The Media Got The Math WRONG - The Golden Ratio
Summary
TLDRThis 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
🔍 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.
Mindmap
Keywords
💡Egyptian Pyramids
💡Parthenon
💡Golden Ratio
💡Phi (φ)
💡Rectangle
💡Ratio
💡Spirals in Nature
💡Controversy
💡Reciprocal
💡Rationalize the Denominator
💡Blog
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
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