Math 105, Topics in Mathematics

Introduction

10.1 Fibonacci Numbers and the Golden Ratio

10.2 Some Geometry

Homework

Introduction

10.1 Fibonacci Numbers and the Golden Ratio

Definition: A list of numbers

a₁, a₂, a₃, ... , a_n, ...

is called a sequence of numbers. The first term of this sequence is a₁, the second term of the sequence is a₂, and so on.

Definition: The sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

is called the Fibonacci sequence, and these numbers are called the Fibonacci numbers.

The first term of the sequence is F1 = 1. The 2nd term of the sequence is F2 = 1. The 3rd term of the sequence is F3 = 2. The 4th term of the sequence is F4 = 3. The 5th term of the sequence is F5 = 5. The 6th term of the sequence is F6 = 8. The 7th term of the sequence is F7 = 13. The 8th term of the sequence is F8 = 21. The 9th term of the sequence is F9 = 34. The 10th term of the sequence is F10 = 55. The 11th term of the sequence is F11 = 89. The 12th term of the sequence is F4 = 144. ... ... ... N) In general, FN denotes the Nth-term of the sequence

What is the value of F_N? To answer that, you have to understand the pattern of the sequence. Note that

1. 1 =F₁
2. 1 =F₂
3. 2 = F₃ =1 + 1 = F₁+ F₂
4. 3 = F₄ =1 + 2 = F₂+ F₃
5. 5 = F₅ =2 + 3 = F₃+ F₄
6. 8 = F₆ =3 + 5 = F₄+ F₅
7. 13 = F₇ =5 + 8 = F₅+ F₆
8. 21 = F₈ = 8 + 13 = F₆+ F₇
9. 34 = F₉ =13 + 21 = F₇+ F₈
10. 55 = F₁₀ = 21 + 34 = F₈+ F₉
11. 89 = F₁₁ =34 + 55 = F₉+ F₁₀
12. 144 = F₁₂ =55 + 89 = F₁₀+ F₁₁

We observe that, except for F₁ and F₂, each term is the sum of the two preceding terms. We have

F1 =1

F2 = 1

FN = FN-2+ FN-1 (for N>2).

The Formula: The Fibonacci numbers can be directly computed by the Binet's Formula

FN= ( ((1 + 51/2 )/2 )N - ((1 - 51/2 )/2)N)/ 51/2

Definitions and remarks

1. The number Φ = (1 + 5^1/2 )/2 is called the golden ratio.
2. According to the calculator, approximately, Φ = 1.618033989.
3. Let us also define another number Θ = (1 - 5^1/2 )/2.
4. According to the calculator, approximately, Θ = - 0.6180339887.
5. Remark: Note that these two numbers Φ, Θ are the two solutions of the quadratic equation x² = x+1.
6. Remark: One can also check that Φ ^N = F_NΦ +F_N-1.
7. Remark: The golden number Φ and the Fibonacci numbers appear in nature, geometry, art, and architecture.

Problems on 10.1: Fibonacci Numbers

Exercise 10.1.1. Compute Fibonacci numbers F13, F14, F15, ... , F19. Solution: F13 = F11 + F12 = 89 + 144 = 233. F14 = F12 + F13 = 144 + 233 = 377.

Exercise 10.1.2. Use Binet's formula (with calculator) to compute F23, F24, F25, F26. Remark: Note that Fibonacci numbers are all whole numbers. Calculators do not understand irrational numbers like 51/2, and it also does some standard rounding. Thus, you may get some approximate values of the Fibonacci numbers, which may not be whole numbers, when you use this method.

10.2 Some Geometry and Gnomons

Recall that two geometric figures (two dimensional) A and B are said to be similar if one is a scaled down version of the other.

Similar Triangles
Two triangles T1 and T2 are similar if their sides are similar. Alternately, two triangles T1 and T2 are similar if the respective angles of T1 and T2 are the same.
Similar Squares
Two squares S1 and S2 are always similar.
Similar Rectangles
Two rectangles R1 and R2 are similar if their sides are proportional. That means, (long side of rectangle R1) / (long side of rectangle R2) = (short side of rectangle R1) / (short side of rectangle R2)
Similar Circles
Two circles C1 and C2 are always similar.
Similar Circular Rings
Two circular rings CR1 and CR2 are similar if their inner and outer radii are proportional. That means, (Outer radius of CR1) / (Outer radius of CR2) = (inner radius of CR1) / (inner radius of CR2)

Definition: A gnomon to a figure A, is a connected figure B, which, when suitably attached to A produces a new figure that is similar to A. If "G&A" is similar to A, then G is a gnomon to A.

Problems on 10.2: Gnomons

Exercise 10.2.1. Problem

Exercise 10.2.2. Problem

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