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## Geometry (all content)

### Course: Geometry (all content)>Unit 1

Lesson 6: The golden ratio

# The golden ratio

An introduction to one of the most amazing ideas/numbers in mathematics. Created by Sal Khan.

## Want to join the conversation?

• I have done a very basic Fibonacci Spiral using the skills I learned in the CS videos. Is the Fibonacci Spiral the same as the Golden Ratio? •   If you keep dividing consecutive terms of the Fibonacci sequence it will eventually get close to the golden ratio. The proof for that uses eigenvalues, but you can check the results yourself picking consecutive larger terms, and its quite cool!
• On , how does "phi=1+1/phi" be squared into ''phi(squared)=phi + 1"? •   It's not "squared into", just multiply both sides by phi. So,
phi = 1 + 1/phi
phi * phi = phi * 1 + phi * 1/phi
phi^2 = phi + phi/phi
phi^2 = phi + 1
• What is a phi? • Why does golden ratio keeps showing up in nature? •  It turns out that the golden ratio is not only an irrational number... it is the most irrational number. And there are places in the natural world were extreme irrationality is the most efficient solution to a problem, so by natural selection living systems tend toward that value where it works best.

Consider a plant that has grown one leaf. If it grows a second leaf in the shadow of the first then that leaf is useless... an evolutionary disadvantage.

If the second leaf is opposite the first then that is good, but the third will be in shadow and useless... same problem.

As this species evolves, the plants whose leaves are most often useful have an advantage and breed more. It turns out that if a plant grows one leaf, then the next phi (the golden ratio) rotations from the first, then the third phi rotations from the second, and the fourth phi rotations from the third, and so on, that process will result in the longest possible time before the newest leaf is in the shadow of any existing leaf. So plants that tend toward this value have an advantage against plants that don't... so the survivors tend toward that value over time.

What we see now is the result of eons of advantaged plants surviving over disadvantaged plants... and passing their advantages on to their offspring.
• At , Sal says that, "we are assuming [Lines a and b] are positive distances..." Does this mean that there can be negative distances? And if so, how can I represent them in real-life incidents? • Think of it this way:
You leave your home for a long journey, realize you left something important, turn around, go back home, and then continue with your trip. Question: how far have you traveled?

Version 1: If you are only concerned with how many kilometers appear on the car's odometer, then you would treat portion of the trip where you drove back home as a positive distance.

Version 2: If you are concerned about how far from home you are, then you would treat the portion of your trip where you drove back home as a negative distance because you were getting closer to your home.
• Can the golden ratio be transferred into a fraction? • At , how did "phi minus one" turn into a positive "one over phi" wouldn't that be negative "one over phi"? • No, if you check out from about , you see that Sal finds (really early on) that Φ = 1 + 1/Φ, so 1/Φ = Φ-1.
``1] Φ = a/b = (a+b)/a             By definition2] Φ = (a+b)/a = a/a + b/a       Separate out the numerator3] Φ = a/a + b/a = 1 + b/a       Simplify a/a4] Φ = a/b, so 1/Φ = b/a         Going back to (1)5] Φ = 1 + 1/Φ                   Substituting (4) into (3)6] 1/Φ = Φ - 1                   Subtract 1 from both sides and swap sides``
• Are the Fibonacci Numbers related to the Golden Ratio?? How?? • Yes, there is a connection. The ratio of one Fibonacci number to the previous in the series gets closer and closer to the Golden Ratio as you get to higher and higher Fibonacci numbers. For example, the 50th Fibonacci number is 20365011074. The 51st is 32951280099. The ratio of the 51st to the 50th is
1.6180339887498948482035085192412
The Golden ratio is:
1.618033988749894848204586834365638....
• So... Phi is irrational, right?
But what I'm confused about is that in the beginning, he says that phi=a/b. So that makes it rational! And what's the difference between a ratio and a fraction? :/ • Yes, Phi is irrational. You must remember that the definition of a rational number is a number that can be written as the ratio of two integers. Sal make's no statement at the beginning, as to what 'a' or 'b' are in phi=a/b. Later when he solves for phi we discover that a=1+√5 an irrational number. So the ratio a/b is not rational since 'a' is not rational.

A ratio is a comparison between two numbers, while a fraction is just a single number. Practically they are indistinguishable most of the time. 