Golden Rectangle 1

Summary

This educational video script demonstrates the construction of a golden rectangle, a geometric figure with aesthetic significance in art and architecture. Starting with a square, the presenter bisects one side and extends it to form a rectangle. The diagonal of the square is used to determine the rectangle's proportions. The golden ratio, derived from the rectangle's dimensions, is approximately 1.618 and is an irrational number. The video concludes by highlighting the importance of the golden ratio in mathematics, even featuring it on a Japanese postage stamp.

Takeaways

• 📏 The presenter starts by constructing a square with sides of length 2 units.
• 📐 A midpoint is found on one side of the square, and the side is bisected into two segments of length 1 each.
• 🔺 The diagonal of the square is drawn, which connects the midpoint to the opposite vertex.
• 📐 The length of the diagonal is extended to form a rectangle.
• 🌟 The resulting rectangle is a golden rectangle, which is significant in mathematics.
• 🔢 The ratio of the longer side to the shorter side in a golden rectangle is a special number known as the golden ratio.
• 📐 By using the Pythagorean theorem, the length of the diagonal is calculated to be the square root of 5.
• 🔢 The golden ratio is mathematically expressed as (1 + √5) / 2, which is an irrational number.
• 📈 The golden ratio is approximately equal to 1.618 when calculated as a decimal.
• 🗸 The concept of the golden ratio is so important that it is featured on a postage stamp from Japan.

Q & A

What is a golden rectangle?

A golden rectangle is a rectangle in which the ratio of the length to the width is the same as the ratio of the whole length to the longer part when the rectangle is divided into two smaller rectangles.

How is a golden rectangle constructed?

A golden rectangle is constructed by starting with a square, bisecting one side to create a midpoint, drawing a diagonal from the midpoint to the opposite vertex, and then extending the other side of the square to meet the diagonal, creating a rectangle.

What is the significance of the golden ratio in mathematics?

The golden ratio is significant in mathematics because it is an irrational number that appears in various natural and artistic contexts, often associated with aesthetically pleasing proportions.

What is the approximate decimal value of the golden ratio?

The golden ratio is approximately equal to 1.61803398875.

Why is the golden ratio considered special?

The golden ratio is considered special because it is an irrational number that appears in many areas of mathematics, art, and architecture, and it is associated with the Fibonacci sequence.

How does the Pythagorean theorem relate to the golden rectangle?

The Pythagorean theorem is used to calculate the length of the diagonal in the golden rectangle, which is the square root of the sum of the squares of the sides of the right triangle formed by the bisected side and the diagonal.

What is the ratio of the length to the width in the golden rectangle?

The ratio of the length to the width in the golden rectangle is (1 + √5) / 2, which is the golden ratio.

Why is the golden ratio represented by the symbol 'φ' (phi)?

The golden ratio is often represented by the symbol 'φ' (phi) because it is the first letter of the name of the sculptor Phidias, who was known for using the golden ratio in his work.

How is the golden ratio derived from the construction of the golden rectangle?

The golden ratio is derived from the golden rectangle by dividing the longer side (which is 1 + √5) by the shorter side (which is 1), resulting in the ratio (1 + √5) / 2.

What is the significance of the golden ratio being on a postage stamp from Japan?

The presence of the golden ratio on a postage stamp from Japan signifies the cultural and historical recognition of the golden ratio's importance in mathematics and its influence on art and design.

Can you provide an example of how the golden ratio is used in art or architecture?

The golden ratio is often used in art and architecture to create aesthetically pleasing proportions. For example, the Parthenon in Greece and the works of Leonardo da Vinci, such as 'The Last Supper,' are believed to incorporate the golden ratio.

Outlines

00:00

📏 Introduction to the Golden Rectangle

The speaker begins by introducing the concept of the golden rectangle, starting with a square of side length 2. They demonstrate how to bisect one side of the square to create two segments of length 1. By drawing a diagonal from the midpoint of the bisected side to the opposite vertex and extending the other side to meet this diagonal, a golden rectangle is formed. The golden rectangle is characterized by a special ratio of length to width, which is explored through the use of a right triangle within the rectangle. The speaker explains that the ratio is derived from the Pythagorean theorem, resulting in an irrational number known as the golden ratio, approximately equal to 1.618. The golden ratio is highlighted as a significant mathematical constant, with the speaker noting its importance by referencing a Japanese postage stamp that features a derivation of the golden rectangle.

Keywords

💡Golden Rectangle

A Golden Rectangle is a rectangle in which the ratio of the length to the width is the same as the ratio of the whole length to the longer side. This ratio is known as the Golden Ratio, which is approximately 1.618. In the script, the presenter demonstrates how to construct a Golden Rectangle by starting with a square and then extending it in a specific way to achieve this ratio. The Golden Rectangle is significant in art, architecture, and design, often associated with aesthetically pleasing proportions.

💡Square

A square is a special case of a rectangle where all four sides are of equal length. In the script, the presenter starts with a square of side length 2 to construct the Golden Rectangle. The square serves as the foundation for the demonstration, illustrating the geometric principles involved in creating the Golden Rectangle.

