Lesson 15: Proportion and the Golden Ratio

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.title[
# Lesson 15: Proportion and the Golden Ratio
]
.author[
### Fei Ye
]
.date[
### May, 2024
]

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## Unit 11C: Proportion and the Golden Ratio

*Primary Source:* PPT for the book "Using & Understanding Mathematics".

---

## The Golden Ratio (1/2)

> Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to gold: the second we may name a precious jewel.
>
> Johannes Kepler (1571-1630)

---

## The Golden Ratio (2/2)

.center[
![](data:image/png;base64,#img/GoldenRatioLine.png)
]

The ratio of the length of the long piece to the length of the short piece is the same as the ratio of the length of the entire line segment to the length of the long piece.

$$
\dfrac{L}{l}=\dfrac{L+l}{L}
$$

For any line segment divided into two pieces according to the golden ratio, `$\phi$`, the ratio of the long piece to the short piece is approximately 8/5.

`$$\phi=\frac{1+\sqrt{5}}{2}=1.61803 \ldots \approx 1.62 \approx \frac{8}{5}$$`

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## Parthenon Proportions and the Golden Ratio

The golden rectangle is a rectangle whose long side is `$\phi$` times as long as its short side.

.center[
![](data:image/png;base64,#img/GoldenRatioRectangle.png)
]

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## Example: Household Golden Ratios (1/2)

Consider the following household items with the given dimensions. Which item comes closest to having the proportions of the golden ratio?

- Standard sheet of paper: 8.5 in `$\times$` 11 in

- 8 `$\times$` 10 picture frame: 8 in `$\times$` 10 in

- HDTV (high-definition television), which comes in many sizes but always with a 16:9 ratio of width to height

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## Example: Household Golden Ratios (2/2)

**Solution:** The ratio of the sides of a standard sheet of paper is `$11/8.5 \approx  1.29$`, which is 20% less than the golden ratio.

The ratio of the sides of a standard picture frame is `$10/8 = 1.25$`, which is 23% less than the golden ratio.

The HDTV ratio is `$16/9 \approx 1.78$` which is about 10% more than the golden ratio.
Of the three objects, the high-definition television is closest to being a golden rectangle.

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## The Golden Ratio in Nature

.center[
A logarithmic spiral

![](data:image/png;base64,#img/Spiral.png)
]

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## The Fibonacci Sequence

The Fibonacci sequence is a sequence that continues to grow with the following pattern:

.center[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...]

The next number in the sequence is the sum of the previous two numbers.

`$$F_{n+1} = F_n + F_{n-1}.$$`

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## Example: Population of Rabbits (1/2)

How many pairs of rabbits will there be at at the beginning of the 8-th month?
.center[
![:resize , 70%](data:image/png;base64,#img/Rabbits.png)
]

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## Example: Population of Rabbits (2/2)

**Solution:** According the the pattern in the picture, the number of pairs forms a Fibonacci sequence. After a month, the number of pairs is the sum of the numbers of pairs in pervious two months.

At the beginning of the 5-th month and the 6-th month, there are 5 and 8 pairs respectively. Then at the beginning of the 7-th month, there will be `$5+8=13$` pairs.

At the beginning of the 8-th month, there will be `$8+13=21$` pairs.

---

## Fibonacci Numbers and the Golden Ratio

.center[
  ![](data:image/png;base64,#img/FibonacciNumbers.png)
]

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## Practice: Divide a Line by Golden Ratio

A line segment of 5 cm is subdivided according to the golden ratio. Find the length of the smaller piece.

---

## Practice: Dimension of Golden Rectangle

The longer side of a golden rectangle has the length 4.5 inches. Find the area of the rectangle.

---

## Practice: Fibonacci Sequence

Based on the pattern of the following sequence,

.center[
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
]

what is the most probable next number in the sequence?