Vibrate geometric forms in Euclidian space

Transcription

1 Vibrate geometric forms in Euclidian space CLAUDE ZIAD BAYEH 1, 1 Faculty of Engineering II, Lebanese University EGRDI transaction on mathematics (003) LEBANON claude_bayeh_cbegrdi@hotmail.com NIKOS E.MASTORAKIS WSEAS (Research and Development Department) Agiou Ioannou Theologou , Zografou, Athens,GREECE mastor@wseas.org Abstract: - The Vibrate geometric form is the study of geometric objects in the Euclidian D space or 3D space that have certain vibrations in their shapes and forms. Usually these kinds of objects are not studied in the known geometry of Euclidian D space or 3D space because they do not exist in mathematics as normal objects such as circle, triangle, quadrilateral form, Rhombus, rectangle, hexagonal form or any other known form in the And usually small vibrations are neglected from the study of shapes in mathematics. But the vibrate geometric forms exist in the nature for example flowers, viruses, sponges and many other forms that are not usually studied in Euclidian D space to determine their exact volume, perimeters and surfaces (areas). The main goal of introducing the Vibrate geometric form is to study these vibrating shapes in Euclidian space and to determine their perimeters, surfaces and volumes in simple methods and formulae. Key-words:-Euclidian space, Geometric forms, Vibration forms, D space, 3D space, Perimeter, Volume. 1 Introduction Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space [1-7]. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment Euclidean geometry set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. A mathematician who works in the field of geometry is called a geometer. form in the nature and how to calculate its perimeter, surface and volume. In this section, we will treat the rectangular vibrating form. The main form is the rectangle, but there exist vibration on the rectangle as mentioned in the figure 1. So to calculate the volume, area and perimeter of this shape we have to follow the following equations. In this paper, the author developed the study of vibrating forms that exist in the nature and tries to extract their volumes, perimeters and surfaces. In this paper, two forms are chosen and treated in order to give an idea about how we can chose a vibrating ISBN:

2 Fig.1: presents the rectangular vibrating form. Fig.3: presents the rectangular vibrating form. Fig.: presents the rectangular vibrating form. xx is the height of the horizontal triangle. It is considering positive if it is inside the rectangle and it is considered negative if it is outside the rectangle. yy is the height of the vertical triangle. It is considering positive if it is inside the rectangle and it is considered negative if it is outside the rectangle. if xx > 0 then yy 0 in order to prevent the overlapping. if yy > 0 then xx 0 in order to prevent the overlapping. The other cases are permitted.. Calculation of the volume of the Volume is equal to the surface multiplied by the height of the rectangle in the third dimension..1 Calculation of the surface of the SS = LL HH xx ddh nn yy mm SS = LL HH xx HH yy LL (1) The equation (1) is the equation that gives the surface of the formed vibrated shape of a rectangle. It is a particular case when the heights of two opposite sides are symmetric as presented in figure 1 and 1.1. If the heights of the opposite sides are not symmetric (such as in figure 3) then the equation will be as following: SS = LL HH xx 1 ddh 1 1 xx ddh yy 1 ddll 1 mm 1 yy ddll mm () Fig.4: presents the rectangular vibrating form in 3D VV = SS. HH ZZ = HH ZZ (LL HH xx ddh nn yy mm) SS = HH ZZ (LL HH xx HH yy LL) (3) The equation (3) is the equation that gives the volume of the formed vibrated shape of a rectangle in 3 dimensions It is a particular case when the heights of two opposite sides are symmetric as presented in figures 1, and 4. If the heights of the opposite sides are not symmetric (such as in figure 3 and 5) then the equation will be as following: ISBN:

3 VV = SS HH ZZ = HH ZZ (LL HH xx 1 ddh 1 1 xx ddh yy 1 ddll 1 mm 1 yy ddll mm ) (4) RR = yy + (6) RR = yy + The perimeter is equal to PP = 4 HH ddh xx + ddh + LL yy + (7) PP = 4 HH xx nn HH LL yy mm LL + 1 (8) With nn, mm, HH, xx, yy and LL are determined values. Fig.5: presents the rectangular vibrating form in 3D.3 Calculation of the perimeter of the In case we have a symmetric height in both sides horizontally and vertically as in figure 6, then the equation can be easily determined as following: Fig.7: presents the rectangular vibrating form in D space which we can calculate its perimeter, volume and surface with simple expressions as mentioned in the previous sections. 3 Circular vibrating form In this section, we will treat the circular vibrating form. The main form is the circle, but there exist vibration on the circle as mentioned in the figure 8. So to calculate the volume, area and perimeter of this shape we have to follow the following equations. Fig.6: presents the rectangular vibrating form. RR = xx + ddh (5) RR = xx + ddh Fig.8: presents a part of a circle vibrating form. ISBN:

4 3.1 Calculation of the surface of the circular vibrating form We consider that the angle is infinitesimal. So RR = rrrrrr 0 = RR with [0,ππ] And the surface of the triangle formed by ABC is equal to ddss AAAAAA = xx AAAA = xx RR sin ( ) Or 1 therefore, sin ( ) ππ SS = RR xx RR 0 SS = ππrr 1 xx (9) RR The equation (9) is the surface of the circle with infinitesimal triangles with, xx is the depth of the triangle. It is considering positive if it is inside the circle and it is considered negative if it is outside the circle. Then, VV = ππ 4γγ 3ππ ππrr 1 xx RR = 4 3 ππππrr 1 xx RR And ππrr 1 xx RR = ππ γγ γγ = RR 1 xx RR VV = 4 3 ππrr3 1 xx RR 3 (10) 3. Calculation of the perimeter of the circular vibrating form Fig. 10: presents a part of a circle vibrating form. Fig.9: presents the circular vibrating form. 3. Calculation of the volume of the sphere formed by the circular vibrating form The volume of a Sphere with vibrating form is equal to: VV = ππ dd SS With dd = 4γγ and γγ = ff(xx, RR) 3ππ The surface of the half circular vibrating form is equal to SS = ππrr 1 xx RR In figure 10, as the angle is infinitesimal then we can consider the following: AAAA = RRRRRR CCCC = xx + AAAA = xx + RRRRRR CCCC = xx + RRRRRR CCCC = xx + RRRRRR The circle contains nn small triangle therefore, PP = nn xx + RRRRRR When nn = ππ nn = ππ PP = 4ππ xx + RRRRRR (11) The equation (11) gives the perimeter of circular vibrating form. In the same manner we can calculate the surface, volume and perimeter of any vibrate object in the 4 Conclusion As conclusion, The Vibrate geometric form is the study of geometric objects in the Euclidian D space or 3D space that have certain vibrations in their shapes and forms. Any object in the nature has a ISBN:

5 vibration in its form, it can be considered as a part of this geometry and it has special equations to calculate its exact form. This is not the case of the traditional Euclidian space in which the calculated shapes and forms have definite and known forms such as circle, triangle, rectangle, quadrilateral form This geometry is a part of the Euclidian geometry but specialized for calculating the vibrating forms in the nature. References: [1] Martin J. Turner, Jonathan M. Blackledge, Patrick R. Andrews, "Fractal geometry in digital imaging". Academic Press, ISBN , (1998). [] Kline, "Mathematical thought from ancient to modern times", Oxford University Press, (197), p [3] J. Friberg, "Methods and traditions of Babylonian mathematics. Plimpton 3, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, (1981), pp [4] Neugebauer, Otto, The Exact Sciences in Antiquity ( ed.). Dover Publications, ISBN , Chap. IV "Egyptian Mathematics and Astronomy", (1969), pp [5] Eves Howard, An Introduction to the History of Mathematics, Saunders, (1990), ISBN , p [6] Kurt Von Fritz, "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics, (1945). [7] James R. Choike, "The Pentagram and the Discovery of an Irrational Number". The Two- Year College Mathematics Journal, (1980). ISBN: