1. The introduction:
Hello my name is Frederick Smith, I will be speaking you about a fascinating thing that is in everything, it’s a part of you, it created you & its not just in you, its all around you. Its also in all plants and in all animals. Take for example an octopus has eight tentacles hence the name “octo’~pus, each one of its tentacles has the exact number of suckers on it and each tentacle is the same length pretty amazing right... (Pause for a break…) and the intricate design on a butterfly. One wing as the exact pattern as the other side in the exact spot adjacent to its counterpart (the other wing) It is exactly the same on one side as it is on the other? Or How does a seashell create a perfect spiral? so how does all this happen… (Another pause…) 2. Thesis statement

In nature there is something not visible bi the untrained eye. It happens because there is something in nature called the golden ratio. (Say softly & clearly…). Think of golden ratio as natures secret un~seen Architect! Although I am not a fan of mathematics, it’s in everything around you (Pause for break, let them think about it for a second)

Have any of you heard about the golden ratio before?
Other names frequently used for the golden ratio are the golden section and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number, 2.1 History

Who came up with this, well it was a mathematician who also liked art, his name was Fibonacci, also known as Leonardo Pisano Bo Gallo he discover a sequence of numbers called the Fibonacci sequence if you take those numbers and u do some math they have a common mean 1.68 , hence the name golden mean, from this mean and number sequence the Fibonacci square is completed using te golden mean as its ratio, what’s cool about this square is if u take the height and fold it down back up ...

...
The Golden Number
1.61803 39887 49894 84820 is by no means a number of memorization. However, it is a recognizable one. Never will you find a combination of numbers that is more significant than this one. This ratio is known as the Golden Number, or the GoldenRatio. This mystery number has been used throughout different aspects of life, such as art, architecture, and of course, mathematics. One may wonder where theGoldenRatio came from? Who thought to discover it? When was it discovered? And how has it been used throughout time? The Goldenratio has been used throughout different aspects of life after being discovered during the ancient times.
About two to three thousand years ago, the GoldenRatio was first recognized and made use by the ancient mathematicians in Egypt. The goldenratio was introduced by its frequent use in geometry. An ancient mathematician, sculptor, and architect named Phidias, who used the goldenratio to make sculptures, discovered it. He lived from sometime around 490 to 430 BC. None of his original works exist, however he was highly spoken of by ancient writers who gave him high praise. Hegias of Athens, Agelades of Argos, and Polygnotus of Thasos were said to have trained him.
Although not much is known...

...Goldenratio ; The Definition of Beauty
“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.” Johannes Kepler, 1571-1630
The goldenratio is present in everyday Life. The golden proportion is the ratio of the shorter length to the longer length which equals the ratio of the longer length to the sum of both lengths. It can be expressed algebraicay like :
This ratio has always been considered most pleasing to the eye. It was named the goldenratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias.
The GoldenRatio is also known as the golden section, golden mean or golden rectangle. The Golden Rectangle has the property that when a square is removed a smaller rectangle of the same shape remains, a smaller square can be removed and so on, resulting in a spiral pattern. It is a unique and important shape in mathematics which also appears in nature, music, and is often used in art and architecture.
Our human eye „sees“ the golden rectangle as a beautiful geometric form.
Leonardo Fibonacci...

...The GoldenRatio
By : Kaavya.K
In mathematics and the arts, two quantities are in the goldenratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The goldenratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for thegoldenratio are the golden section and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number, and mean of Phidias. The goldenratio is often denoted by the Greek letter phi, usually lower case (φ).
[pic]
The golden section is a line segment divided according to the goldenratio: The total length a + b is to the longer segment a as a is to the shorter segment b.
The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:
[pic]
This equation has one positive solution in the algebraic irrational number
[pic]
At least since the Renaissance, many artists and architects have proportioned their works to approximate the...

...The GoldenRatio
Body, art, music, architecture, nature – all connected by a simple irrational number – the GoldenRatio. According to Posamentier & Lehmann in their work The
(Fabulous) Fibonacci Numbers, there is reason to believe that the letter φ (phi) was
used because it is the first letter of the name of the celebrated Greek sculptor Phidias (490-430 BCE). He produced the famous statue of Zeus in the Temple of Olympia and supervised the construction of the Parthenon in Athens Greece (Posamentier & Lehmann, 2007). In constructing this masterpiece building, Phidias used the GoldenRatio to create a masterpiece of work.
Figure 1: This is a model of Zeus in the Temple of Olympia. The red lines show the use of the GoldenRatio. (www.scarletcanvas.com/)
Phidias brought about the beginning of the one of the most universally recognized form of proportion and style used throughout history (Posamentier & Lehmann, 2007). The irrational number Phi, also known as the GoldenRatio, has had tremendous importance. To properly understand this mathematical concept, it is important to explore the definition, history, and the relations to architecture, art, music and the Fibonacci sequence.
Figure 2: This model shows the line segments in the GoldenRatio....

...Introduction
In this investigation we are going to examine the Fibonacci sequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonacci sequence
The Fibonacci sequence can be defined as the following recursive function:
Fn=un-1+ un-2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below:
F2=F1-F0F2=1+0=1
We are able to calculate the rest of the terms the same way:
F0 | F1 | F2 | F3 | F4 | F5 | F6 | F7 |
0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 |
Segment 2: The Goldenratio
In order to define the goldenratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB
The ratio described above is called the goldenratio.
If we assume that AP=x units and PB=1 units we can derive the following expression:
x+1x=x1
By solving the equation x2-x-1=0 we find that: x=1+52
Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence:
φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value of x to the formula of the goldenratio we discussed before. Therefore:
φ=x+1x φ=1+52+11+52 φ=1+52
Thus...

...GoldenRatio
In mathematics, two quantities are in the goldenratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b,
Where the Greek letter phi (φ) represents the goldenratio. Its value is:
Thegoldenratio is also called the golden section (Latin: sectio aurea) or golden mean. Other names include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.
Some twentieth-century artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the goldenratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the goldenratio—believing this proportion to be aesthetically pleasing (see Applications and observations below). Mathematicians since Euclid have studied the properties of the goldenratio, including its appearance in the dimensions of a regular pentagon and in a golden...

...The History of Algebra and
The GoldenRatio in Nature
By: Lauren Pressley
Introduction to Statistics
Throughout history algebra has changed in words through etymology. Etymology is an account of the history of a particular word or elements of a word. The word “algebra” is derived from Arabic writers. Algebra is a method for finding solutions of equations to the simplest possible form. Different cultures have come up with different types of names to classify algebra. Al Khwarizmi and Fibonacci contributed talented mathematic systems that shaped algebra.
Al Khwarizmi was born in the town of Khwarizm in Khorason. He achieved most of his work between 813 a.d and 833 a.d. Khwarizmi contributed logical approaches to algebra and trigonometry. He came up with ways of solving linear and quadratic equations. Khwarizmi was not the only person who contributed to algebra; Fibonacci contributed to algebra has well.
one by adding a number to sum up the two numbers that precedes the previous two numbers. He used this method to tie nature and mathematic together. It is formed by using a triangle whose sides’ measure one number of the Fibonacci
Fibonacci contributed the decimal number system which is known as the Fibonacci sequence. The Fibonacci sequence is closely related to the goldenratio that uses the number
number of the Fibonacci...

...The GoldenRatio: Natures Beautiful Proportion
At first glance of the title, many may wonder: What is the GoldenRatio? There are many names the GoldenRatio has been called including the Golden Angle, the Golden Section, the Divine Proportion, the Golden Cut, the Golden Number et cetera, but what is it and how is it useful for society today? One may have heard of the number π (Pi 3.14159265…) but less common is π’s cousin Φ (Phi 1.61803399…). Both Φ and π are irrational numbers, meaning they are numbers that cannot be expressed as a ratio of two whole numbers as well as the fact that they are never-ending, never-repeating numbers. The GoldenRatio is the ratio of 1:Φ (1.61803399…). The GoldenRatio is a surprising ratio that is based on the research of many composite mathematicians spanning over 2300 years, and it is found in many areas of everyday life including art, architecture, beauty and nature.
Euclid, a Greek mathematician who taught in Alexandria around 300 B.C., was one of the first to discover and record the bases for the GoldenRatio back 2300 years ago. Euclid’s discovery was that if one takes a line and divide it into two unequal sections in such a way that the longer section is the same...

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