Golden Ratio

¶ … Real Life Golden Ratio: The golden ratio is acknowledged as the divine proportion golden mean, or golden section is represented as a number mostly confronted while considering the ratios of distances in simple geometric diagrams like pentagram, decagon and dodecagon. It is indicated by the symbol 'phi'. The concept 'golden section' was first used by Martin Ohm in the 1835 in his book Die Reine Elementar-Mathematik. The first ever English use was seen in the article of James Sulley in 1875 which appeared in the 9th edition of the Encyclopedia Britannica.

The symbol 'phi' was first used by Mark Barr at the inception of the 20th century in commemoration of the Greek sculptor Phidias, who was an extensive user of golden ratio in his works. Phi has surprising linkage with the continued fractions and the Euclidean algorithm for enumerating the Greatest Common Divisor of two integers and is also known as the Pisot Number. (Golden Ratio: mathworld.wolfram.com) Conventionally, Golden Ratio was well understood by the Egyptians who applied it for construction of their pyramids.

However, it was extensively popular under the application by the Greek geometers. Till 1946 the very concept of 'Golden Ratio' was however, not originated, when the mathematician Friar Pacioli published a paper named 'De Divna Proportione' where he made a mention about the ratio as a divine number, one notices everywhere in nature. (Inter.View to George Cardas - Cardas Cables - a brief introduction to Golden Ratio) Most of us are coming across the number 'Pi' and take it to be the most omnipresent irrational number ever known to man.

However, the 'Phi' is taken to be another irrational number and has the same propensity for popping up and is not as popular as 'Pi'. 'Phi' is also visible in several geometrical shapes, however, rather than indicating it as an irrational number, we can visualize it as a ratio in the manner described in the diagram. Taking a line segment we can conveniently split it into two segments a and B.

is such a manner that the length of the entire segment is to the length of the segment a as the length of segment a is to the length of segment B. A calculation of such ratios derives a close approximation to the Golden Ratio. Another geometrical diagram related to the Phi is the Golden Rectangle. This particular rectangle has sides a and B. that are in the ratio equivalent to the golden ratio.

(Phi: That Golden Number) Historically, the proportion of length to width of the rectangles of 1.61803 39887 49894 84820 is visualized to be the most enjoyable. The distance between the columns of the structure created by Greek sculptor Phidias and it presently exists in Athens, Greece which forms golden rectangles. Phidas extensively applied the golden ratio in his sculpture. The external dimensions of the Parthenon in Athens, constructed in about 440 BC constituted a kind of perfect golden rectangle.

Even the dimensions of great Pyramid of Giza believed to have been built about 4600 years ago much before that of the Greeks were based on Golden Ratio and Golden Rectangle. Many artists who prevailed after Phidias have applied this proportion; Leonardo Da Vinci acknowledged it to be the 'divine proportion' and adopted it in many of his paintings. The famous art 'Mona Lisa' can be examined minutely to find out that the measurements of the rectangles drawn around her face are in golden proportions.

(the Golden Ratio: (http://www.geom.uiuc.edu/) In addition to the application of Golden Ratio in Architecture, it is also applicable to HiFi and Music. To illustrate the standard AES listening room is also known as a Golden Cuboid, where the dimensions are in golden ratio to each other. The cabinets of the most of loudspeakers have golden ratio 'inner' dimensions. More generally, whenever, it is felt to reduce or maximize harmonic resonances the Golden ratio is considered as effective way out.

(Inter.View to George Cardas - Cardas Cables - a brief introduction to Golden Ratio) Fractals: Fractals refer to geometrical diagrams as like the rectangles, circles and squares. However, the fractals represent unique properties those are not prevalent in case of these geometrical figures. (What are Fractals? A Fractals Unit for Elementary and Middle School Students) Fractals are fascinating images to which the people are quickly attracted. Fractal geometry blends art with mathematics to represent that equations are more than mere collection of numbers.

The fractal geometry assists us in modeling visually what is demonstrated in nature; the most acknowledged being coastlines and mountains. The fractals are applied to model soil erosion and to examine seismic patterns also. However, in addition to its extensive application in narrating the complex natural patters the fractals can assist varying the common impression of the students that mathematics is dry and not reachable and may assist to induce the mathematical discovery in the classroom.

(the Fractal Microscope: A Distributed Computing Approach to Mathematics in Education) The fractal geometry was popularized under the Benoit B. Mandelbrot in terms of Mandelbrot set who formulated the concept fractal in 1975 from the Latin fractus or 'to break'. The Mandelbrot set is the set of all the points that continue to be bounded for every iteration of z = z*z + c on the complexity level where the originating value of z = 0 and c is permanent.

With the assistance of NCSA supercomputers and two programs written by Michael South and Dr. Robert M. Panroff functioning with the Education Group at NCSA, it is quite feasible to find out many common elementary mathematical principles while exploring the Mandelbrot set. A program has been devised termed as Fractal Microscope that permits anyone to zoom in and out of the Mandelbrot set very quickly and easily by simply indicating and clicking under the Macintosh platform.

Another program Starstruck has been developed the path generated through the Madlebrot set by iteration. (the Fractal Microscope: A Distributed Computing Approach to Mathematics in Education) The prolonged history of fractals reveals that these structures were found out even before the actual coinage of the concept. Karl Weirstrass discovered an example of a function with non-intuitive property in 1872 which is continuous every where but not differentiable anywhere, the graph of which is presently known as fractal.

