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Golden Ratio

- Mike Pericht and John Murphy
- December 6, 2006

Golden Ratio

- An irrational number 1.618
- Many shapes have been defined by the G.R. in

Western Cultures - Reflects natures balance between symmetry and

asymmetry and the ancient Pythagorean belief - Numbers were not units, but expressions of ratio

Golden Ratio defined algebraically

- Greek symbol for Golden Ratio
- Algebraic Formulas

History

- In use for the last 2500 years
- Mathematicians, such as Pythagoras and Euclid,

spent numerous hours over the ratio and its

properties - Ancient Greeks study the G.R. because of its

constant appearance in Geometry - Pentagrams and Pentagons

History (cont.)

- Euclid gives first definition for the G.R.-
- 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 (Wikipedia.com)

Luca Pacioli

- Created ratio which captured the imagination of

artists, architects, scientists - Wrote a book called Divina Proportione (Divine

Proportion) - Describes ratios divinity by stating Just

like God cannot be properly defined, nor can it

be understood through words. Likewise this

proportion of ours cannot ever be designated

through intelligible numbers, nor can it be

expressed through any rational quantity, but

always remains occult and secret. It is thus

called irrational by mathematicians.

Golden Timeline

- Phidias (490-430 BC) made the Parthenon statues

that seemed to embody the golden ratio - Plato (427- 347 BC) describes five possible

regular (or Platonic) solids related to the

Golden Ratio Tetrahedron, cube, octahedron,

dodecahedron, and icosahedron - Euclid (circa 325- 265 BC) gave the first

definition (extreme and mean ratio) - Fibonacci (1170-1250 AD) discovered numerical

series which is closely related to the G.R. - Pacioli (1445- 1517) defines G.R. as divine

proportion - Johannes Kepler (1571- 1630) describes the G.R.

as a precious jewel

Timeline (cont.)

- Charles Bonnet (1720- 1793) states that in the

spiral phyllotaxis of plants going clockwise and

counter-clockwise were frequently two successive

Fibonacci series - Martin Ohm (1792- 1872) believed to be the first

to formally describe this theory as golden

ratio - Edouard Lucas (1842- 1891) names Fibonaccis

sequence its proper name - Mark Barr (20th Cent.) assigns Phi as name for

G.R. - Roger Penrose (1931- ) finds symmetry in the

field of periodic tilings using the G.R.

Golden Rectangle

- A Golden Rectangle is one whose side lengths are

in the Golden Ratio 1f, that is, approximately

11.618 - A distinctive feature of this shape is that when

a square section is removed, the remainder is

another golden rectangle, Square removal can be

repeated infinitely, which leads to an

approximation of the golden spiral - According to Mario Livio, since Luca Pacioli

wrote Divina Proportione in 1509, the Golden

Ratio started to become available to artists in

theoretical theories, that were not too

complicated. Many artists and architects have

proportioned their works to approximate the form

of the golden rectangle

Golden Spiral

- A Golden Spiral is a logarithmic spiral whose

growth factor b is related to the golden ratio

(f) - The Golden Spiral widens by a factor of f every

quarter-turn. A full turn widens by a factor of f

4. - The equation for the Golden Spiral is
- r ab?

Where to Find the Golden Ratio

- Paintings
- Architecture
- The human body

Golden Ratio in Art

- Da Vincis Mona Lisa is not only one of the most

famous paintings in the world, but also an

excellent example of the Golden Ratio in art. - If you draw a rectangle around Mona Lisa's face,

that rectangle will turn out to be golden. - The dimensions of the painting itself also form a

golden rectangle. - As well, the proportions of Mona Lisa's body

exhibit several golden ratios. - A golden rectangle can be drawn from her neck to

just above the hands.

Mona Lisa

Golden Rectangle in Architecture

- Studies have showed that the proportions of the

Parthenon approximate the golden ratio - The many facades of the Parthenon can be

circumscribed by Golden rectangles

Parthenon

The Golden Ratio in the Human Body

- The Golden Ratio is found all over the human body
- Examples The Face, Head, Teeth, Arm, Body, and

Ear

The Golden Ratio in the Face

- The blue line defines a perfect square of the

pupils and outside corners of the mouth. The

golden section of these four blue lines defines

the nose, the tip of the nose, the inside of the

nostrils, the two rises of the upper lip and the

inner points of the ear. The blue line also

defines the distance from the upper lip to the

bottom of the chin. - The yellow line, a golden section of the blue

line, defines the width of the nose, the distance

between the eyes and eye brows and the distance

from the pupils to the tip of the nose. - The green line, a golden section of the yellow

line defines the width of the eye, the distance

at the pupil from the eye lash to the eye brow

and the distance between the nostrils. - The magenta line, a golden section of the green

line, defines the distance from the upper lip to

the bottom of the nose and several dimensions of

the eye.

Golden Ratio in other Places

- The ear reflects the Golden Spiral
- Your teeth are based on Phi
- The front two incisor teeth form a golden

rectangle, with a phi ratio in the length to the

width. The ratio of the width of the first tooth

to the second tooth from the center is also phi. - The ratio of the width of the smile to the third

tooth from the center is phi as well.

Conclusion

- The Golden Ratio may be a tiny little number, but

this tiny little number - Is accountable for years of research and

discovery from several mathematicians - Has been found in the construction of

architecture, and in many artists paintings - Is located all over the human head/ face

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