The Golden MeanThe Golden Mean

The Mathematical Formula ofThe Mathematical Formula of

LifeLife

Adapted from Adapted from Grace Hall, Wilkes Central High School in Wilkesboro, NC

The poet John Keats, in his The poet John Keats, in his Ode on a Grecian UrnOde on a Grecian Urn, , said: "Beauty is truth, truth said: "Beauty is truth, truth beauty," - that is all Ye beauty," - that is all Ye know on earth, and all ye know on earth, and all ye need to know.need to know.

What does he mean?What does he mean? This statement is the basis This statement is the basis

for the short story for the short story Beauty Beauty is Truthis Truth..

Ode to a Grecian UrnOde to a Grecian Urn

Keats is not the first person to consider the Keats is not the first person to consider the concepts or themes of beauty and truth. concepts or themes of beauty and truth.

A short story with the theme of beauty and truth A short story with the theme of beauty and truth is trying to make meaning out of what makes is trying to make meaning out of what makes something beautiful or true. something beautiful or true.

Today, we’ll look at the following: Today, we’ll look at the following: What is beauty? What is beauty? How is beauty measured? How is beauty measured?

Goal: To understand the Golden Mean as a Goal: To understand the Golden Mean as a means of organizing a work of art.means of organizing a work of art.

Beauty and TruthBeauty and Truth

The golden mean is the most mysterious of all The golden mean is the most mysterious of all compositional strategies. We know that by compositional strategies. We know that by creating something visual based on the golden creating something visual based on the golden mean will be more likely to appeal to the mean will be more likely to appeal to the human eye, but we don’t know why.human eye, but we don’t know why.

Some scholars argue that the Egyptians applied Some scholars argue that the Egyptians applied the golden ratio when building the great the golden ratio when building the great pyramids, as far back as 3000 B.C.pyramids, as far back as 3000 B.C.

Golden MeanGolden Mean

The Golden Mean ProportionsThe Golden Mean Proportions The “beautiful” Golden Mean Proportions and the The “beautiful” Golden Mean Proportions and the

mysterious Fibonacci numbers have fascinated mysterious Fibonacci numbers have fascinated philosophers for thousands of years and are still philosophers for thousands of years and are still the subject of inquiry in the field of art, the subject of inquiry in the field of art, architecture, music, botany, biology, astronomy, architecture, music, botany, biology, astronomy, physics, and more.physics, and more.

So far, science has documented its existence, but So far, science has documented its existence, but no one has fully understood the complete mystery. no one has fully understood the complete mystery.

Let's watch (6:57 min)Let's watch (6:57 min)

The Golden Mean The Golden Mean The golden ratio is 1.618034 (or the Greek letter Phi The golden ratio is 1.618034 (or the Greek letter Phi

Φ). Φ). The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ... The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ...

(add the last two to get the next).(add the last two to get the next). There’s a mathematical ratio commonly found in There’s a mathematical ratio commonly found in

nature—the ratio of 1 to 1.618—that has many names. nature—the ratio of 1 to 1.618—that has many names. Most often we call it the Golden Section, Golden Most often we call it the Golden Section, Golden Ratio, or Golden Mean, but it’s also occasionally Ratio, or Golden Mean, but it’s also occasionally referred to as the Golden Number, Divine Proportion, referred to as the Golden Number, Divine Proportion, Golden Proportion, Fibonacci Number, and Phi.Golden Proportion, Fibonacci Number, and Phi.

Have You Seen This?Have You Seen This?

Note that each Note that each new square has a new square has a side which is as side which is as

long as the sum of long as the sum of the latest two the latest two

square's sides. square's sides.

The Golden RectangleThe Golden Rectangle The golden ratio is typically depicted The golden ratio is typically depicted

as a single large rectangle formed by as a single large rectangle formed by a square and another rectangle.a square and another rectangle.

You can repeat the sequence You can repeat the sequence infinitely and perfectly within each infinitely and perfectly within each section. If you take away the big section. If you take away the big square on the left, what remains is yet square on the left, what remains is yet another golden rectangle, and so on.another golden rectangle, and so on.

What’s unique about this is that you What’s unique about this is that you can repeat the sequence infinitely and can repeat the sequence infinitely and perfectly within each section.perfectly within each section.

The Golden TriangleThe Golden Triangle The golden ratio is also The golden ratio is also

depicted as a triangle depicted as a triangle formed by a triangle formed by a triangle and another triangle.and another triangle.

You can repeat the You can repeat the sequence infinitely and sequence infinitely and perfectly within each perfectly within each section. If you take section. If you take away the big triangle, away the big triangle, what remains is yet what remains is yet another golden triangle, another golden triangle, and so on.and so on.

The ParthenonThe Parthenon ““Phi” was named for Phi” was named for

the Greek sculptor the Greek sculptor Phidias. Phidias.

The ancient Greeks also The ancient Greeks also used the golden ratio used the golden ratio when building the when building the Parthenon, in Athens, Parthenon, in Athens, in about 440BC. The in about 440BC. The exterior dimensions, exterior dimensions, form a perfect golden form a perfect golden rectangle.rectangle.

Artists often use the Artists often use the Golden Mean in the Golden Mean in the creation of great works. creation of great works.

