Thursday, February 27th: Symmetry & the Golden Ratio
Symmetry & Body Pattern
The Fibonacci Sequence in nature
The Fibonacci Sequence can be found in patterns within plants such as the positions of leaves, the branches on trees, spirals on pineapples, and the seeds in a sunflower. This is no coincidence, though! The pattern maximizes the number of leaves that are exposed to sunlight and the number of seeds that can fit in a space.
The following three videos by Vi Hart explore and explain the Fibonnacci Sequence and Golden Ratio in detail (and at a great speed!).
The following three videos by Vi Hart explore and explain the Fibonnacci Sequence and Golden Ratio in detail (and at a great speed!).
Make a Fibonnacci spiral: 1:08-2:21
Pineapple spirals: 3:39-4:00 |
Advantages to plants: 0:30-3:26
|
And more!
|
The Golden Ratio
From the Fibonacci Sequence comes the Golden Ratio, also called the Golden Section, or Golden Proportion.
Take the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ...
If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers:
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
34/21 = 1.61904...
Each number gets closer and closer to 1.618… an irrational number, also known as the Greek letter Phi Φ.
Take the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ...
If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers:
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
34/21 = 1.61904...
Each number gets closer and closer to 1.618… an irrational number, also known as the Greek letter Phi Φ.
The Golden Ratio on a number line:
If you divide a line into two parts so that: the longer part divided by the smaller part is also equal to the whole length divided by the longer part, then you will have the golden ratio!
It is the ratio of 1:1.618. |
Click the image above to visit a simulation where you can explore the proportions of a rectangle and make a golden ratio.
|
Fibonacci Numbers are the building blocks of the Golden Ratio
Take the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …
Start with a 1x1 square. Add another 1x1 square. Then add a square with side lengths equal to the length longest rectangle side- 2x2. What remains is a golden rectangle. Then add a square with side lengths equal to the length longest rectangle side- 3x3. What remains is a golden rectangle. Then add a square with side lengths equal to the length longest rectangle side- 5x5. What remains is a golden rectangle. Then add a square with side lengths equal to the length longest rectangle side- 8x8. This could go on forever! We can build larger and larger golden rectangles by building on smaller ones with Fibonacci numbers. If you think about where the squares are getting added, they are wrapping around the first two squares, building a spiral.
Start with a 1x1 square. Add another 1x1 square. Then add a square with side lengths equal to the length longest rectangle side- 2x2. What remains is a golden rectangle. Then add a square with side lengths equal to the length longest rectangle side- 3x3. What remains is a golden rectangle. Then add a square with side lengths equal to the length longest rectangle side- 5x5. What remains is a golden rectangle. Then add a square with side lengths equal to the length longest rectangle side- 8x8. This could go on forever! We can build larger and larger golden rectangles by building on smaller ones with Fibonacci numbers. If you think about where the squares are getting added, they are wrapping around the first two squares, building a spiral.
The Golden Spiral
The Golden Spiral is made by drawing a smooth arc from corner to corner though the squares in the golden section. It is also found in spiral galaxies, nautilus shells, in the shape of hurricanes, in the cochlea of the inner ear, and in the proportions of animal bodies.
The Golden Ratio brings beauty t0 both Art & Architecture
Artists and architects have used the golden proportion to make objects more appealing to the human eye for centuries.
Can you find the golden ratio in your favorite works of art?
Can you find the golden ratio in your favorite works of art?
Download copies of handout here:
|
|