The Golden Section
Studying about Fibonacci numbers and the golden ratio makes an excellent project for high school to write a report on. Besides algebra, it ties in with geometry, botany and art at least. Students do projects and reports in history and English and other school subjects - why not do one or two in math too?
The discussion below covers the basics of golden section. I'll try to keep it short.
Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.
Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).
This is how we get this golden section or golden cut:
Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.
From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).
And the golden ratio is the ratio Phi:1.
Golden rectangle
Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.
Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.
And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.
My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.
P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.
The discussion below covers the basics of golden section. I'll try to keep it short.
Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.
Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).
This is how we get this golden section or golden cut:
Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.
S:L = L:W
From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).
And the golden ratio is the ratio Phi:1.
Solving the equation - more details Skip this box if you so wish. Solving this simple-looking equation of golden cut requires using the formula for quadratic equations, so it is a nice exercise for high-schoolers. S:L = L:W is usually written in the form of S/L = L/W Since S+L = W, we can substitute that for W and get: S/L = L/(S+L) And another trick is, since this is just a general line, we can choose for the shorter part S to be 1. After that, the equation looks simple enough: 1/L = L/(1+L) Solving that using the quadratic formula, and discarding the negative root, you get L = (√5 + 1)/2 |
Golden rectangle
Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.
here is one golden rectangle |
Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.
And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.
My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.
- The Golden section in architecture
- Dividing the line in golden section using compass and ruler
- Phi in pentagons and pentagrams
- Where Fibonacci numbers are found in nature
- Where golden section is found in nature
P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.
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