Golden ratio
From Wikipedia, the free encyclopedia
In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.[1]
At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.
The golden ratio is often denoted by the Greek letter phi (Φ or φ). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:
This equation has as its unique positive solution the algebraic irrational number
Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, and the Greek letter phi (Φ).[2][3][4] Other terms encountered include extreme and mean ratio,[5] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[6] and mean of Phidias.[7][8][9]
1. Construct a unit square (red).
2. Draw a line from the midpoint of one side to an opposite corner.
3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle.
Contents[show] |
[edit] Calculation
| List of numbers γ - ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ | |
| Binary | 1.1001111000110111011… |
| Decimal | 1.6180339887498948482… |
| Hexadecimal | 1.9E3779B97F4A7C15F39… |
| Continued fraction |  |
| Algebraic form |  |
Two quantities a and b are said to be in the golden ratio φ if:
This equation unambiguously defines φ.
The right equation shows that a = bφ, which can be substituted in the left part, giving
Dividing out b yields
Multiplying both sides by φ and rearranging terms leads to:
The only positive solution to this quadratic equation is
[edit] History
The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years:
Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa (Fibonacci) and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal.
In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.
—Mario Livio, The Golden Ratio: The Story of Phi, The World's Most Astonishing Number
Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. The Greeks usually attributed discovery of this concept to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.
Euclid's Elements (Greek: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less."[5] Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio.[10] Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.[11] Some of these propositions show that the golden ratio is an irrational number.
The name "extreme and mean ratio" was the principal term used from the 3rd century BC[5] until about the 18th century.
The modern history of the golden ratio starts with Luca Pacioli's Divina Proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.
The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as "about 0.6180340," was written in 1597 by Prof. Michael Maestlin of the University of Tübingen in a letter to his former student Johannes Kepler.[12]
Since the twentieth century, the golden ratio has been represented by the Greek letter Φ or φ (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τομή—meaning cut).
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.[38] Zeising wrote in 1854:
[The Golden Ratio is a universal law] in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.
OR
a +b is to a as a is to b
WOW....MY HEAD HURTS.
one looks at the equation and says....so what?
another one either accepts and understands the equation and spends hours and days and years researching its beauty
still another one simply wonders what all the fuss is about
and finally another one (of which I am a long standing member) just simply gets a headache and needs to take a nap.
Math solutions and my brain are so far apart that sometimes it actually startles me....and then I look at some of my artwork and I see proportion and spacing mixed with curving geometric scale and I simply get a headache and need to take a nap.
It all comes down to nap time....which proves nothing has really changed since I was kindergarten....except the size of the cot.
The Golden Ratio
Hummmmmm........
MIchael Timothy McAlevey
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