Welcome !

Welcome on our blog !
Here, we're going to speak about mathematics, architecture and harmony. You'll discover to what extent the use of mathematics enables harmony in architectures, specifically in English-speaking countries. We'll begin to talk about the golden number and its role in architecture, then we'll see that harmonious architectures, for example a geodesic dome, can be linked with mathematics.

Our work can be divided into two parts : 

1. The Golden Number, or the main link between maths and architecture.
2. The Geodesic Dome, or an other proof that mathematics enable harmony.

Before beginning, we must talk about harmony. What is it ? You can't read our blog if you don't agree with us on its meaning. For us, harmony is a kind of beauty. With beauty, there are two factors : the "viewer" and the object. With harmony, there is a third parameter, which allows the object to be more beautiful, hence the sentence : "It's in harmony with ...", with  "it" representing the object and "..." the third parameter. This can be the environment (in most cases), but also other things, as you'll see at the end of the blog.

We hope you'll discover much information, and you'll enjoy your reading !

Introduction about the Golden Number

Here, we're going to make a small introduction about the golden number and its history. The links you could find in this paragraph are linked with our articles about the golden number.

The golden number has been present through the ages, and through the world.
It appeared for the first time at the beginning of Antiquity, in Ancient Egypt. Indeed, we find it, for example, in the Cheops Pyramid.
We find it again in the Greek civilization. The Pythagoricians studied this number because of its frequent apparition in geometry, and we can find it in the well-known Parthenon.
Later, in the Middle-Age, Fibonacci talked about it in his sequence.
We also find it in art, in the 19th century, for example in the famous "Mona Lisa" and "The Birth of Venus". For painters, this is a sort of "number of art" : it operates without a conscious intervention of the artist. It's the number of perfect proportions.
It takes its name of "golden number" in 1932, with Matila Ghyka.

So, it was a small introduction about this legendary number which is the golden number, and about our work about it. Now, we're going to precise that !

The Golden Number

We're going to begin with different calculations about the golden number. First, you have to know that the golden number is a ratio. You'll see that this parameter will have an importance after. It can be defined with two numbers, a and b, such as :

The value of the golden number is :

This value can be found in the construction of the golden rectangle.  

The golden number has some properties that no-one has, like :

You'll see that we can find this number in a lot of architectures : the CN Tower is a great example of the omnipresence of this number.

 The CN Tower

Built in 1976, the CN Tower is the Canadian's national Tower, and the emblem of Toronto. It's a very tourist place, because it's the third tallest tower and the second self-supporting tower in the world. "self-supporting" means that it's an architectural structure whose stability is due only to its shape. It's also one of the seven wonders of the modern world (just with architectures which are built during the XXIth century). Furthermore, the CN Tower was built with the proportions of the golden number.

In the CN Tower, it appears with :

In fact, we approach the ratio :

Here, it's equal to 1.58. We agree that it's not exactly the value of the golden number, but the difference is minimal.

Calculations

There are many calculations which exist to calculate the golden number. We're going to make you discover the most important ones.

 With the formula : (1/x)+1

As the calculus progresses, the result will be more precise and it approaches the golden number. In each step, x is equal to the result of the previous calculation. The first value of x can be any value.
For example, if we begin with x is equal to 2 :

• ( 1/ 2 ) + 1 = 1,5
• ( 1 / 1,5 ) + 1 = 5/3 ≈ 1,6666
• ( 1 / 5/3 ) + 1 = 1,6
• ( 1 / 1,6 ) + 1 = 1,625

It can also be put in practice thanks to continued fractions, which is finally the same calculation. It also prove that the beginning number is the one we want : this number would be on the bottom right, and you can see that the continued fraction is always equal to the golden number

With the Fibonacci sequence 

    1. Explanations with "rabbits"

Leonardo Pisano, nicknamed Fibonacci, was one of the most famous mathematician of the beginning of the 13 century. He discovered in particular the Fibonacci sequence, which he explained with the problem below : if we put out a pair of rabbits in a place surrounded by a wall, how many pairs of rabbits can be produced in a year, if it's supposed that every month each pair produces a new pair which can breed from the second month ?

In fact, the Fibonacci sequence is the number of pair of rabbits in each month. So, it's begin with : 1 ; 1 ; 2 ; 3 ; 5 ...

    2. Explanations with calculations

There is an other explanation for this problem, with calculation. We find this sequence when we add a number with the one before it. We must start with 1.

• 1                    1
• 0 + 1 = 1        1
• 1 + 1 = 2        2
• 1 + 2 =.3        3
• 2 + 3 = 5        5
• 3 + 5 = 8        8
• 5 + 8 = 13     13

So, the Fibonacci sequence starts with 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34 ; 55 ...

There is a link with the golden number : if you divide one number of the sequence by the number before it, you obtain an approximate value of the golden number.

• 3 / 2 = 1.5       
• 5 / 3 = 1.666       
• 8 / 5 = 1.6        
• 13 / 8 = 1.625   

The bigger value of the Fibonacci sequence we use, the better approximate value of the golden number we obtain.

The Epidaurus Theatre, built 1700 years before Fibonacci, is the best preserved theatre of the Ancient Greece. He was constructed with 13 stairs, 21 rows in the superior half, and 34 in the inferior half. In total, the Epidaurus Theatre is composed of 55 rows.
This numbers are consecutive numbers in the Fibonacci sequence.
We can emit the hypothesis that when constructors of the Epidaurus Theatre built it, they researched something harmonious, and came by chance across the Fibonacci sequence.

 The Cheops pyramid

We can find the golden number in an other architecture : the Cheops pyramid.

Ancient Egypt is the period of gorgeous construction : the Great Sphinx of Giza, the pyramids... But let's focus on one of them : the Cheops pyramid (or Great pyramid), one of the seven Wonders of the world. It would have taken 30 years and 100,000 slaves to have built it. It would have taken over 2,300,000 blocks of stone with an average weight of 2.5 tons each. We hardly imagine how much work was involved. 

Calculation :

First, you have to know that the pyramid has a square base.
The link with the golden number can be seen in the picture presented below.
To calculate, we have to use 3 constants.

• "h", which is the height of the pyramid (481,5 feet)
• "s", which is the half of one side of the base (378 feet)
• "a", which is the apothem, or the slope from the summit to the half of the base (612 feet)

The Pyramid has got a square base, and h is its height. It means that h is perpendicular with the base of the pyramid, so with s. That's why the triangle in yellow is a rectangle triangle (the angle between h and s measures 90° ). So, we can use the Pythagorean theorem :



Thus, we can found the golden number in the Cheops pyramid. 

Golden Number : Golden Rectangle

The first thing you have to learn about the golden number is that it's just a story of proportion. Thus, it's possible to find it in many fields, like architecture. In fact, architectures possess the Golden Number when they possess geometrical shapes which have the golden ratio, like a rectangle or a triangle. Here, we talk about rectangles.

The Parthenon

  The Parthenon is an emblematic construction, built during the peak of the Greek civilization, during the Antiquity (from 447 BC to 438 BC). It was a dedicate to the goddess Athena, who is considered like their patron deity (the god who protect them). The Parthenon was considered like one of the world's greatest cultural monuments.

 It has the special feature to possess the golden number : it contains "golden rectangles". But what is a golden rectangle ?

The golden rectangle

• Definition :

A golden rectangle is a figure which has its length and its width proportional to a rectangle which have a length equals 1 and a width equals :

• How to build this ? 

First, you have to build a square ABCD with a side equals 1.

Place I such as it's the middle point of [AD].

Next, trace a circle with centre I and radius [BI]. F is the intersection point between this circle and the line (AD). With the Pythagorean theorem, we can say that [BI] is equal to the square root of 5 divided by 2.

Thus, we have :

Place E, the intersection point between (BC) and the perpendicular to (AB) passing by F. The golden rectangle is ABEF.

In the Parthenon, we can find a lot of golden rectangle, as you can see here :

There is an other construction, a modern one, which applies the principle of the golden rectangle : the United Nations Headquarters, in New York. It possess principal organs of the United Nations, like the General Assembly and the Security Council.
Indeed, it's composed of 3 golden rectangles.

It was built from 1948 to 1952, under the direction of the Swiss architect Le Corbusier. He was very interested in the golden number. In his theory of the Modulor (1943), which derive its name from the contraction of "Module" and "nombre d'or", he integrates the notion of harmonic measurement in the human scale, which is applicable in architecture. The harmony of measurement pass through the use of the golden number.

