The Media Got The Math WRONG - The Golden Ratio

Summary

This video clarifies the confusion surrounding the golden ratio, a mathematical constant symbolized by the Greek letter phi (𝜶). It's typically expressed as (√5 + 1) / 2, approximately 1.618, representing the ratio of a longer side to a shorter side in a golden rectangle. However, it can also be written as (√5 - 1) / 2, about 0.618, which is the reciprocal and represents the shorter side to the longer side. The video explains how both expressions are correct and equivalent, using a rectangle with sides 2 and 1 as an example. It also addresses a recent museum controversy and media frenzy over the correct representation of the golden ratio.

Takeaways

• 🔍 The Egyptian pyramids, Parthenon, and natural spirals are all connected by the golden ratio, denoted by the Greek letter phi.
• 📏 The golden ratio, φ (phi), is approximately 1.618 and is expressed as the square root of 5 plus 1 over 2.
• 🤔 A 15-year-old noticed a discrepancy in the representation of the golden ratio at the Boston Science Museum, which sparked a media frenzy.
• 🔄 The museum initially admitted an error but later clarified that both the positive and negative forms of the golden ratio are correct.
• 📐 The golden ratio can be expressed in two ways: as the ratio of the longer side to the shorter side or vice versa.
• 📈 The reciprocal of the golden ratio, denoted by Φ (capital phi), is approximately 0.618 and is the ratio of the shorter side to the longer side.
• 📝 Mathematically, the golden ratio and its reciprocal are proven to be reciprocals of each other by multiplying them together, which equals one.
• 🧩 Understanding the golden ratio involves recognizing it can be represented in two equivalent forms, both related to the ratio of sides in a golden rectangle.
• 📚 The script explains that the golden ratio is a famous mathematical constant with significant applications in architecture, art, and nature.
• 📈 The video aims to clear up confusion about the golden ratio, emphasizing that both expressions are valid and mathematically equivalent.
• 📘 The video is part of a series on math and game theory, suggesting a broader exploration of mathematical concepts beyond the golden ratio.

Q & A

What is the golden ratio and how is it represented?

The golden ratio, denoted by the lowercase Greek letter phi (φ), is a mathematical constant approximately equal to 1.618. It is represented as the square root of 5 plus 1 divided by 2.

Why was there confusion about the golden ratio at the Boston Science Museum?

A 15-year-old visitor noticed that the golden ratio was written as the square root of 5 minus 1 over 2, which he believed was an error due to the negative sign. The museum initially agreed with the correction but later stated that their representation was also correct, causing confusion.

How can the golden ratio be expressed in two different ways?

The golden ratio can be expressed as the ratio of the longer side to the shorter side (approximately 1.618) or as the reciprocal, the ratio of the shorter side to the longer side (approximately 0.618).

What is the relationship between the two expressions of the golden ratio?

The two expressions of the golden ratio are reciprocals of each other. When you multiply them together, the result is 1, confirming their reciprocal relationship.

What is a golden rectangle and how is it related to the golden ratio?

A golden rectangle is a rectangle where the ratio of the longer side to the shorter side is the golden ratio, approximately 1.618. It is a rectangle that is aesthetically pleasing and found in various aspects of art, architecture, and nature.

Why are the golden ratio and its reciprocal considered equivalent?

They are equivalent because they represent the same ratio from different perspectives: one is the longer side to the shorter side, and the other is the shorter side to the longer side. Mathematically, they are reciprocals and equal to each other when multiplied.

How does the script illustrate the concept of reciprocals in the context of the golden ratio?

The script explains that the golden ratio and its reciprocal are two numbers that multiply together to equal one, which is the definition of reciprocals.

What is the significance of the golden ratio in nature, art, and architecture?

The golden ratio is believed to be aesthetically pleasing and is found in various natural phenomena, such as the spirals in flowers, and has been used in the design of famous structures like the Egyptian pyramids and the Parthenon.

What is the significance of the golden ratio in the Parthenon?

The golden ratio is believed to be present in the dimensions of the Parthenon, with the width of the building to its height reflecting the golden ratio, contributing to its harmonious and aesthetically pleasing proportions.

How can one verify that the two expressions of the golden ratio are indeed reciprocals?

One can verify this by multiplying the two expressions together and simplifying to show that the result is 1, or by taking the reciprocal of the golden ratio and rationalizing the denominator to arrive at the other expression.

