is, .

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45. What is the sequence at the end of the sixth month?

46. Explain why we can never have two s next to each other.

47. Explain why we can never have three s in a row.

48. Show that from the fourth month on, the sequence for the current month consists of the sequence for last month followed by the sequence for two months ago.

• Inflation: Replace each by and each by . For example, the inflation of ()()() would be , where we have inserted parentheses for clarity.
• Deflation: Replace each by and each lone by . For example, the deflation of ()()() would be , where we have inserted parentheses for clarity.

49. (a) Start with just a single and repeat the inflation process, showing the stages, until you reach a stage with 21 tiles.

(b) How many tiles are there at each stage that you reached? (If you continue this process forever, you tile a half-line to the right; you could tile the entire line by reflecting this right half-line over to cover the left half-line. The result is called a Fibonacci tiling of the line because of the appearance of the Fibonacci sequence in the numbers of tiles.)

(c) If a line segment contains copies of the tile and copies of the tile, how many tiles will the inflation of the segment contain?

50. (a) Apply deflation repeatedly to your answer to Exercise 49a, which has 21 letters, until you can deflate no further. What do you end up with?

(b) Apply deflation repeatedly instead to the periodic sequence …. What do you end up with?

(c) Devise your own periodic sequence of and tiles. Apply the deflation process to it repeatedly. What do you end up with? What do you conjecture?

(d) Explain how the Fibonacci tiling is quasiperiodic.

51. Let a lone be considered the first stage of inflation. Show that at the nth stage of inflation, for , there are (the th Fibonacci number—see Section 19.1 on page 780) symbols in the sequence, of which are s and are s. (Hint: Check it for , and 4.)

52. Explain why inflation and deflation preserve musicality: If we inflate or deflate a musical sequence, we get another musical sequence. (Another way to think of this relationship is that a musical sequence is self-similar under inflation and deflation.)

53. Explain why no musical sequence contains .

54. Explain why no musical sequence contains .

55. Explain why no musical sequence ends in .

56. Explain why no musical sequence ends in .

57. Show that apart from the lone sequence , every musical sequence is an initial subsequence of all the musical sequences that are successive inflations of it.

58. Exercises 53–56 give necessary conditions for a sequence to be musical. Are those conditions sufficient? That is, if a sequence does not contain or , and it does not end in or , must it be a musical sequence? (This is the kind of question that mathematicians ask in an effort to pin down the characteristics of a musical sequence.)

59. We show how to check whether a finite block of s and s can be a part (subsequence) of some musical sequence or else is never a part of any musical sequence. We do so by deflating the block: If the deflation arrives at a single symbol, the block is part of a musical sequence; if the deflation cannot arrive at a single symbol, the block is not part of any musical sequence. However, we use a slightly modified form of deflation:

Check the two blocks below. Is either a part of a musical sequence?

60. (For students who have covered Chapter 19.) From Exercise 53, we know that each application of inflation to a musical sequence simply extends it. By successive inflation, then, we can build an infinite sequence. Argue that as we approach this limiting sequence, the ratio of s to s tends toward the golden ratio φ of page 780.

61. (For students who have covered Chapter 19.)

Conclude from Exercise 56 that the sequence cannot be periodic or settle into a periodic repetition after a finite “burn-in” period. Thus, the sequence is nonperiodic. (Hint: φ is not a rational number; that is, it cannot be represented as a ratio / of whole numbers and .)

62. (For students who have covered Chapter 19.) After applying deflation a large number of times, a large area of a Penrose pattern will contain some number kites and darts. At each deflation, a kite is broken down into two kites and two halves of darts, and a dart is broken down into a single kite and two halves of darts. Thus, each of the kites gives rise to 2 kites, and each of the darts gives rise to kites, for a total of new kites. Similarly, there will be new darts. The ratio of kites to darts will become

Assume that the ratio eventually stabilizes as the number of darts and kites gets larger and larger. What value does it tend to? Hint: Set and solve for .

Chapter Review

63. Specify a formula for the measure of an exterior angle of a regular -gon.

64. Give a numerical reason why a semiregular tiling could not include both regular polygons with 12 sides and regular polygons with 8 sides (with or without any regular polygons with other numbers of sides).

65. The Translation Criterion for tiling of the plane by translations alone offers two possibilities that guarantee such a tiling. State the details of one of these possibilities.

66. State the Conway Criterion in full detail: What kind of tilings does it refer to? What is the condition for such tiling?

67. What is a quasicrystal? What is the connection of quasicrystals to Penrose tilings? What was so revolutionary about Penrose tilings?