2. Excursions in Modern Mathematics, 7e: 9.4 - 2Copyright © 2010 Pearson Education, Inc. 9 The Mathematics of Spiral Growth 9.1Fibonacci’s Rabbits 9.2Fibonacci Numbers 9.3 The Golden Ratio 9.4Gnomons 9.5Spiral Growth in Nature

3. Excursions in Modern Mathematics, 7e: 9.4 - 3Copyright © 2010 Pearson Education, Inc. The most common usage of the word gnomon is to describe the pin of a sundial– the part that casts the shadow that shows the time of day. The original Greek meaning of the word gnomon is “one who knows,” so it’s not surprising that the word should find its way into the vocabulary of mathematics. Gnomons

4. Excursions in Modern Mathematics, 7e: 9.4 - 4Copyright © 2010 Pearson Education, Inc. In this section we will discuss a different meaning for the word gnomon. Before we do so, we will take a brief detour to review a fundamental concept of high school geometry–similarity. Gnomons

5. Excursions in Modern Mathematics, 7e: 9.4 - 5Copyright © 2010 Pearson Education, Inc. We know from geometry that two objects are said to be similar if one is a scaled version of the other. (When a slide projector takes the image in a slide and blows it up onto a screen, it creates a similar but larger image. When a photocopy machine reduces the image on a sheet of paper, it creates a similar but smaller image.) The following important facts about similarity of basic two- dimensional figures will come in handy later in the chapter: Similar Figures

6. Excursions in Modern Mathematics, 7e: 9.4 - 6Copyright © 2010 Pearson Education, Inc. Two triangles are similar if and only if the measures of their respective angles are the same. Alternatively, two triangles are similar if and only if their sides are proportional. In other words, if Triangle 1 has sides of length a, b, and c, then Triangle 2 is similar to Triangle 1 if and only if its sides have length ka, kb, and kc for some positive constant k. Similar Figures - Triangles

7. Excursions in Modern Mathematics, 7e: 9.4 - 7Copyright © 2010 Pearson Education, Inc. Two squares are always similar. Two rectangles are similar if their corresponding sides are proportional. Similar Figures - Squares - Rectangles

8. Excursions in Modern Mathematics, 7e: 9.4 - 8Copyright © 2010 Pearson Education, Inc. Two circles are always similar. Any circular disk (a circle plus all its interior) is similar to any other circular disk. Two circular rings are similar if and only if their inner and outer radii are proportional Similar Figures - Circles - Disks - Rings

9. Excursions in Modern Mathematics, 7e: 9.4 - 9Copyright © 2010 Pearson Education, Inc. In geometry, a gnomon G to a figure A is a connected figure that, when suitably attached to A, produces a new figure similar to A. By “attached,” we mean that the two figures are coupled into one figure without any overlap. Gnomon

10. Excursions in Modern Mathematics, 7e: 9.4 - 10Copyright © 2010 Pearson Education, Inc. Informally, we will describe it this way: G is a gnomon to A if G & A is similar to A. Here the symbol “&” should be taken to mean “attached in some suitable way.” Gnomon

11. Excursions in Modern Mathematics, 7e: 9.4 - 11Copyright © 2010 Pearson Education, Inc. Consider the square S. The L-shaped figure G is a gnomon to the square–when G is attached to S as shown, we get the square S’. Example 9.3Gnomons to Squares

12. Excursions in Modern Mathematics, 7e: 9.4 - 12Copyright © 2010 Pearson Education, Inc. Consider the circular disk C with radius r. The O-ring G with inner radius r is a gnomon to C. Clearly, G & C form the circular disk. Since all circular disks are similar, C’ is similar to C. Example 9.4Gnomons to Circular Disks

13. Excursions in Modern Mathematics, 7e: 9.4 - 13Copyright © 2010 Pearson Education, Inc. Consider a rectangle R of height h and base b. The L-shaped figure G can clearly be attached to R to form the larger rectangle. This does not, in and of itself, guarantee that G is a gnomon to R. Example 9.5Gnomons to Rectangles

