The golden ration can occur anywhere. 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. The golden ratio is a term used to describe proportioning in a piece. In a work of art or architecture, if one maintained a ratio of small elements to larger elements that was the same as the ratio of larger elements to the whole, the end result was pleasing to the eye. The ratio for length to width of rectangles is 1.61803398874989484820. The numeric value is called "phi". The Golden Ratio is also known as the 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. The Golden Rectangle is a unique and important shape in mathematics. The Golden Rectangle appears in nature, music, and is often used in art and architecture. Some thing special about the golden rectangle is that the length to the width equals approximately 1.618

Golden Ration = Length = 1.6
Width

The golden rectangle has been discovered and used since ancient times. Our human eye perceives the golden rectangle as a beautiful geometric form. The symbol for the Golden Ratio is the Greek letter Phi.

The Fibonacci Series was discovered around 1200 A.D. 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.

The Golden Ratio in Architecture and Art
Throughout the centuries, artists have used the golden ratio in their own creations. An example is "post" by Picasso. When using a golden mean gauge you can see that the lines are spaced to the Golden...

...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...

...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...

...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 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...

...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...

...Secret of golden mean ratio:
1.618~(or its inverse 0.681~): number of goldenratio is a mystery of ka`aba. Unbelievely, the closer a ratio to this number in an object or a system of objects ,the more beautifull it is manisfied.
Phi constant: 1.681~; superior design number of mathematics, is a repairing decimal , which never ends. This formula is b/a.
The creator has always used the very same number in numerous events in the universe even in our bodies. The aspect ratio of DNA spiral, I dodecahedron, phylotaxy , I the snowflake crystals,in the spiral structure of numerous galaxies. The creator used the same number;the number of goldenratio which is 1.681~.
As a result of his 25 years long study, aesthetic Dr.Steven Markout proves that each of human faces and bodies , created pursuant to this ratio, are compeletly beautifull. If the realative ratio is 1.681 for the components of any structure ,then this form will be convenient to goldenratio, the perfect design.
It is determined that this ratio has been used for the design of various reputable architecture structures, even including pyramids in Egypt. Famous astronomer kepler defined this number as a great treasury.
So, where is the goldenratio point of the world?
The propotion of distance between Mecca and...

...The GoldenRatio
The goldenratio is a number used in mathematics, art, architecture, nature, and architecture. Also known as, the divine proportion, golden mean, or golden section it expresses the relationship that the sum of two quantities is to the larger quantity as is the larger is to the smaller. It is also a number often encountered when taking the ratios of differences in different geometric figures.
Represented mathematically as approximately 1.618033989, and by the Greek letter Phi, the number tends to show up frequently in geometrical shapes. For example, the goldenratio is the basis for the construction of a pentagram. This shape looks like a regular star; five straight lines form a star with five points. The pentagon within the star in the center is proportional to the points of the star by a ratio of 1: 1.618.
The goldenratio appeared so much in Geometry, as stated above with the pentagram example, that it intrigued the Ancient Greeks. They studied the ratio for most of the same reasons mathematicians study it today. They found it to have unique and interesting properties. It is said that the Parthenon, among other Greek architecture have many proportions approximate to the goldenratio. Other classical buildings and structures have been...