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Golden Section

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Golden Section


Golden rectangle


Fig. 2 Circle and square based on Golden Section


Fig. 2b Squaring the circle. Image on the right: Squaring the circle: the areas of this square and this circle are equal. Image on the left: Circumference of the circle equals the perimeter of the square.

RH 9 14 186 124 88 160 114 16:17, September 23, 2012 (UTC)

Human Proportion

These drawings are based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architecture. Vitruvius described the human figure as being the principal source of proportion among the Classical orders of architecture. His drawing is named in honor of the architect.

A method to construct a golden rectangle. The square is outlined in red. The resulting dimensions are in the golden ratio

Mathematician Mark Barr proposed using

the first letter in the name of Greek

sculptor Phidias, phi, to symbolize the

golden ratio. Usually, the lowercase

form (φ) is used. Sometimes, the uppercase

form (Φ) is used for the reciprocal of the

golden ratio,

'A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship. Two quantities a and b are said to be in the golden ratio 'φ Two quantities are said to be in the golden ratio, if "the whole (i.e., the ... b² b. φ = 1 ± √5. 2. a + b = a. a b. a = φ. b. φ = 1 + √5 ≈ 1.618. 2. x² + bx + c = 0, two quantities are in the golden ratio if the ratio of the sum of ... a/b=(a+b)/a=(a+b+a)/(a+b)=phi. (the golden ratio). One method for finding the value of 'φ' is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/'φ',

Phi 1 618 davinci

1.618 phi

Fig. 2b 'shows a circle with Radius = 1 and a square with side = 1.571. The Circumference of the Circle = 6.28... [2 x Pi = 6.28] The square with side 1.571 has perimeter equal 6.28 [4 x 1.571 = 6.28].

'Fig. 2b shows a circle with Radius = 1 and a square with side = 1.772. The Area of the circle is 3.14 [as determined by pi multiplied by the radius squared]. The area of the square is also 3.14... [1.772 x 1.772].

The "Golden Ratio" is a mathematical ratio of 1.618:1, and the number 1.618 is called "Phi" Phi is the Golden Section of the Greeks. It was said to be the first section in which the One became many,The Golden Ratio can be expressed as 1.618 and 0.618 and is known as Phi and phi, respectively; phi being the reciprocal of Phi Pi = 3.1416... Pi is found in any circle, A geometric construction of the calculation of phi based on the square root of five.

Also referred to as the "'Golden section'" and the GOLD MEAN the Golden mean is an ancient fine arts formula that mathematically defines a rectangle of specific proportions, the golden section is a line segment divided according to the golden ratio (approximately 1.6180339887): The total length a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b

a+b is to a as a is to b

the blend of art and science during the Renaissance and provides the perfect example of Leonardo's keen interest in proportion. In addition, this picture represents a cornerstone of Leonardo's attempts to relate man to nature. Encyclopedia Britannica online states, "Leonardo envisaged the great picture chart of the human body he had produced through his anatomical drawings and Vitruvian Man as a cosmography Del minor Mondo (cosmography of the microcosm). He believed the workings of the human body to be an analogy for the workings of the universe." It is also believed by some that Leonardo symbolized the material existence by the square and spiritual existence by the circle.


Fig. 3 The simplest way to describe the geometrical construction of the Vitruvian Man.

The simplest composition is based on a square, which is duplicated and rotated 45º to form an octagram. The distance between the base line of the first square and the apex of the rotated one simply represents the diameter of the circle.

If a circle has radius = 1 unit, square side is equal to:

1.656 For Vitruvian Man 1.618 for Golden section construction 1.571 for the condition: circumference of the circle = perimeter of the square 1.772 for the condition: area of the circle = area of the square

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.
Step 1 draw a square and circle radius r1 as shown on fig 4

Step 1 draw a square and circle (radius R1) as shown on Fig 4

Vitruvian Man - methods of geometrical construction of the circle and the square

Step 2 move the circle so point a overlaps with point b see fig 5

Step 2 move the circle so point a overlaps with point b see Fig 5

Step 3 locate centre last circle point 0 by dividing distance ab in half draw new circle with radius r2-0a see fig 6

Step 3 locate centre of last circle point 0 by dividing distance ab in half, draw new circle with radius R2-0a see Fig 6

Fig 7

Vitruvian Man

1681 is a perfect square

41 X 41 = 1681 same numbers as Phi divided by 1000

1679 divided by 1038 = 1.618 phi

1618/1.618 phi = 1000 1000/1000 = 1 and is a Factor of 1679

1679 is an Aricebo number 1679/104.937 = 16.


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