14. Excursions in Modern Mathematics, 7e: 9.4 - 14Copyright © 2010 Pearson Education, Inc. The rectangle R’ [with height (h + x) and base (b + y)] is similar to R if and only if their corresponding sides are proportional, which Example 9.5Gnomons to Rectangles requires that This can be simplified to

15. Excursions in Modern Mathematics, 7e: 9.4 - 15Copyright © 2010 Pearson Education, Inc. There is a simple geometric way to determine if the L-shaped G is a gnomon to R–just extend the diagonal of R in G & R. If the extended diagonal passes through the outside corner of G, then G is a gnomon; if it doesn’t, then it isn’t. Example 9.5Gnomons to Rectangles

16. Excursions in Modern Mathematics, 7e: 9.4 - 16Copyright © 2010 Pearson Education, Inc. Let’s start with an isosceles triangle T, with vertices B, C, and D whose angles measure 72º, 72º, and 36º, respectively. On side CD we mark the point A so that BA is congruent Example 9.6A Golden Triangle to BC. (A is the point of intersection of side CD and the circle of radius BC and center B.)

17. Excursions in Modern Mathematics, 7e: 9.4 - 17Copyright © 2010 Pearson Education, Inc. Since T’ is an isosceles triangle, angle BAC measures 72º and it follows that angle ABC measures 36º. This implies that triangle T’ has equal angles as triangle T and thus Example 9.6A Golden Triangle they are similar triangles.

18. Excursions in Modern Mathematics, 7e: 9.4 - 18Copyright © 2010 Pearson Education, Inc. “So what?” you may ask. Where is the gnomon to triangle T? We don’t have one yet! But we do have a gnomon to triangle T’ – it is triangle BAD, labeled G’. After all, G’ & T’ Example 9.6A Golden Triangle is a triangle similar to T’. Note that gnomon G’ is an isosceles triangle with angles that measure 36º, 36º, and 108º.

19. Excursions in Modern Mathematics, 7e: 9.4 - 19Copyright © 2010 Pearson Education, Inc. We now know how to find a gnomon not only to triangle T’ but also to any 72-72-36 triangle, including the original triangle T: Attach a 36-36-108 triangle, G, to one of the longer sides of T. 72-72-36 and 36-36-108 Triangles

20. Excursions in Modern Mathematics, 7e: 9.4 - 20Copyright © 2010 Pearson Education, Inc. If we repeat this process indefinitely, we get a spiraling series of ever increasing 72-72- 36 triangles. 72-72-36 and 36-36-108 Triangles

21. Excursions in Modern Mathematics, 7e: 9.4 - 21Copyright © 2010 Pearson Education, Inc. It’s not too far-fetched to use a family analogy: Triangles T and G are the “parents,” with T having the “dominant genes;” the “offspring” of their union looks just like T (but bigger). The offspring then has offspring of its own (looking exactly like its grand-parent T), and so on ad infinitum. 72-72-36 and 36-36-108 Triangles

22. Excursions in Modern Mathematics, 7e: 9.4 - 22Copyright © 2010 Pearson Education, Inc. Example 9.6 is of special interest to us for two reasons. First, this is the first time we have an example in which the figure and its gnomon are of the same type (isosceles triangles). Second, the isosceles triangles in this story (72-72-36 and 36-36-108) have a property that makes them unique: In both cases, the ratio of their sides (longer side over shorter side) is the golden ratio.These are the only two isosceles triangles with this property, and for this reason they are called golden triangles. Golden Triangles

23. Excursions in Modern Mathematics, 7e: 9.4 - 23Copyright © 2010 Pearson Education, Inc. Consider a rectangle R with sides of length l (long side) and s (short side), and suppose that the square G with sides of length l is a gnomon to R. Example 9.7Square Gnomons to Rectangles

24. Excursions in Modern Mathematics, 7e: 9.4 - 24Copyright © 2010 Pearson Education, Inc. If so, then the rectangle R’ must be similar to R, which implies that their corresponding sides must be proportional (long side of R’ / short side of R´ = long side of R / short side of R): Example 9.7Square Gnomons to Rectangles

