Golden Rectangle 1
Summary
TLDRThis 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
- π The presenter starts by constructing a square with sides of length 2 units.
- π A midpoint is found on one side of the square, and the side is bisected into two segments of length 1 each.
- πΊ The diagonal of the square is drawn, which connects the midpoint to the opposite vertex.
- π The length of the diagonal is extended to form a rectangle.
- π The resulting rectangle is a golden rectangle, which is significant in mathematics.
- π’ The ratio of the longer side to the shorter side in a golden rectangle is a special number known as the golden ratio.
- π By using the Pythagorean theorem, the length of the diagonal is calculated to be the square root of 5.
- π’ The golden ratio is mathematically expressed as (1 + β5) / 2, which is an irrational number.
- π The golden ratio is approximately equal to 1.618 when calculated as a decimal.
- πΈ 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 + β5) / 2, which is the golden ratio.
Why is the golden ratio represented by the symbol 'Ο' (phi)?
-The golden ratio is often represented by the symbol 'Ο' (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 + β5) by the shorter side (which is 1), resulting in the ratio (1 + β5) / 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
π 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.
Mindmap
Keywords
π‘Golden Rectangle
π‘Square
π‘Midpoint
π‘Diagonal
π‘Pythagorean Theorem
π‘Irrational Number
π‘Golden Ratio
π‘Right Triangle
π‘Postage Stamp
π‘Aesthetics
π‘Mathematics
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
Browse More Related Video
The Media Got The Math WRONG - The Golden Ratio
The Golden Ratio: The Divine Beauty of Mathematics by Gary B. Meisner
Mathematics - Fibonacci Sequence and the Golden Ratio
Why is 1.618034 So Important?
The Golden Ratio: Nature's Favorite Number
GEC104 Video Lecture 1 - Mathematics in our World (Part 2): Fibonacci Sequence and Golden Ratio
5.0 / 5 (0 votes)