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 golden ratio“. 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 golden ratio in everydays objects such as tables, couches, doors,posters, books and etc.

Because it is very pleasing to the eye, the golden ratio is used alot in art. Leonardo da Vinci used the golden ratio 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 golden ratio 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 Golden Ratio which looks stunning. In the human body, if we’d took a good look in the mirror we would notice that most of our body parts follow the numbers one, two, three and five. You have one nose, two eyes, three segments to each limb and five fingers on each hand. The proportions and measurements of the human body can also be divided up in terms of the golden ratio. Plenty of them think that the closer you are to the golden ratio, the more beautiful you are. If you divide the DNA spiral by its diameter, you get the ratio again. In nature we can find the goldren ratio in leaves,flowers... One trunk grows and produces a branch, resulting in two growth points. The main trunk then produces another branch, resulting in three...

...
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 about Phidias’s life, he is...

...GOLDENRATIO- maths project
Index
Serial no. chapter
1 | Introduction |
2 | History |
3 | In nature |
4 | In human body |
5 | In architecture |
6 | In art |
7 | In day to day life |
8 | SIGNIFICANCE |
ACKNOWLEDGEMENT
I would like to express my special thanks of gratitude to my teacher sonali durgam on the topic goldenratio, which also helped me in doing a lot of Research and I came to know about so many new things. I am really thankful to her.
Secondly I would also like to thank my parents and friends who helped me a lot in finishing this project’s information finding work.
I am making this project not only for marks but to also increase my knowledge.
THANKS AGAIN TO ALL WHO HELPED ME
Introduction
The goldenratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron. It is denoted , or sometimes .
The designations "phi" (for the goldenratio conjugate ) and "Phi" (for the larger quantity ) are sometimes also used (Knott), although this usage is not necessarily recommended.
The term "golden section" (in German, goldener Schnitt or der goldene Schnitt) seems to first have...

...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...

...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...

...has never been deciphered. Until, the discovery of a natural ratio, that changed the game of beauty in all aspects. “Many of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kelper, to present day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties, called the goldenratio”(Livio, 2007). This can be located in thousands of natural and manmade objects, and is believed to hold the key to the secret of beauty. Still, no one knows who first discovered the goldenratio, but it is known that the Egyptians used it in the creation of the Great Pyramids at Giza. In addition, Phidias applied the goldenratio to the design of the Parthenon (Livio, 2007). These two marvelous structures were the first known cases of where the goldenratio had been used. The goldenratio started to make its mark on the world when it was first applied to the arts.
Modern artists do not receive the fame or glory from their artwork, and some of the artists are forgotten. However, the artist of the renaissance period are still praised and remembered well after their time, for their work. Their works of art whether it be a...

...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,...