19.1 Fibonacci Numbers and the Golden Ratio

1. Examine the “scales” on the surface of a pineapple, which are arranged in spirals around the fruit in three distinct directions. For each direction, how many spirals are there?

2. Repeat Exercise 1, but for a pinecone from your area.

3. Repeat Exercise 1, but for a sunflower. (If you can’t obtain one, then count the spirals in Figure 19.1b.)

4. Here are two primitive models of natural increase of biological populations, similar to those that Fibonacci hypothesized around the year 1200, based on the following situation: A pair of newborn male and female rabbits is placed in an enclosure to breed.

5. Put the golden ratio into the memory of your calculator.

6. The golden ratio satisfies the equation . Show that also satisfies the equation, so that is the other solution to the equation.

815

7. The geometric mean has interpretations in both arithmetic and geometry.

8. What is the geometric mean of 3, 9, 81, and 243?

9. What is the geometric mean of 2, 4, 8, 16, and 32? (Such a sequence, in which each successive number is the same constant times the previous one, is called a geometric sequence.)

10. Using its recursion rule, extend the Fibonacci sequence to the left. For example, the value of must satisfy , or , so . Using the same idea, find .

11. Another sequence closely related to the Fibonacci sequence is the Lucas sequence, which is formed using the same recursive rule but different starting numbers. The th Lucas number is given by

What do you notice?

12. Repeat Exercise 11, but start with the pair of numbers 1 and 4.

13. Repeat Exercise 11, but start with a pair of numbers of your choice. Based on your result and those in Exercises 11 and 12, what is your hunch?

14. For a sequence specified by a recursive rule, finding an explicit expression for the th term is not easy, nor is the form necessarily simple. An exact expression for the th term of the Fibonacci sequence is given by the Binet formula:

15. For two positive numbers and , show that the arithmetic mean is always greater than or equal to the geometric mean. Try some values for and and convince yourself, then demonstrate algebraically that it is true in general. When does equality hold? [Hint: Suppose that the claim is false, so that . Square both sides of the inequality, bring all terms to one side, factor, and observe a contradiction.]

16. You may remember having to work problems like, “If Joe can dig a ditch in 3 days, and Sam can dig it in 4, how long will it take the two of them working together?” The answer is related to the harmonic mean of 3 and 4. The Environmental Protection Agency uses the harmonic mean to calculate the “average” fuel economy of the fleet of cars from a manufacturer. The formula for the harmonic mean of two numbers and is

816

17. New houses are to be built along one side of a street (Leonardo’s Lane), divided into equal-sized lots. Each house is either a single-family detached house, taking up one lot, or a duplex, taking up two lots. Suppose that there are lots on the street. How many different arrangements (orderings) of houses are there, for , and in general? (This exercise was inspired by a puzzle by Paul Dixon at the website by Ron Knott, cited in the Suggested Websites at the end of this chapter.)

18. When it snows in the winter, the local school district superintendent must decide by 5 A.M. whether to declare a snow day and cancel school. The 900 faculty and staff are now notified by a robocall broadcast, but formerly a binary “telephone tree” was used, in which the superintendent called two people and each person who received a call called two others. Suppose that each call takes exactly 1 minute.

(This exercise was inspired by a puzzle at the website by Ron Knott.)

19. Here is a trick to “prove” that you can calculate faster than a person with a calculator. Turn your back and ask a friend to write down any two positive integers, then add them to get a third, then add the second and third to get a fourth, and so on, adding each time the last two integers until there are 10 numbers. Have your friend show you the list, whereupon you write down right away the total of all 10, while your friend begins to add them up on the calculator (to prove that you’re right). The secret: The total is always 11 times the seventh number, and multiplying by 11 is fairly easy to do in your head—just add each pair of neighboring digits, carrying if necessary. Suppose that your friend writes down and as the first two numbers. Show that indeed the total of all 10 numbers is 11 times the seventh number. (Adapted from Martin Gardner, Mathematical Circus, Knopf, New York, 1979.)

