Chapter 13 Fibonacci Numbers and the Golden Ratio
Chapter 13: Fibonacci Numbers and the Golden Ratio Section 13. 3: Gnomons ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 1
Similarity We know from geometry that two objects are said to be similar if one is a scaled version of the other. The following important facts about similarity of basic two-dimensional figures will come in handy later in the chapter: • Triangles: 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 corresponding sides are proportional. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 2
Similarity (cont) 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 called the scaling factor (see figure). When k > 1, triangle 2 is larger than triangle 1; when 0 < k < 1, triangle 2 is smaller than triangle 1. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 3
Similarity (cont) • Squares: Two squares are always similar. • Rectangles: Two rectangles are similar if their corresponding sides are proportional (see figure). ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 4
Similarity (cont) • Circles and disks: Two circles are always similar. Any circular disk (a circle plus all its interior) is similar to any other circular disk. • Circular rings: Two circular rings are similar if and only if their inner and outer radii are proportional (see figure). ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 5
Similarity (cont) Always Similar – Ratios of Sides/radii are always matching Squares, Circles, Equilateral Triangles Sometimes Similar – Correspondnig Angles need to match AND Ratios of corresponding sides need to match. (Does not always happen) Triangles, Rectangle, Rings, etc… ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 6
Gnomons 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. Informally, we will describe it this way: G is a gnomon to A if G & A is similar to A (see figure). Here the symbol “&” should be taken to mean “attached in some suitable way. ” ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 7
Example – Gnomons to Squares Consider the square S in Fig. a. The L-shaped figure G in Fig. b is a gnomon to the square—when G is attached to S as shown in Fig. c, we get the square Sʹ. Note that the wording is not reversible. The square S is not a gnomon to the L-shaped figure G, since there is no way to attach the two to form an L-shaped figure similar to G. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 8
Example – Gnomons to Circular Disks Consider the circular disk C with radius r in figure a. The O-ring G in figure b with inner radius r is a gnomon to C. Clearly, G & C form the circular disk Cʹ shown in figure c. Since all circular disks are similar, Cʹ is similar to C. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 9
Example – Gnomons to Rectangles Consider a rectangle R of height h and base b as shown in figure a. The L-shaped figure G shown in figure b can clearly be attached to R to form the larger rectangle Rʹ shown in figure c. This does not, in and of itself, guarantee that G is a gnomon to R. The rectangle Rʹ [with height (h + x) and base (b + y)] is similar to R if and only if their corresponding sides are proportional, which requires that b/h = (b + y)/ (h + x) With a little algebraic manipulation, this can be simplified to b/h = y/x. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 10
Example – A Golden Triangle In this example, we are going to do things a little bit backwards. Let’s start with an isosceles triangle T, with vertices B, C, and D whose angles measure 72 o, and 36 o, respectively, as shown in figure a. On side CD we mark the point A so that BA is congruent to BC [Fig. b]. (A is the point of intersection of side CD and the circle of radius BC and center B. ) For convenience, we will call the triangle ABC [the light blue triangle in figure b] Tʹ. Since Tʹ is an isosceles triangle, the measure of angle BAC equals the measure of angle BCA (72 o), so it follows that angle ABC measures 36 o. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 11
Example – A Golden Triangle (cont) This implies that triangle Tʹ has angles equal to those of triangle T, and thus they are similar triangles. “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 Tʹ—it is triangle BAD, labeled Gʹ in figure c. After all, Gʹ & Tʹ give T—a triangle similar to Tʹ. Note that Gʹ is an isosceles triangle with angles that measure 36 o, and 108 o. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 12
Example – A Golden Triangle (cont) We now know how to find a gnomon not only to triangle Tʹ but also to any 72 -72 -36 triangle, including the original triangle T: Attach a 36 -36 -108 triangle to one of the longer sides [Fig. a]. If we repeat this process indefinitely, we get a spiraling series of ever-increasing 72 -72 -36 triangles [Fig. b]. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 13
Example – A Golden Triangle (cont) This example 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 3636 -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. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 14
Example – Square Gnomons to Rectangles Consider a rectangle R with sides of length B and S [Fig. a], and suppose that the square G with sides of length B shown in figure b is a gnomon to R. If so, then the rectangle Rʹ shown in figure c must be similar to R, which implies that their corresponding sides must be proportional: . If this proportion looks familiar, it’s because it is the divine proportion we first discussed in section 13. 2 and whose only solution is B/S = ϕ. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 15
Example – Square Gnomons to Rectangles (cont) This example tells us that the only way that a rectangle can have a square gnomon is if its sides are in a divine proportion (i. e. , B/S = ϕ where B and S are the lengths of the bigger and shorter sides, respectively. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 16
Golden and Fibonacci Rectangles 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 B and S satisfying B/S = ϕ. A close relative to a golden rectangle is a Fibonacci rectangle—a rectangle whose sides are consecutive Fibonacci numbers. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 17
Example – Golden and Almost Golden Rectangles The figures on this slide and the next show an assortment of rectangles (please note that the rectangles are not drawn to the same scale). Some are golden, some are close. • The rectangle in figure a has B = 1 and S =. Since , this is a golden rectangle. • The rectangle in figure b has B = ϕ+ 1 and S = ϕ. Here Since ϕ + 1 =ϕ 2, this is another golden rectangle. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 18
Example – Golden and Almost Golden Rectangles (cont) • The rectangle in figure c has B = 8 and S = 5. This is a Fibonacci rectangle, since 5 and 8 are consecutive Fibonacci numbers. The ratio of the sides is B/S = 8/5 = 1. 6, so this is not a golden rectangle. On the other hand, the ratio 1. 6 is reasonably close to ϕ= 1. 618. . . , so we will think of this rectangle as “imperfectly golden. ” • The rectangle in figure d with B = 89 and S = 55 is another Fibonacci rectangle. Since 89/55 = 1. 61818. . . , this rectangle is as good as golden—the ratio of the sides is the same as the golden ratio up to three decimal places. ALWAYS LEARNING Copyright © 2018, 2014, 2010 Pearson Education Inc. Slide 19
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