What are the implications of understanding the golden ratio in mathematics and design?

Understanding the golden ratio can provide insights into the principles of proportion and harmony in design, as well as its applications in mathematics, art, architecture, and even nature.

Outlines

00:00

🔍 The Golden Ratio Controversy

This paragraph discusses a mathematical controversy surrounding the golden ratio, denoted by the Greek letter phi (φ), which is approximately 1.6. The golden ratio is a mathematical constant found in various natural and man-made structures, such as the Egyptian pyramids, the Parthenon, and spirals in nature. The paragraph explains two equivalent expressions for the golden ratio: the square root of 5 plus 1 over 2, and the square root of 5 minus 1 over 2. The confusion arose when a 15-year-old pointed out what he believed was an error in the Boston Science Museum's representation of the golden ratio, leading to media frenzy and subsequent clarification by the museum that both expressions are correct due to the nature of reciprocal ratios. The explanation includes a mathematical demonstration of why the two expressions are reciprocals and thus equivalent.

Keywords

💡Egyptian Pyramids

The Egyptian Pyramids are ancient structures built as tombs for pharaohs and their consorts during the Old and Middle Kingdom periods. They are renowned for their massive scale and precise construction. In the video, the script mentions a mathematical connection between the pyramids and the golden ratio, suggesting that the proportions of the pyramids may have been influenced by this mathematical constant.

💡Parthenon

The Parthenon is a former temple on the Athenian Acropolis in Greece, dedicated to the goddess Athena, and is considered a key landmark of classical architecture. The script discusses the ratio of the width to the height of the Parthenon in relation to the golden ratio, indicating a possible intentional use of this mathematical principle in its design.

💡Golden Ratio

The golden ratio, often denoted by the Greek letter phi (φ), is a mathematical constant approximately equal to 1.618. It is widely found in art, architecture, and nature, and is believed to be aesthetically pleasing. The video explains the controversy and the mathematical basis for the golden ratio's different expressions, emphasizing its significance in the context of the Egyptian pyramids, the Parthenon, and natural spirals.

💡Phi (φ)

Phi (φ) is the lowercase Greek letter used to represent the golden ratio. It is a mathematical constant that has been the subject of fascination due to its prevalence in various natural and man-made structures. The video script clarifies the confusion around the golden ratio's representation and its connection to the Egyptian pyramids and the Parthenon.

💡Rectangle

A rectangle is a quadrilateral with four right angles. The script uses the concept of a rectangle to illustrate the golden rectangle, which has sides in proportion to the golden ratio. This concept is central to explaining the golden ratio's application in design and nature.

💡Ratio

A ratio is a mathematical expression that compares two quantities. In the context of the video, the ratio is used to describe the relationship between the dimensions of the golden rectangle and the proportions found in the Egyptian pyramids and the Parthenon. The script explains how different ratios can be used to express the same relationship.

💡Spirals in Nature

Spirals in nature refer to the growth patterns found in various organisms, such as the arrangement of leaves on a stem or the shape of a galaxy. The video suggests that these natural spirals are connected by the golden ratio, indicating a universal mathematical principle at work in the natural world.

💡Controversy

The term 'controversy' in the script refers to the confusion and debate over the correct representation of the golden ratio, particularly the incident at the Boston Science Museum. The video aims to clear up this controversy by explaining the mathematical equivalence of the two expressions of the golden ratio.

💡Reciprocal

A reciprocal is a number which, when multiplied by the original number, yields the product of one. In the script, the reciprocal relationship between the two expressions of the golden ratio is discussed, showing that they are indeed equivalent despite their different appearances.

💡Rationalize the Denominator

Rationalizing the denominator is a mathematical process used to eliminate the radical from the denominator of a fraction. The script mentions this process in the context of verifying the reciprocal relationship between the two expressions of the golden ratio.

💡Blog

A blog is a personal or informational website consisting of discrete, often informal diary-style text entries. In the video's closing remarks, the creator mentions their blog 'Mind Your Decisions,' which is a platform where they share content related to math and game theory.

Highlights

The Egyptian pyramids, the Parthenon, and Athens are mathematically connected through the golden ratio found in their dimensions.

Natural phenomena such as spirals in flowers also exhibit the golden ratio.

The golden ratio is denoted by the lowercase Greek letter phi (φ) and is approximately equal to 1.6.