Helge Von Koch being discontented with the very abstract and analytic definition of Weirstrass and propounded a more geometrical view in terms of Koch Snowflake. During 1938 the initiative for construction of self-similar curves was advanced by Paul Pierre Levy in his paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole in terms of Levy C curve.

Fractals can be categorised under three groups: Iterated function systems indicating fixed geometric replacements; the fractals generated by recurrence relation at each point in a space; and Random fractals created by stochastic instead of deterministic processes. (Fractal. Wikipedia, the free encyclopedia) Fractals emerged to be the significant questions twice prior to invention of the computers. For the first time when the British map makers invented the problem with quantifying the distance of coastal area of Britain.

It was evident that the more zoomed in the maps were visualized the more detailed and longer the coastline got. However, rarely it was understandable to be a property of fractals- the circumstances of finite area being bounded by infinite line. The second incidence of emergence of the pre-computer fractals was indicated by the French Mathematicians Gaston Julia.

He was amazed when a complex polynomial function would visualize like one named after him in form of z^2 + c, where c is a complex constant with numbers which are real and imaginary. (What are fractals: (www.jracademy.com) The inherent essence behind the formula is that x and y coordinates of a point is considered and connected them into z in the form of equation x +y*i where I is the square root of negative one, this number is squared and then c is added as a constant.

The resulting pair of real and imaginary number is plugged back into z, the equation is again run and this is continued until the result is greater than some number. The number of times the equation is run to take out of its orbit a color is assigned and then the pixel (x, y) becomes that color, unless those coordinates cannot get out of their orbit, in that case they are made black. The underlying attribute of fractals represent a large magnitude of self similarity.

This indicates that they normally involve miniature copies of themselves included within the original. This also involves minute details. (What are fractals: (www.jracademy.com) Thus the fractals are taken to be the geometric shapes which have complexity and are minutely detailed. An attempt to zoom in on a section will reveal as much detail as the entire fractal.

One of the procedure to think of fractals for a function f (x) is to take into account x, f (x), f (f (x)), f (f (f (x))), f (f (f (f (x)))), and so on. One of the applications of Fractal is seen in respect of Seirpinski's Triangle. It is initiated as a triangle and the new iteration of it generates a triangle with the mid points of the other triangles of it. Another illustration of fractal is Koch Snowflake.

It initiates with a triangle and go on adding triangles to its trisection points that exert outward points for all infinity. (Chaos Theory and Fractals) Platonic Solids: Platonic Solids indicates to the polyhedron where all the faces are congruent regular polygons. Evidentially, same number of faces converges at every vertex here. (the Platonic Solids: The University of Utah) the platonic solids are also known as regular solids or regular polyhedra and taken to be the convex polydedra with same number of faces consisted of convex regular polygons.

Euclid in the last proposition of the Elements afforded to prove exactly five such solids like cub, dodecahedron, icosahedron, octahedron and tetrahedron. Cromwell in 1997 applied the term cosmic figures to mean collectively to both platonic solids and Kepler-poinsot solids. The platonic solids were understood by the Greeks and were narrated by Plato in his magnum opus Timaeus ca. 350 BC.

Plato in this work exemplified tetrahedron to be fire, the cube to be the earth, the icosahedrons to be the water, the octahedron with air and the dodecahedron to be the stuff of which the constellations and heavens were generated. Schlafli during 1852 could establish that there existed exactly six regular bodies with Platonic properties in four dimensions three in five dimension and three in all higher dimensions. (Platonic Solid: mathworld.wolfram.com) The Greeks could acknowledge that there existed only five platonic solids.

The crucial surveillance was that the internal angles of the polygons converging at a vertex of a polyhedron add to less than 360 degrees. Taking this into account it can be perceived that if such polygons converge in a plane the interior angles of al the polygons converging at a vertex would add to exactly 360 degrees. Considering all the possibilities of number of faces converging at a vertex of a regular polyhedron all the possibilities can be found out.

Triangles: since the interior angle of an equilateral triangle is 60 degree, there is possibility of only 3, 4 or 5 triangles that can converge on a vertex. In case of more than 6 triangles their angles would add up to at least 360 degrees which would be impossible. When three triangles converge at each vertex this gives rise to Tetrahedron. (the Platonic Solids: The University of Utah) Similarly four triangles converge to result in an Octahedron, 5 triangles converge to result in an Icosahedrons.

Squares: As the interior angle of the square is 90 degrees a maximum of three squares can converge at a vertex. This is of course possible and it results in hexahedron or cube. Pentagons: as in case of cubes there is the possibility of conversion of only three pentagons at a vertex which results in a Dodecahedron. Hexagons: or regular polygons cannot have more than six sides to form the faces of a regular polyhedron since their interior angles are at the minimum of 120 degrees.

(the Platonic Solids: The University of Utah) the shapes are sometimes used to make dice since dice of such shapes can be made fair, 6 sided dice are very normal, but the other numbers are normally applied in role-playing games. Such dice are normally referred to as D. along with the number of faces it contains like d8, d20 and so on. (Platonic solid: Wikipedia, the free encyclopedia) Escher: The artifice of M.C. Escher is quite amazing. However, most of the so called impracticable drawings.