20. The game of Fibonacci Nim begins with counters. Two players take turns removing at least one counter, but no more than twice as many as the opponent just did. The winner is the player who takes the last counter. One other rule: The first player may not win immediately by taking all the counters on the first turn. (Adapted from Martin Gardner, Mathematical Circus, Knopf, New York, 1979.)

21. Shari Lynn Levine, as a high school student, published an article in The Fibonacci Quarterly that investigated the “Beta-nacci” sequence that results if, instead of bearing one pair of baby rabbits per month, mature rabbits bear two pairs every month, starting when they reach two months of age. Here, we ask you to rediscover some of Shari’s results.

22. Generalize Exercise 21, parts (a) through (d), as follows:

817

23. Use the quadratic formula to find expressions in terms of square roots for the silver, bronze, copper, and nickel means, and approximate these to three decimal places. Find a general expression in terms of a square root for the th metallic mean.

24. Just as the golden mean arises as the limiting ratio of consecutive terms of the Fibonacci sequence, each of the metallic means arises as the limiting ratio of consecutive terms of generalized Fibonacci sequences. A generalized Fibonacci sequence can be defined by

where and are positive integers. The Fibonacci sequence itself is the case .

19.2 Rosette and Strip Patterns

25. Determine whether each of the following statements is always or sometimes true. Drawing some sketches may be helpful.

26. Determine whether each of the following statements is always or sometimes true. Drawing some sketches may be helpful.

27. Which of the capital letters of the alphabet, when drawn in the most symmetrical way, have the following symmetries? For example, assume that the upper and lower loops of B are the same size.

28. Repeat Exercise 27 for the lowercase letters.

29. In The Complete Walker III (3d ed., Knopf, New York, 1984, p. 505), Colin Fletcher’s answer to “What games should I take on a backpacking trip?” is the game he calls “Colinvert”: “You strive to find words with meaningful mirror (or half-turn) images.” Some of the words he found are as follows:

30. Repeat Exercise 29, but for words written vertically instead of horizontally.

31. For each of the following patterns, identify the rigid motions that preserve the pattern:

32. Repeat Exercise 31, but for the following patterns:

19.3 Notation for Patterns

33. What is the notation for the symmetry pattern of a regular pentagon (which has all five sides equal)—, , or something else?

818

34. What is the notation for the symmetry pattern of a snowflake with six symmetrical arms, with each arm having mirror symmetry?

35. Give the notation (such as or ) for the symmetry patterns of the rosettes in hubcaps (a) through (c) below, disregarding the logos in the centers.

36. Repeat Exercise 35 for hubcaps (d) through (f).

37. Repeat Exercise 35 for corporate logos (a) through (c) below. (Bonus: Can you identify the corporations?)

38. Repeat Exercise 35 for automobile logos (d) through below.

39. Identify the rosette pattern for

40. Which rosette pattern would result if the prototype, unlike the one above, has reflection symmetry across its diagonal from the top left to the lower right and

41. Use the flowchart in Figure 19.13 (page 799) to identify the notation for the types of strip patterns from the pottery and basketry shown in the illustrations on the next page.

819

42. In each of the four accompanying examples, two adjacent triangles of an infinite strip are shown. (Contributed by Margaret A. Owens, California State University, Chico.)

For each example:

43. Repeat Exercise 41 for the accompanying eight strip patterns, all of which appear on the brass straps for a single lamp from 19th-century Benin in West Africa. (From H. Ling Roth, in Great Benin.)

Pottery and basketry from the Americas. (a), (b) Mexico, modern.

(c) Lower Central America, pre-Columbian.

(d) Pomo people, California, early 20th century.

(e) Woodland Indians, central North America, early 20th century.

(f) Pomo people, California, mid-20th century; originally from the collection of Dr. Herbert Zim, editor of the Golden Guides series of nature books.

(g) Woodland Indians, central North America, early 20th century.