A 15-year-old corrected a museum's representation of the golden ratio, sparking a media frenzy.

The Boston Science Museum initially admitted an error in the golden ratio's representation but later claimed correctness.

The golden ratio can be represented in two ways: as the square root of 5 plus 1 over 2, or as the square root of 5 minus 1 over 2.

A rectangle with a width of 2 and a height of 1 demonstrates the concept of ratio representation.

A golden rectangle has sides in the ratio of the golden ratio to 1, or vice versa.

The golden ratio's reciprocal is sometimes denoted by the capital Greek letter Phi and equals approximately 0.618.

Both expressions of the golden ratio are correct due to the ability to express ratios in two ways: longer to shorter or shorter to longer.

Multiplying the two expressions of the golden ratio results in one, confirming they are reciprocals.

The reciprocal of phi, when rationalized and simplified, results in the expression of the golden ratio as the square root of 5 minus 1 over 2.

The golden ratio is typically written as the square root of 5 plus 1 over 2, representing the longer side to the shorter side.

The video explains the confusion around the golden ratio and clarifies its two correct expressions.

The video encourages viewers to subscribe to the channel for more content on math and game theory.

The presenter invites viewers to follow their blog 'Mind Your Decisions' on various social media platforms.

The presenter, Preshtalwalker, is active on social media and has a book linked in the video description.

Transcripts

there is a mathematical connection

between the egyptian pyramids the

parthenon and athens and spirals in

flowers in nature

the ratio of the long side of the

pyramid to its base

the ratio of the width of the parthenon

to its height

and the spirals are all connected by the

golden ratio

this is denoted by the lowercase greek

letter phi and it's equal to the square

root of 5 plus 1 over 2 or approximately

1.6

the golden ratio is a very famous

mathematical constant

recently a 15 year old at the boston

science museum saw the golden ratio was

written as the square root of five minus

one over two

believing this was an error he told the

museum about it and the museum admitted

that oh this negative sign should

actually be a positive sign

this was picked up by the media and

everyone went crazy but what's even more

interesting

is that recently the museum said no

actually we are correct this is the

golden ratio it's a different way to

write the golden ratio

so now everyone is confused about what

the golden ratio is and why these two

things would be equivalent so let me try

and clear up the controversy

let's look at a rectangle which has a

width of 2 and a height of 1. if i ask

you what's the ratio of the two sides

there are actually two different ways

you could tell me what the ratio is

if you told me the ratio of the longer

side to the shorter side you would tell

me the ratio of the sides is two

but you could also tell me the ratio of

the shorter side to the longer side in

which case you would say the ratio is

one-half or 0.5

now consider a golden rectangle where

one side is the golden ratio and the

other side is one

if you told me the longer side to the

shorter side

you would have the square root of five

plus one over two is the ratio of the

sides or approximately 1.618

but you could also tell me the ratio of

the shorter side to the longer side

which is the reciprocal

this is sometimes denoted by the capital

letter phi and that's equal to 0.618

so there are two different ways that you

could express the ratio of the sides and

that's why both of these are correct

ways of expressing the golden ratio

so just to give one technical aside i

said these two are reciprocals of each

other they do not look like reciprocals

so let's verify this mathematically if

you multiply the two numbers together

and simplify you'll find out they're

equal to one

two numbers that multiply together to

equal one are reciprocals

another way you could verify this is by

taking the reciprocal of

phi

you then need to rationalize the

denominator

and then if you do a little bit of

algebra you'll find out you'll get to

the capital greek letter phi which is

the square root of five minus one over

two

so what all this means is that the

golden ratio is typically written as the

square root of five plus one over 2

which is the ratio of the longer side to

the shorter side

this is approximately 1.618

but you could equivalently look at the

golden ratio as the capital greek letter

phi as the square root of 5 minus 1 over

2 and this is the ratio of the shorter

side to the longer side and that's

approximately 0.618

so both ways are correct ways of

expressing the golden ratio

and that's because you can express

ratios

either as the longer side to the shorter

side or as the shorter side to the

longer side

thanks for watching this video please

subscribe to my channel i make videos on

math and game theory you can catch me on

my blog mind your decisions which you

can follow on facebook google plus and

patreon you can also catch me on social

media at preshtalwalker

and if you like this video please check

out my books there's a link in the video

description