💡Midpoint

The midpoint of a line segment is the point that divides the segment into two equal parts. In the script, the presenter locates the midpoint of one side of the square and uses it as a reference point to bisect the side, which is a crucial step in constructing the Golden Rectangle.

💡Diagonal

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. In the script, the presenter draws a diagonal from the midpoint to the opposite vertex of the square. This diagonal plays a key role in extending the square to form the Golden Rectangle.

💡Pythagorean Theorem

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The presenter uses this theorem to calculate the length of the diagonal in the right triangle formed during the construction of the Golden Rectangle.

💡Irrational Number

An irrational number is a number that cannot be expressed as a simple fraction, meaning its decimal representation is non-repeating and infinite. The Golden Ratio, which is approximately 1.618, is an example of an irrational number. The script mentions that the ratio of the length to the width of the Golden Rectangle is an irrational number, emphasizing the mathematical complexity and uniqueness of the Golden Ratio.

💡Golden Ratio

The Golden Ratio, often symbolized by the Greek letter phi (φ), is a mathematical constant approximately equal to 1.618. It is found in various aspects of nature, art, and architecture where proportions are considered aesthetically pleasing. The script explains how the ratio of the length to the width of the Golden Rectangle is the Golden Ratio, highlighting its significance in mathematics and design.

💡Right Triangle

A right triangle is a triangle that has one angle measuring 90 degrees. In the script, the presenter identifies a right triangle within the square used to construct the Golden Rectangle. The properties of this right triangle are essential for applying the Pythagorean Theorem to determine the diagonal's length.

💡Postage Stamp

A postage stamp is a small piece of paper issued by postal authorities for the purpose of franking mail. The script mentions a postage stamp from Japan that features the Golden Ratio, indicating the cultural and historical significance of this mathematical concept and its recognition even in everyday objects.

💡Aesthetics

Aesthetics is the study of beauty and the principles that make something pleasing to the senses. The Golden Rectangle is associated with aesthetics because its proportions are considered harmonious and pleasing to the eye. The script suggests that the Golden Ratio's presence in the Golden Rectangle is why it is often used in design and art to create visually appealing compositions.

💡Mathematics

Mathematics is the study of numbers, quantities, and shapes, and their relationships. The script is centered around a mathematical concept, the Golden Ratio, and its application in creating the Golden Rectangle. The use of geometric shapes, the Pythagorean Theorem, and the calculation of irrational numbers all fall under the realm of mathematics, showcasing its role in understanding and creating beauty in the world around us.

Highlights

Introduction to the concept of the golden rectangle and its significance in mathematics.

Demonstration of constructing a golden rectangle starting with a square of side length 2.

Explanation of bisecting the square's side to create two segments of length 1.

Drawing the diagonal of the square to create a right-angled triangle.

Extension of the square's side to meet the dropped length from the diagonal.

Identification of the resulting rectangle as the golden rectangle.

Calculation of the ratio of length to width in the golden rectangle.

Use of the Pythagorean theorem to find the length of the diagonal.

Derivation of the golden ratio as the ratio of the lengths in the golden rectangle.

Explanation that the golden ratio is an irrational number.

Approximation of the golden ratio as 1.618.

Discussion on the importance of the golden ratio in various fields.

Mention of the golden ratio's representation on a Japanese postage stamp.

Highlighting the cultural and mathematical significance of the golden ratio.

Illustration of the golden ratio's derivation process on the postage stamp.

Reflection on the golden ratio's impact on mathematical and artistic design.

Transcripts

I'm going to show you the golden

rectangle I'm going to start here with a

square that's two on each side so a

square of side two I want to go to the

midpoint of this side and kind of bisect

it here so that length is 1 and that

length is 1 then I'm going to draw this

diagonal in let me get a ruler here draw

this diagonal in that goes from this

midpoint up to that vertex of that

square and then what I want to do is

drop that down that length down and

extend this side right here to meet that

now when I do that then the rectangle

that I get is called the golden

rectangle let me see if I can just

quickly draw it in right here

there's my golden rectangle and so what

I want to do is find the ratio of the

length to width in the golden rectangle

because in mathematics that's a very

special number so if I look in this

little right triangle right here I see

that this side is 1 this side is 2

because that was my original square and

so by the Pythagorean theorem I know

that this side is going to be square

root of 1 squared plus 2 squared square

root of 1 plus 4 or square root of 5 so

this length is square root 5 that means

that this length right here is square

root 5 and this little length is 1 so if

I want the ratio of link to width in the

golden rectangle the length is 1 plus

square root 5 and the width is this

width right here which is just 2 so this

number right here 1 plus square root 5

over 2 it's an irrational number but

it's also called the golden ratio very

special number in mathematics if you

work this out on a calculator it's

approximately equal to one point six one

eighth of course I can't write it as a

decimal because it's an irrational

number so the golden rectangle when I

find the ratio of

length to width and that I end up with

what's called the golden ratio now I

want to show you a little postage stamp

from Japan that kind of this little

derivations on it as you can see sort of

vertical rather than horizontal the way

we did it but very important derivation

in mathematics so important that they

actually put it on a postage stamp