•  Study about the "aestheticism" of the golden rectangle

A study was done. On a page, students drew 6 rectangles and one of them was the golden rectangle. They questioned people in the street, and asked them which rectangle they found more aesthetic. Most of them chose the golden rectangle. It shows that the golden rectangle has got perfect proportions for the eyes.

The Golden Triangle

The Pentagon :

One of the best-known construction in the US is the Pentagon, which is the United States Department of Defence (located in Virginia). Built in 1943, its architect is Georges Bergstrom. With its 5 floors and 28 km of corridors, the Pentagon is the largest office block in the world. It was of course built from a pentagon : it's a polygon with 5 equal sides.

But where is the golden number ?

Pentagons and Golden triangles

First, you must find that all we're going to do use regular pentagons, that means pentagons with 5 equal segments.
A pentagon can be made with a star with five branches, where each summit of the branches is linked with the one near it. This star is called a pentagram. In Antiquity, the pentagram was considered like the universal symbol of perfection. We can find the golden number in it :

Naturally, all is a story of proportionality.

In fact, when we find the golden number in the pentagon, we find golden triangles. But, what are golden triangles ?

A golden triangle is an isosceles triangle such as the ratio of a side a over an other side b which is different of a is  :

Here is the method to build a golden triangle :
First, you have to take a system (O, OB, OA ).

 Next, you have to trace the circular arch, length [AI], center I, which cuts the abscissa axis. The point of intersection is C. Take the length [OC] and report it in A, to place the point J and S1, like in the picture below. It makes your first isosceles triangle, which is a golden triangle.

 There are two "types" of golden triangles, which are define by their angles : they measure respectively 36° or 108° at the summit, and 72° or 36° at the base.

How to make a pentagon ?

Now, we're going to take an example of pentagon to prove you that it possess the golden number.
The golden triangle traced before
Trace the circle of center O and radius [OA], which will be the circumscribed circle of the pentagon. After, trace the parallel of the abscissa axis passing by S1. Thus, you find S5, which is the intersection point with the circle. After, you have to trace the line passing by S5 and J : the intersection point with the circle is S2. You do it again with S1, in order to find S4. S3 is just the intersection point between the circle and [AO].

If you want to have a pentagram, you just have to trace the diagonals :

 The small pentagon is regular too :  all sides are equal.

Where can we find the golden number ?
There are many ratios whose the result is a power of the golden number (power -1, 1 or 2). In fact, when the result is phi squared (2.61), you can find the inverse of phi (0.61) switching the numerator and the denominator of the initial fraction.
The ratios made come from golden triangles. For example, in the three first cases, we have the golden triangles LKS2, S1S2S5 and KS1S2.

LK / KS2 ≈ 1,61...
S1 S5 / S2 S5 ≈ 1,62...
K S2 /  S1 S2 ≈ 1,61...
S2 S5 / M S2 ≈ 2,61...
J S2 / K S2 ≈ 2,61...

1,61 is an approximate value for the golden number, and 2,61 is an approximate value of phi squared.
It does not always give the same values, because values on the picture are not exact. 
You must now that we find also golden triangles in decagons.

Conclusion about the Golden Number

As you could see, the golden number is THE number of architecture and harmony. It represents by itself the many links existing between mathematics and architecture. We can find it into different patterns, like the triangle or the rectangle, and in architectures of any age, from the Cheops Pyramid to the United Nations Headquarters, passing by the Parthenon. We found it many times in the Antiquity, but more recently, men like Le Corbusier updated it. There are many ways to calculate it, but finally, the golden number is just a story of proportion, and its use creates the golden proportion, the perfect proportions of beauty and harmony.

Introduction about the Geodesic Dome

In this part, we'll talk about the Biosphere, in Montreal (Canada).
The Biosphere is a geodesic dome which contains a museum about environment.
We have no doubt you're wondering : where is the relation between a geodesic dome, mathematics and the harmony of architectures ? We're going to explain it to you, demonstrating the link between mathematics and a geodesic dome, and after showing that a geodesic dome can be harmonious.

We'll follow this plan :

1. Parameters of a geodesic dome
2. How to build a geodesic dome ? And what is finally a geodesic dome ?
3. The second shape of a geodesic dome
4. Calculations and properties in a geodesic dome
5. Geodesic dome and harmony

The three first steps will explain the "rules" of the construction of the geodesic dome, whereas the fourth is about calculations and the last is a kind of conclusion with a link with harmony.

So, let's go !

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