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ATINER CONFERENCE PAPER SERIES No: MAT2013-0695
Athens Institute for Education and Research
ATINER's Conference Paper Series
Global Positioning System as it is
Related to Trisecting Angles
Miami Dade College
ATINER CONFERENCE PAPER SERIES No: MAT2013-0695
Athens Institute for Education and Research
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ISSN 2241-2891 5/11/2013 2 ATINER CONFERENCE PAPER SERIES No: MAT2013-0695 An Introduction to ATINER's Conference Paper Series ATINER started to publish this conference papers series in 2012. It includes only the papers submitted for publication after they were presented at one of the conferences organized by our Institute every year. The papers published in the series have not been refereed and are published as they were submitted by the author. The series serves two purposes. First, we want to disseminate the information as fast as possible. Second, by doing so, the authors can receive comments useful to revise their papers before they are considered for publication in one of ATINER's books, following our standard procedures of a blind review.
Dr. Gregory T. Papanikos President Athens Institute for Education and Research 3 ATINER CONFERENCE PAPER SERIES No: MAT2013-0695
This paper should be cited as follows:
Olivier, F. (2013) "Global Positioning System as it is Related to Trisecting Angles" Athens: ATINER'S Conference Paper Series, No: MAT2013-0695.
4 ATINER CONFERENCE PAPER SERIES No: MAT2013-0695 Global Positioning System as it is Related to Trisecting Angles Frantz Olivier Lecturer Miami Dade College USA
AbstractMany problems of geometry seem to have persisted over the years. The most famous: “the three problem of antiquity”. 1 We are only going to consider the problem of trisecting angles.
a) Dividing an arbitrary angle into three equal parts subject to one main restriction
b) You are allowed to use only a straight edge and compass as tools in your construction.
Since 450 BC the initial search by Hippias of Eliss on the trisecting problem, a valid purely geometric solution was not available until Gauss suggested a way with his study of regular polygons.2In the article Modal logic the author introduce us to necessity as a mother of modality. 3 Here we view mathematics as a set of worlds accessible to each other thus the treatment of trisecting an angle is proposed using calculus. We need to define the geometric series as a set of aggregates elements of regions that converge to a total covering of limit 1/3. Each element c1c2c3…cn converges closer to 1/3 as the series of circles degenerate in magnitude, namely Georg Cantor’s disappearing table. Thus we will reach a critical region that is no bigger in magnitudes than a point, theorem 1.
4 it is impossible to trisect a 600 angle.4 We will construct an indirect trisection of angle 600. This is where Global positioning system plays an essential role. In order to correctly locate a point in space, standard algebraic equations, combined with measuring equipment, geometry and a known point are used. A GPS method needs to move a known point in the opening of the 600 degree angle.
• This can be achieved by using vector Projection( Theorem 2)
• The equation cos (3ϴ) =4cos3(ϴ) − 3cos(ϴ) which the corner stone on how trisecting was proven not to be possible. However, we may be able to shed
more light on the subject at hand. We can define angle as a dynamic notion:
1 Number Theory and its History by Oystein Ore page 340 2 “Angular Unity” The case of the missing Theorem, P 17 by Leon O. Romain 3 Modal logic should say more than it does. P113 computational logic Lassez & plotkin 4 Experiencing Geometry Euclidean and Non-Euclidean with History 3rd Edition by David W.
Henderson ;Daina Taimina page 216
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Angle as movement.1 The overall strategy is as the angle change position the vectors already in fixed position will eventually intersect with the angle in motion. This will create the environment where any arbitrary angle can be divided into three equal parts. Theorem 3.
• In a Heptagon, One of the angles that we found in our transformation triangle is 510428. Currently it is accepted that a heptagon cannot be built using straight edge and compass however the latter can be built with a mark ruler.2 This implies for us that we need to build a unit for the construction of the angle. Vector Mapping thru projection of unit 1/6 across a line of unity. The Star of David and a pentagon must be built as a piece wise graphic function to achieve the Central Angle of a Heptagon (7sides polygon 510.428). The question that one may ask is certainly: How did we get there? The methods used in the past to tackle trisecting have not worked. We posit that Model logic offers a better option. Per this model, the science of mathematics is viewed as a set of worlds accessible to each other, and we propose that the treatment of trisecting an angle can be achieved using calculus. The worlds in question are that of geometry and calculus, where a difference of reality exists, whereas limits or sequences exist in calculus but not in the geometry. These ideas will be further explored.
1 Experiencing Geometry Euclidean and Non-Euclidean with History 3rd Edition by David W.
Henderson ;Daina Taimina page 38 2 Geometry Our cultural Heritage by Audun Holme page 93.
Sequence Concept and Consequences A number can be represented in two ways: as a finite point or as a sequence. A sequence is defined as a list of rational numbers that either converges to a point or diverges. A finite point is a number. Extrapolating further on this definition, the problem is envisioned as stated: Can a computer be formatted to convey the difference between information and knowledge? If information is seen as a sequence and knowledge as a limit, a computer can possibly project a correlation between these two pieces of data. Subsequently, limit could be interpreted as new information which could lead to new knowledge. This process, replicated over and over again, will eventually lead to the emergence of artificial information as a concept. Based on this proposition, a tool that can perform the trisecting of regions can be conceived and built.
Review of the Greek Methods
The Greeks, in planar geometry, introduced us to the concept of points, lines, planes, bisection of angles, and so on. These concepts were explained most of the time through the ideas of intersection because the Greek understood that trough intersection they could achieve accuracy in measurements. Henceforth, geometry emerged as a precise science. Let us
analyze this approach further:
A point could be explained as the intersection of two line segments. A line segment is a portion of a line, which means it has a beginning point and no end point. A line is the intersection of two planes. This is obvious that the concept of intersection plays a central role in the Greek concepts of planar geometry.
Looking back at these fundamental concepts one must realize that, in the past, celestial bodies were used as point of references in travelling by sea or by land. Astrology was also used in the prediction of life cycles and so on. In keeping with tradition, to solve the problem of trisection of angles, one should analyze the concept of a global positioning system.
A tool to achieve trisection is critically needed. Fortunately, there’s a sequence that can achieve this, if the angle to be trisected is viewed as a space that can be divided into an infinite partition. Namely, ¼ +1/16+1/64+1/256…This infinite partition behaves like an aggregate and areas which have a limit of 1/3 will be obtained. This implies that the limit of the series is 1/3 (¼ div (1-1/4) =1/4 div ¾=1/3 the limit L=1/3). Essentially the sequence of partition is formally stated as 1/4+1/16+ 1/64…1/4n. The geometric series are defined as a set of aggregates elements of regions that converge to a total covering of limit 1/3. Each element c1c2c3…cn converges
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closer to 1/3 as the series of circles degenerate in magnitude i.e. each partial sum contains a limit point that is dense. If the space under study is finite each sub-region is greater than the remaining region in the covering. And a critical region will be reached that is no bigger in magnitude than a point.
The resulting point will contain the remaining elements of the infinite set of aggregated elements. This is significant since trisecting a 600 angle will exhibit the same geometrical structure that stated here, namely the trisecting of a 600 angle will be completed in space allocation long before the same precision numerically is achieved. The two sets are equivalents except that the space collapses to a point. However, the point is small omega infinite in nature1. The concept of Georg Cantor disappearing table2 is closely linked to this global activity.
Illustration of partial covering and implication of the infinite geometric
Let S1=1/4→1/4; ¼ is what % of 1/3? →75.7575% 2 S2=S1+1/4 →1/4+1/16=5/16; 5/16 is what percent of 1/3? → 93.4375% S3=S2+1/64→5/16+1/64=21/64; 21/16 is what percent of 1/3? → 98.4375 % S4=S3+1/256→21/64+1/256=85/256; 85/256 is what % of 1/3? → 99.60937%
Now that “covering” is explained, some facts about the sequence can be examined. The trisecting sequence began with a ratio r=an//an-1 which led to the geometrical sequence. And the nth derivative of the sequence of the last (n) can be formulated as an equation y=an commonly known as an exponential function. It behaves as y=ex thus the rate of change d/dx(ex) = ex. Thus, if we extrapolate d/d(n) (1/4n)→(1/4)n. When applying the rate of change to the 4th power→ (.25)4=0.00391, implies that 99.61% of the region is covered leaving 1 See page 12 Graphic proof for an arbitrary angle trisection 2 see Mathematic Monthly 1985 November edition volume 5 Cantor’s disappearing table page 398 by Larry E. Knop Hamilton College, Clinton, NY
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the remaining partition of the disappearing table analogy known as the Cantor’s theorem. Obviously the complement gave us the anticipated result outlined earlier in our calculation of percentile of the degenerated circle with respect to the limit 1/3, namely 1-0.00391=0.99609 this is significant because one can control the level of accuracy of the sequence dependent of the target area.
The present research project supports a theorem subtitled Theorem 1 that demonstrates that “A trisecting sequence is a subset of an infinite geometric sequence a1(rn-1+rn-2+…) where r=1/4, n= 2,3,…Eq2 presented in the form of aggregated partition in a partial covering of degenerated circles magnitude which converges to a limit point L=1/3.1.” This position is not supported by Theorem 1.5.
4 that posits the impossibility of trisecting a 600 angle with a compass and an unmarked straight edge sequence 2 as already cited.
The main theorem is best served when the angle under study is large. If the angle to be trisected is acute, say 600, the issue is more complex. We want to address the issue that is raised when we deal with an acute angle. We know the main theorem is best served when the angle under study is large. By construction, the respective radii are diminishing in magnitude by a ratio of ¼.
An indirect trisection of angle 600 will be done using Global positioning system as mentioned earlier. Currently, there are many methods of positioning.
The most widely used are: Global Positioning System (GPS), Triangulation, Resection, Multilateration, and Euclidean distance. In order to correctly locate a point in space, standard algebraic equations, combined with measuring equipment, geometry and a known point are used. For example, GPS method uses a satellite as a known point. Let us therefore construct an indirect trisection of angle 600. The motivation here is to show that angle 600 is practical, if we put the latter in a natural environment as an equilateral triangle.
The basic argument is that if angle 200 was achievable in the trisection of an equilateral triangle then the other triangle will have the following measurement of 200; 600; 1000 Given that we can built the Star of David, a six side regular polygon, we are now in the position to built a 1000 angle which is obtuse using theorem1 directly
1) As stated earlier, the geometric series is defined as a set of aggregates elements of regions that converge to a total covering of limit 1/3. Each element c1c2c3…cn converges closer to 1/3 as the series of circles degenerate in magnitude.
2) This will form an angle of 400
3) Dropping a 900 at the foot of the 1200using the new found 400.
The net difference will achieved 500. Now using a central angle we can easily make a 1000 degree 1 Frantz Olivier Major Theorem of trisecting any arbitrary angle Graphic representation page 12 Fig1 2 Experiencing Geometry Euclidean and Non-Euclidean with History 3rd Edition by David W.
Henderson ;Daina Taimina page 216
Furthermore, indirect trisection can be achieved in an ABC triangle if you do the following post constructing the 1000 angle1
1) Used a 600 triangle already in standard position. Triangle ABC
2) Let an angle 900 be constructed such that the angle 1000 be constructed at the feet of the 900 angle call it D