Note that the patterns are roughly carved, so you will need to discern the intent of the artist.

820

44. Repeat Exercise 41 for the accompanying patterns from San Ildefonso Pueblo, New Mexico.

45. The following table shows comparative data about the frequency of occurrence of strip designs of various types on Chinese porcelain and smoking pipes (Begho, in what is now Ghana) from two continents.

A-41

46. For the Nigerian Yoruba cloths (a) and (b) in the following illustration, use the flowchart in Spotlight 19.9 (page 797) to identify (by notation) the type of wallpaper pattern.

821

Patterns on Yoruba (West Africa) adire cloth, made by starching a pattern onto white cloth, then dyeing the cloth before rinsing out the starch, so that the starched portion remains as a white design against a colored background.

47. For the Nigerian Yoruba cloths (c) and (d) in the accompanying illustration, use the flowchart in Spotlight 19.9 (page 797) to identify (by notation) the type of wallpaper pattern.

48. For the floor tilings (a) and (b) below, use the flowchart in Spotlight 19.9 (page 797) to identify (by notation) the type of wallpaper pattern.

49. For the floor tilings (c) and (d) that follow, use the flowchart in Spotlight 19.9 (page 797) to identify (by notation) the type of wallpaper pattern.

50. The triangles in the grid at the top of the following figure show the beginning steps in forming instances of several of the wallpaper patterns by putting together a vertical motion and a horizontal motion.

51. Which of the 17 wallpaper patterns can be formed by the technique used in Exercise 50?

52.Figure 20.10 (page 837)

53.Figure 20.11 (page 838)

54.Figure 20.16 (page 841)

55. The figure in Spotlight 19.9 (page 836)

822

56. The hexagonal regular tiling at upper right in Figure 20.5 (page 831)

57. The convex hexagon tiling of type 3 in Figure 20.9 (page 835)

58. Visit the website escher.epfl.ch/escher/, which features an interactive Java program called Escher Web Sketch. Experiment with choosing wallpaper patterns using crystallographic notation. For each one, draw on the screen a colored design for the motif; the program will reproduce the motif using the pattern.

19.4 Symmetry Groups

59. For positive integers and , the expression mod means the remainder when is divided by . Thus, , because , and we say that “23 is equivalent to 3 modulo 4.” (See Chapter 17 for further details about this modular arithmetic.) Every positive integer is equivalent to 0, 1, 2, or 3 modulo 4. Consider the collection of elements {0, 1, 2, 3} and the operation on them defined by mod 4. Show that under this operation, the collection forms a group.

60. Explain, by referring to the properties of a group, whether the collection of all real numbers is a group under the operation of (a) addition and (b) multiplication.

61. Explain why the table for the operation * below shows that the elements indicated do not form a group under *.

62. Consider the table below for an operation #.

63. Construct the table for the mattress group of Example 6 on page 804, putting down the side of the table the first turn made and, across the top of the table, the subsequent turn.

64. The mattress group is a commutative group, meaning that the order of the turns doesn’t make any difference. How can you tell that from the table of Exercise 63? [A commutative group is sometimes called an abelian group, after Niels Henrik Abel (1801–1829), a Norwegian mathematician who died of tuberculosis at a young age.]

65. A problem about mattress turning is that people usually don’t remember the immediately previous position or immediately previous turn that they made months ago. So the next time, they could wind up just turning the mattress back to the position that it was in only two seasons ago. Show that the mnemonic “Spin in spring, flip in fall” (courtesy of Bill Sandidge of Atlanta, Georgia), in fact, cycles the mattress through its four positions.

66. A king mattress is officially , which is almost square. Make yourself a “square mattress” from a sheet of paper and investigate its group.

67. Show that the collection of numbers {1, 3, 5, 7} under multiplication modulo 8 (see Exercise 59) has the structure of the Klein 4-group (page 805).

68. Show that the dihedral rosette group has the structure of the Klein 4-group.

823

69. Show that successive tire rotations using scheme A form a group. Is it the Klein 4-group (page 805)?

70. Observe that repeating scheme B will never take the front right tire to the back right wheel. Is there any combination of schemes A and C that will produce the result of scheme B?

71. A tire rotation scheme is designed so that no tire will remain where it was. Such a rearrangement (permutation) of objects is called a derangement. Two more tire rotation schemes are shown below. Are there still more schemes that are derangements? One way to record a tire rotation is to label the original tire positions clockwise from the left front: 1 for left front, 2 for right front, 3 for right rear, and 4 for left rear. Record the results of a scheme by writing in turn where each tire goes; for example, scheme D below produces the derangement 4321 because tire 1 goes to position 4, tire 2 goes to position 3, and so forth.

72. Do the five tire rotation schemes A, B, C, D, and E, plus the identity rotation and any others that you found in Exercise 69, form a group? Why or why not?

73. For the traditional North American beadwork shown below, answer the following questions.

74. Repeat Exercise 73 for the Plains Indian embroidery shown below.

75. What are the elements for the group of symmetries of a square?

76. Using the notation of rosette patterns, describe the group of symmetries of

77. (a) Give a numerical example to show that the operation of subtraction on the integers is not associative.

(b) Repeat part (a), but for division on the positive real numbers.

78. What are the elements of the group of symmetries of

79. What are the elements of the group of symmetries of

80. What are the elements of the group of symmetries of the dihedral pattern ? (See the flower in Figure 19.10a on page 794.)

81. What are the elements of the group of symmetries of the cyclic pattern ?

82. What is the group of symmetries of a cube?

83. What is the group of symmetries of a general rectangular solid (its length, width, and height are all unequal)?

19.5 Fractal Patterns and Chaos

84. Explore Sprott’s Fractal Gallery at sprott. sprott.physics.wisc.edu/fractals.htm, which features a “Fractal of the Day” and accompanying fractal music. There are various “rooms” in the gallery—including “Iterated Function Systems,” “Natural Fractals” (we particularly like “Broccoli” and “Trees”), and “Publication Quality Attractors” (“SMKBNZQA” is our favorite)— together with PC programs for generating such fractals. What are your favorites, and why?

85. Explain how the pattern of the following illustration is fractal.

86. The website csdt.rpi.edu/African/MANG_DESIGN/culture/mang_homepage.html offers information about a fractal-patterned ivory hatpin from the Mangbetu culture in Africa. The site includes a tutorial on producing similar designs using reflection, rotation, translation, and scaling. Work your way through the tutorial and then create a Mangbetu-style artifact.

87. Cornrow hairstyles are fractal in nature. At the website csdt.rpi.edu/african/CORNROW_CURVES/, you can see how and why, including a tutorial on designing cornrow hairstyles using reflection, rotation, translation, and scaling. Work your way through the tutorial and then create a hairstyle. The website also includes instructions for actual braiding, with a short video.

88. Download fractal-creation software and accompanying documentation and use the software to create your own fractal. Links to software, most of it free, are available at www.Nahee.com/PNL/Fractal_Software.html and at fractalfoundation.org/resources/fractal-software/. Recommended software:

For Windows: Fractint, from www.Nahee.com/spanky/www/fractint/fractint.html

For Macintosh: XaoS, a fractal zoomer, from fractalfoundation.org/resources/fractal-software/.

Chapter Review

89. You have 42 in. of string that you want to use as the perimeter of a rectangle whose sides have lengths in inches that are Fibonacci numbers. What are the dimensions of your rectangle?

90. Identify the rigid motions that preserve the pattern of DDDDDDDD.

91. What is the notation for the symmetry pattern of DDDDDDDDD?

92. What is the notation for the symmetry pattern of the pinwheel below?

93. For the floor tiling below, use the flowchart in Spotlight 19.9 (page 797) to identify (by notation) the type of wallpaper pattern.

94. Do the real numbers form a group under division?

95. What are the elements of the group of symmetries of rosette pattern ?

825