25. Excursions in Modern Mathematics, 7e: 9.4 - 25Copyright © 2010 Pearson Education, Inc. After some algebraic manipulation the preceding equation can be rewritten in the form Example 9.7Square Gnomons to Rectangles

26. Excursions in Modern Mathematics, 7e: 9.4 - 26Copyright © 2010 Pearson Education, Inc. Since (1) l/s is positive ( l and s are the lengths of the sides of a rectangle), (2) this last equation essentially says l/s that satisfies the golden property, and (3) the only positive number that satisfies the golden property is , we can conclude that Example 9.7Square Gnomons to Rectangles

27. Excursions in Modern Mathematics, 7e: 9.4 - 27Copyright © 2010 Pearson Education, Inc. We can summarize all the above with the following conclusion: A rectangle with sides of length l and s (long side and short side, respectively) has a square gnomon if and only if Rectangles-Squares and Gnomons

28. Excursions in Modern Mathematics, 7e: 9.4 - 28Copyright © 2010 Pearson Education, Inc. A rectangle whose sides are in the proportion of the golden ratio is called a golden rectangle. In other words, a golden rectangle is a rectangle with sides l (long side) and s (short side) satisfying l/s = . A close relative to a golden rectangle is a Fibonacci rectangle–a rectangle whose sides are consecutive Fibonacci numbers. Golden and Fibonacci Rectangles

29. Excursions in Modern Mathematics, 7e: 9.4 - 29Copyright © 2010 Pearson Education, Inc. This rectangle has l = 1 and s = 1/ . Since l/s = 1/(1/  ) = , this is a golden rectangle. Example 9.8Golden and Almost Golden Rectangles

30. Excursions in Modern Mathematics, 7e: 9.4 - 30Copyright © 2010 Pearson Education, Inc. This rectangle has l =  + 1 and s = . Here l/s = (  + 1)/ . Since  + 1 =  2, this is a golden rectangle. Example 9.8Golden and Almost Golden Rectangles

31. Excursions in Modern Mathematics, 7e: 9.4 - 31Copyright © 2010 Pearson Education, Inc. This rectangle has l = 8 and s = 5. This is a Fibonacci rectangle, since 5 and 8 Example 9.8Golden and Almost Golden Rectangles are consecutive Fibonacci numbers. The ratio of the sides is l/s = 8/5 = 1.6 so this is not a golden rectangle. On the other hand, the ratio 1.6 is reasonably close to so we will think of this rectangle as “almost golden.”

32. Excursions in Modern Mathematics, 7e: 9.4 - 32Copyright © 2010 Pearson Education, Inc. This rectangle has l = 89 and s = 55 and is a Fibonacci rectangle. The ratio of the sides is l/s = 89/55 = 1.61818…, in theory this is not a golden rectangle. In practice, this Example 9.8Golden and Almost Golden Rectangles rectangle is as good as golden–the ratio of the sides is the same as the golden ratio up to three decimal places.

33. Excursions in Modern Mathematics, 7e: 9.4 - 33Copyright © 2010 Pearson Education, Inc. This rectangle is neither a golden nor a Fibonacci rectangle. On the other hand, the ratio of the sides (12/7.44 ≈ 1.613) is very close to the golden ratio. It is safe to say that, sitting on a supermarket shelf, that box of Corn Pops looks temptingly golden. Example 9.8Golden and Almost Golden Rectangles

34. Excursions in Modern Mathematics, 7e: 9.4 - 34Copyright © 2010 Pearson Education, Inc. From a design perspective, golden (and almost golden) rectangles have a special appeal, and they show up in many everyday objects, from posters to cereal boxes. In some sense, golden rectangles strike the perfect middle ground between being too “skinny” and being too “squarish.” Golden Rectangles

35. Excursions in Modern Mathematics, 7e: 9.4 - 35Copyright © 2010 Pearson Education, Inc. A prevalent theory, known as the golden ratio hypothesis, is that human beings have an innate aesthetic bias in favor of golden rectangles, which, so the theory goes, appeal to our natural sense of beauty and proportion. Golden Ratio Hypothesis