Golden Ratio Term Paper

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 of Escher can be acknowledged as the real physical objects. The artifice of Escher resembles such objects when viewed from a definite angle. Many types of three dimensional models were designed and built adopting geometric modeling and computer graphics tools. The few of them can be grouped as under- Convention: the figures are frequently presented in pairs in such a manner that the left figure in each pair is the front view in the direction of the Escher's drawing while the right figure provides a general view. When a real and tangible model has been generated it will be demonstrated as a second pair to the right of the pair of computer rendered images. (Escher for Real)

The Penrose Triangle was independently invented by Oscar Reutersvard. The specific shape of Penrose Triangle is constructed as a C^0 continuous sweep surface with a square cross section that rotates as we travel along the edges. The Penrose Triangle plays a significant part in MC Escher's drawing. Similarly, the structures in terms of Penrose Rectangle, Penrose Pentagon, Penrose Triangle II, Escher's cube, Escher's Moebius Ring Ducks and Escher's Moebius Ring- Ants, Escher's Waterfall, Escher's Belvedere, Escher's relativity, etc. are formed. (Escher for Real) the crucial phenomenon of Escher drawing involves- domain of the artist depends upon but rests beyond, craft. Escher's art generates perceptions into a pluralistic concept of the world and Escher's concept of plurality does not indicate chaos but order -- a contact between structures and acknowledgeable motifs. Further the art of Escher is concerned with illusions of reality and the design uses mathematical and geometric properties. (M.C. Escher: Beyond the Craft)

Bibliography

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