The Last SupperThe Last SupperArtists throughout history, like Botticelli and Leonardo daVinci, have used the Artists throughout history, like Botticelli and Leonardo daVinci, have used the golden rectangle, or variations of it, as the basis for their compositions.golden rectangle, or variations of it, as the basis for their compositions.Here’s da Vinci’s painting, The Last Supper, with golden sections highlighted.Here’s da Vinci’s painting, The Last Supper, with golden sections highlighted.

Golden rectangles are still the most visually pleasing rectangles known, and Golden rectangles are still the most visually pleasing rectangles known, and although they’re based on a mathematical ratio. The appearance of this ratio although they’re based on a mathematical ratio. The appearance of this ratio in music, in patterns of human behavior, even in the proportion of the human in music, in patterns of human behavior, even in the proportion of the human body, all point to its universality as a principle of good structure and design.body, all point to its universality as a principle of good structure and design.

Mona LisaMona Lisa

Many artists who lived Many artists who lived after Phidias have used after Phidias have used this proportion. this proportion. Leonardo Da Vinci Leonardo Da Vinci called it the "divine called it the "divine proportion" and proportion" and featured it in many of featured it in many of his paintings, for his paintings, for example in the famous example in the famous "Mona Lisa.”"Mona Lisa.”

Da Vinci’s “Vitruvian Man”Da Vinci’s “Vitruvian Man”

Leonardo Da Vinci depicts Leonardo Da Vinci depicts ratios of the lengths of body ratios of the lengths of body parts in his “Vitruvian Man,” parts in his “Vitruvian Man,” which illustrates that the which illustrates that the human body is proportioned human body is proportioned according to the Golden according to the Golden Ratio—similar to Euclid in Ratio—similar to Euclid in 300 B.C. and before that, 300 B.C. and before that, Pythagoras around 500 B.C.Pythagoras around 500 B.C.

The golden ratio involve The golden ratio involve seashell shapes, seashell shapes,

branching plants, flower branching plants, flower petals and seeds, leaves petals and seeds, leaves

and petal arrangements. and petal arrangements.

TheThe Golden Mean in NatureGolden Mean in Nature

TheThe Golden Mean in NatureGolden Mean in Nature

The Golden Mean The Golden Mean spirals are spirals are evident in an evident in an ordinary ordinary pinecone seen pinecone seen from its base from its base where the stalk where the stalk connects it to the connects it to the tree. tree.

Golden Mean in Ideal BeautyGolden Mean in Ideal Beauty

From Queen Nefertiti From Queen Nefertiti in 1350 BC to in 1350 BC to Angelina Jolie today, Angelina Jolie today, you can see that they you can see that they both share similar both share similar facial proportions facial proportions conforming to this conforming to this ideal beauty. Beauty ideal beauty. Beauty may be timeless after may be timeless after all. all.

Golden Mean is EverywhereGolden Mean is Everywhere• Understand how you can use the golden mean in Understand how you can use the golden mean in

your visual projects for home, school, college, your visual projects for home, school, college, and future career. and future career.

• Consider how advertisers use the golden mean to Consider how advertisers use the golden mean to tap into your subconscious attraction to an ideal tap into your subconscious attraction to an ideal beauty and aesthetic (or beautiful) proportions. beauty and aesthetic (or beautiful) proportions.

• Look for the golden mean everywhere. Look for the golden mean everywhere.

Golden RatioGolden Ratio Today, we looked at the following: Today, we looked at the following:

What is beauty? What is beauty? How is beauty measured?How is beauty measured? Goal: To understand the Golden Ratio as a Goal: To understand the Golden Ratio as a

means of organizing a work of art.means of organizing a work of art. Next, we’ll do an activity to practice what Next, we’ll do an activity to practice what

we’ve learned. we’ve learned.

Activity Activity 1. Take out a sheet of paper. 1. Take out a sheet of paper.

2. Find the artwork you used during the Beautiful 2. Find the artwork you used during the Beautiful Activity. Activity. 3. Take one transparency. See if you can find a pattern 3. Take one transparency. See if you can find a pattern in the artwork that matches part or all of the grid. Note in the artwork that matches part or all of the grid. Note any patterns. any patterns.

4. Switch transparencies with another student with a 4. Switch transparencies with another student with a different grid. Note any patterns. different grid. Note any patterns.

5. How many of you found that the artists were 5. How many of you found that the artists were following the golden mean to design their artwork? following the golden mean to design their artwork?

Quick WriteQuick Write Take out a sheet of paper. Take out a sheet of paper. Take out your Anticipation Guide hand out.Take out your Anticipation Guide hand out. Pick one statement you agree or disagree with.Pick one statement you agree or disagree with. Write one or more paragraphs describing why you Write one or more paragraphs describing why you

agree or disagree with the statement. agree or disagree with the statement. Use evidence from your experience with the Beautiful Use evidence from your experience with the Beautiful

Activity, Golden Mean Activity, and reading of Beauty Activity, Golden Mean Activity, and reading of Beauty is Truth to support your agreement or disagreement.is Truth to support your agreement or disagreement.

When you’re done, put your name, period, and date When you’re done, put your name, period, and date on your sheet of paper and turn it in. on your sheet of paper and turn it in.