Golden ratio ; 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 golden ratio 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 golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias.

The Golden Ratio 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 discovered the unusual properties of the numeric series, that’s how it was named. It is not proven that Fibonacci even noticed the connection between the Golden Ratio meaning and Phi. The Renaissance used the Golden Mean and Phi in their sculptures and paintings to achieve vast amounts balance and beauty. Throughout the centuries, artists have used the golden ratio in their own creations. An example is “post” by Picasso. The Golden Ratio also appears in the Parthenon in Athens. It was built about 440 B.C.; it forms a perfect Golden Rectangle. Another example of the Golden Ratio is shown in Egyption Pyramids. Ancient Egyptions used the Golden Ratio to build their pyramids. The pyramids show one of the first examples of using the golden ratio in...

...rederick smith
The GoldenRatio
March 31 2011
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 goldenratio. (Say softly & clearly…). Think of goldenratio 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 goldenratio before?
Other names frequently used for the goldenratio are the golden section and...

...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 φ2=1+522 φ2=3+52 and F2φ+F1=1+52+1=3+52
Therefore:
φ2=F2φ+F1
We can simplify other powers of φ the same way, thus:
φ3=2+5 and φ4=35+72
In order to from a conjecture...

...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...

...Golden Patterns
It is a common misconception to believe that mathematics is only found in books, written on paper, expressed as variables, functions, shapes and alphanumerical characters or as a real world application of a mathematical problem found in real life now worded and written into text. As a student progresses through grade school a student can’t help but feel no connection between the world’s nature and mathematical work, this concept however could not be further from the truth. As the student journeys through college and higher division mathematics, the student finds that mathematical patterns and sequences can be found naturally and without the need of human influence and that furthermore, patterns influence nature heavily. Patterns have always been of interest in mathematics, after all it can be said that mathematics is the science of patterns. In nature there are many simple yet elegant patterns present. Let’s consider the patterns composed by the scales on a pineapple, or the ones found on an acorn, is there a possible way to model these patterns mathematically? We can most certainly answer this question with a “yes” given that patterns have always been of interest as stated above. Furthermore, the amount of research and time that has gone into finding mathematical representations for these patterns is phenomenal and I will be using just a few of these works to give some insight on some of the most important patterns in mathematics, that is,...

...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. (Wikipedia.org)
As is with any new...

...THE GOLDENRATIO IN THE HUMAN BODY
GABRIELLE NAHAS
IBDP MATH STUDIES
THURSDAY, FEBRUARY 23rd 2012
WORD COUNT: 2,839
INTRODUCTION:
The GoldenRatio, also known as The Divine Proportion, The Golden Mean, or Phi, is a constant that can be seen all throughout the mathematical world. This irrational number, Phi (Φ) is equal to 1.618 when rounded. It is described as "dividing a line in the extreme and mean ratio". This means that when you divide segments of a line that always have a same quotient.
When lines like these are divided, Phi is the quotient:
When the black line is 1.618 (Phi) times larger than the blue line and the blue line is 1.618 times larger than the red line, you are able to find Phi.
What makes Phi such a mathematical phenomenon is how often it can be found in many different places and situations all over the world. It is seen in architecture, nature, Fibonacci numbers, and even more amazingly,the human body.
Fibonacci Numbers have proven to be closely related to the GoldenRatio. They are a series of numbers discovered by Leonardo Fibonacci in 1175AD. In the Fibonacci Series, every number is the sum of the two before it. The term number is known as ‘n’. The first term is ‘Un’ so, in order to find the next term in the sequence, the last two Un and Un+1 are added. (Knott).
Formula: Un + Un+1 =...

...What is the GoldenRatio?
Most people are familiar with the number Pi because it can be found in so many different math problems and equations. There is, however, another irrational number like Pi. This number isn¡¦t as well known as Pi however. This number is called Phi. This number is also called the goldenratio. The goldenratio is equal to the square root of five plus one, divided by two. If you work this out it comes out as 1.618033988749895. This is also the only number that if squared, is equal to itself plus one. Mathematically speaking, Phi^2 = Phi + 1. Also if you find the reciprocal of Phi, it is equal to itself minus one, Phi^-1 = Phi ¡V 1.
The GoldenRatio is the basis for many things in nature. Even ones fingers use the GoldenRatio. First measure the length of the longest finger bone. Then measure the shorter one next to it. Finally if you divide the longer one by the shorter one, you should get a number that is close to 1.168 which is really close to the GoldenRatio. Most parts of the human body are proportional to the GoldenRatio.
The GoldenRatio can even be traced back into the times of the Romans and Pyramids. For example, the Great Pyramid of Giza, which was built in 2560 BC, is one of the earliest ways the...

...The GoldenRatio
The theory of the Italian mathematician Leonardo Pisano is extremely present today. While he was trying to sort out the number of rabbits that mated in a year, he discovered a series of numbers, that are profoundly consistent in man, nature & animals. This discovery was extraordinary, but he also found that the ratio always resulted in 1.618. Although it is called differently, this ratio is often called „the goldenratio“. It's marked with the Greek letter phi. It's just amazing how we've used it to create beauty in art & architecture, today you may find the goldenratio in everydays objects such as tables, couches, doors,posters, books and etc.
Because it is very pleasing to the eye, the goldenratio is used alot in art. Leonardo da Vinci used the goldenratio in many paintings including The Vetruvian Man"(The Man in Action)" The Annuncation, The Mona Lisa, St. Jerome, Micahelangelo in Holy Family, Raphael in Crucifixion, Rembrandt in the self-portrait by and other art works. The goldenratio was especially used in the Renaissance and by the greeks and the romans. Various important proportions of Michelangelo’s amazing sculpture, David, are carved in the GoldenRatio...