DOCSLIB.ORG

Delian Problem and Other Construction Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Delian Problem and Other Construction Problems Delian problem and other construction problems by LIU Rui Ling (19220642) A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science (Honours) in Mathematics and Statistics at Hong Kong Baptist University 14 January 2021 Acknowledgements Thanks to my supervisor, Dr. Sun Pak Kiu provided some useful reading materials for me so that I have a clear idea of how completing this thesis. Also, I am much obliged to Dr. Sun for his friendly reminder and support. ______________________ Signature of Student LIU Rui Ling Student Name Department of Mathematics Hong Kong Baptist University Date: 14 January 2021 1 Delian problem and other construction problems LIU Rui Ling (19220642) Department of Mathematics Abstract In this thesis, I will discuss the problem of duplication of a cube, which is one of the three geometrical construction problems in ancient Greek mathematics. The problem of doubling the cube refers to whether we can use a straightedge with no scale and compass to construct a cube so that its volume is equal to twice the original one. In chapter one, I will introduce the history of the problem of doubling the cube. In chapter two, I will discuss the straightedge and compass construction. Therefore, I will explain that how to construct numbers by using straightedge and compass construction. In chapter three, I will discuss the fields. In chapter four, I will discuss the field extensions, including the vector space, the field extensions and the algebraic extension. In chapter five, I will prove that why we cannot solve the problem of doubling the cube by using the straightedge and compass construction. 2 Table of Contents Acknowledgements ........................................................................................................ 1 Abstract .......................................................................................................................... 2 Chapter 1The history of the problem of doubling the cube ........................................... 4 Chapter 2 Constructible Numbers .................................................................................. 5 2.1 The straightedge and compass construction .......................................................... 5 Chapter 3 Fields ........................................................................................................... 10 Chapter 4 The field extensions .................................................................................... 11 4.1 The vector space .................................................................................................. 11 4.2 The field extensions ............................................................................................. 13 4.3 The algebraic extension ....................................................................................... 15 Chapter 5 Doubling the cube ....................................................................................... 17 Chapter 6 Conclusion ................................................................................................... 19 References .................................................................................................................... 20 3 Chapter 1 The history of the problem of doubling the cube Delian problem, also called the problem of doubling the cube, is one of the three geometric construction problems raised by the ancient Greeks 2,400 years ago. The problem of doubling the cube is to construct a cube which its volume is equal to twice the original one. The reason why this question is difficult to solve is that there are restrictions on the drawing tools. The ancient Greeks emphasized that only straightedge with no scales and compasses could be used for geometric drawing. There is a myth about the origin of the problem of doubling the cube. [1] The myth is that a plague broke out in the island of Delos. A quarter of the population of Delos died of the plague. The islanders went to the temple to ask for Apollo's will. The oracle replied that if they want to stop the plague, they should double the volume of altar in the Apollo’s temple. The islanders doubled the length of each side. The new altar was finished, and the plague continued to spread. They felt confused. They asked the famous philosopher and mathematician Plato about the reason why the oracle was not working. Plato pointed out the problem. They doubled each side of the altar, so that the volume would become eight times the original size instead of doubling. At this time, everyone suddenly realized. So, if the volume is to be doubled, how many should the length of each side increase? This is the problem of doubling the cube among the three famous construction problems in ancient Greece. Since then the problem of doubling the cube also called the Delian problem. Since the appearance of the problem of the doubling the cube, it has attracted many mathematicians to study. In 1837, a French mathematician, Pierre Wantzel proved that we could not use the straightedge and compass construction to solve the Delian problem. 4 Chapter 2 Constructible Numbers 2.1 The straightedge and compass construction Straightedge and compass construction is a mathematical problem that originated in ancient Greece. Only compasses and straightedges can be used to solve a given problem in a limited number of steps. In the straightedge and compass construction, the definition of straightedge and compass as below: A straightedge is unmarked, it can only be used for drawing a line segment through any two constructed points. A compass is used for drawing a circle with a constructed point O as the center and a constructed line segment r (r>0) as the radius. For two arbitrary points, point A and point B, we define |AB | as their distance. First, finding two points A and B. The line determined by these two points is the x- axis. Second, drawing a circle with A as the center and |AB | as the radius by using a compass. Final, drawing a circle with B as the center and |AB | as the radius by using a compass. Two circles which we had drawn intersect at point C and point D, and the straight line determined by these two points is the y-axis. The two line segments, that are AB and CD intersect at a point O, which is the origin of the coordinate system. We define the length of |OA| as 1 unit, and the coordinate of O is (0,0). So, we can draw a line segment is 1 unit of length at the beginning. See Figure 2.1. Definition 2.1.1 The constructible number: The constructible number refers the real number that can be drawn in a limit number of steps by using a straightedge. In the case of a given |α| (α∈ ℝ) unit length, if the line segment could be drawn by using a straightedge, then the number α is a constructible. [2] 5 All the constructing steps can be reduced to the following five basic steps: Definition 2.1.2 The constructible point: The constructible point refers to as below: (a) An intersection point of 2 non-parallel constructible segments, or (b) An intersection point of 1 constructible segment and a constructible circle, or (c) An intersection point of 2 constructible circles. Definition 2.1.3 The constructible segment: The constructible segment refers to a line segment which two endpoints are constructible points. Definition 2.1.4 The constructible circle: The constructible circle refers to a circle with a constructible point as the center and a constructible segment as the radius. The length of each line segment is equivalent to a real number. Here we assume that "given a positive real number" means "given a line segment, and the length is equivalent to a given positive real number. Therefore, given two positive real numbers, we can construct the sum of the two numbers and difference. Given two positive real numbers and a line segment of length 1unit, then the product, quotient and square root of the two numbers could be constructed. So, for a given constructed line segment, we can obtain a constructible number by the five mathematical operation: addition, subtraction, multiplication, division and taking a square root. Let α, β>0, the five operation as show below: 1. α + β For two given line segments, line segment AB and line segment CD which the length of α and β respectively. Constructing a circle with point B as the center and line segment CD as the radius by a compass. Constructing a line segment AB, and the two intersection points of line segment AB and a constructed circle are point M and N respectively. See Figure 2.1.2. 6 2. α - β Two numbers that are α-β and α+β have a similar construction, but the line segment of CD is constructed in the opposite side. See Figure 2.1.3. 3. α x β For three given line segments, OP and OA which the length of α and βrespectively, and a line segment of length 1unit. Constructing a line segment OA with a length of β by using a compass. Let |OR|=1 unit, and point R lies on the line segment OA. Constructing a line l through point O and do not contain OA. Let |OP|=α, and a point P in the line l. Using the straightedge to construct a line segment RP and construct a line segment AQ from point A, such that a line segment PR and a line segment AQ is a pair of parallel lines, and the line segment AQ intersects the line l at point Q. Now, we have a pair of similar triangles, the triangle OPR is similar to the triangle OQA. According to the properties of similar triangle, we have 1 β = . 훼 |푂푄| So |OQ|=αβ. See Figure 2.1.4. 7 4. α / β Two numbers that are α/β and αβ have a similar construction. For three given line segments, OQ and OA which the length of α and β (whereβ≠0) respectively, and a line segment of length 1unit. Constructing a line segment OA with a length of β by using a compass. Let |OR|=1 unit, and point R lies on the line segment OA. Constructing a line l through point O and do not contain OA. Let |OQ|=α, and point Q lies on the line l. Using the straightedge to construct a line segment AQ and a line segment RP from a point R, such that the line segment AQ is parallel to the line segment RP, and the line segment RP intersects the line l at point P.
Load more

Recommended publications
  • Extending Euclidean Constructions with Dynamic Geometry Software
    Proceedings of the 20th Asian Technology Conference in Mathematics (Leshan, China, 2015) Extending Euclidean constructions with dynamic geometry software Alasdair McAndrew [email protected] College of Engineering and Science Victoria University PO Box 18821, Melbourne 8001 Australia Abstract In order to solve cubic equations by Euclidean means, the standard ruler and compass construction tools are insufficient, as was demonstrated by Pierre Wantzel in the 19th century. However, the ancient Greek mathematicians also used another construction method, the neusis, which was a straightedge with two marked points. We show in this article how a neusis construction can be implemented using dynamic geometry software, and give some examples of its use. 1 Introduction Standard Euclidean geometry, as codified by Euclid, permits of two constructions: drawing a straight line between two given points, and constructing a circle with center at one given point, and passing through another. It can be shown that the set of points constructible by these methods form the quadratic closure of the rationals: that is, the set of all points obtainable by any finite sequence of arithmetic operations and the taking of square roots. With the rise of Galois theory, and of field theory generally in the 19th century, it is now known that irreducible cubic equations cannot be solved by these Euclidean methods: so that the \doubling of the cube", and the \trisection of the angle" problems would need further constructions. Doubling the cube requires us to be able to solve the equation x3 − 2 = 0 and trisecting the angle, if it were possible, would enable us to trisect 60◦ (which is con- structible), to obtain 20◦.
    [Show full text]
  • Växjö University
    School of Mathematics and System Engineering Reports from MSI - Rapporter från MSI Växjö University Geometrical Constructions Tanveer Sabir Aamir Muneer June MSI Report 09021 2009 Växjö University ISSN 1650-2647 SE-351 95 VÄXJÖ ISRN VXU/MSI/MA/E/--09021/--SE Tanveer Sabir Aamir Muneer Trisecting the Angle, Doubling the Cube, Squaring the Circle and Construction of n-gons Master thesis Mathematics 2009 Växjö University Abstract In this thesis, we are dealing with following four problems (i) Trisecting the angle; (ii) Doubling the cube; (iii) Squaring the circle; (iv) Construction of all regular polygons; With the help of field extensions, a part of the theory of abstract algebra, these problems seems to be impossible by using unmarked ruler and compass. First two problems, trisecting the angle and doubling the cube are solved by using marked ruler and compass, because when we use marked ruler more points are possible to con- struct and with the help of these points more figures are possible to construct. The problems, squaring the circle and Construction of all regular polygons are still im- possible to solve. iii Key-words: iv Acknowledgments We are obliged to our supervisor Per-Anders Svensson for accepting and giving us chance to do our thesis under his kind supervision. We are also thankful to our Programme Man- ager Marcus Nilsson for his work that set up a road map for us. We wish to thank Astrid Hilbert for being in Växjö and teaching us, She is really a cool, calm and knowledgeable, as an educator should. We also want to thank of our head of department and teachers who time to time supported in different subjects.
    [Show full text]
  • Construction of Regular Polygons a Constructible Regular Polygon Is One That Can Be Constructed with Compass and (Unmarked) Straightedge
    DynamicsOfPolygons.org Construction of regular polygons A constructible regular polygon is one that can be constructed with compass and (unmarked) straightedge. For example the construction on the right below consists of two circles of equal radii. The center of the second circle at B is chosen to lie anywhere on the first circle, so the triangle ABC is equilateral – and hence equiangular. Compass and straightedge constructions date back to Euclid of Alexandria who was born in about 300 B.C. The Greeks developed methods for constructing the regular triangle, square and pentagon, but these were the only „prime‟ regular polygons that they could construct. They also knew how to double the sides of a given polygon or combine two polygons together – as long as the sides were relatively prime, so a regular pentagon could be drawn together with a regular triangle to get a regular 15-gon. Therefore the polygons they could construct were of the form N = 2m3k5j where m is a nonnegative integer and j and k are either 0 or 1. The constructible regular polygons were 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, ... but the only odd polygons in this list are 3,5 and 15. The triangle, pentagon and 15-gon are the only regular polygons with odd sides which the Greeks could construct. If n = p1p2 …pk where the pi are odd primes then n is constructible iff each pi is constructible, so a regular 21-gon can be constructed iff both the triangle and regular 7-gon can be constructed.
    [Show full text]
  • The Angle Trisection Solution (A Compass-Straightedge (Ruler) Construction) Kimuya .M
    I S S N 2 3 4 7 - 1921 Volume 13 Number 4 Journal of Advance in Mathematics The Angle Trisection Solution (A Compass-Straightedge (Ruler) Construction) Kimuya .M. Alex1, Josephine Mutembei2 1Meru University of Science and Technology, Kenya, Department of Physical Sciences, Faculty of Physics 2Meru University of Science and Technology, Kenya, Department of Mathematics 1Cell; +254 704600418 2Cell; +254 721567967 [email protected] [email protected] ABSTRACT This paper is devoted to exposition of a provable classical solution for the ancient Greek’s classical geometric problem of angle trisection [3]. (Pierre Laurent Wantzel, 1837), presented an algebraic proof based on ideas from Galois theory showing that, the angle trisection solution correspond to an implicit solution of the cubic equation; , which he stated as geometrically irreducible [23]. The primary objective of this novel work is to show the possibility to solve the ages old problem of trisecting an arbitrary angle using the traditional Greek’s tools of geometry (classical compass and straightedge), and refute the presented proof of angle trisection impossibility statement. The exposed proof of the solution is theorem , which is based on the classical rules of Euclidean geometry, contrary to the Archim edes proposition of using a marked straightedge construction [4], [11]. Key Words Angle trisection; An arbitrary angle; A particular angle; Compass; Ruler (Straightedge); Classical geometry; GeoGebra Software; Plane geometry; Subset; Superset; Singular angle Notations Notation for an angle Denotes a straight line segment and a length Two Dimensions Three Dimensions Four significant figures Denotes a member or an element of a particular set Subset Not a member or an element of a particular set divides Cosine of angle decimal places Academic Discipline And Sub-Disciplines Mathematics (Geometry) 1.0 INTRODUCTION Compass and straightedge problems have always been the favorite subject o f classical geometry.
    [Show full text]
  • A Brief History of Impossibility
    VOL. 81, NO. 1, FEBRUARY 2008 27 A Brief History of Impossibility JEFF SUZUKI Brooklyn College Brooklyn, NY jeff [email protected] Sooner or later every student of geometry learns of three “impossible” problems: 1. Trisecting the angle: Given an arbitrary angle, construct an angle exactly one-third as great. 2. Duplicating the cube: Given a cube of arbitrary volume, find a cube with exactly twice the volume. 3. Squaring the circle: Given an arbitrary circle, find a square with the same area. These problems originated around 430 BC at a time when Greek geometry was advanc- ing rapidly. We might add a fourth problem: inscribing a regular heptagon in a circle. Within two centuries, all these problems had been solved (see [3, Vol. I, p. 218–270] and [1] for some of these solutions). So if these problems were all solved, why are they said to be impossible? The “im- possibility” stems from a restriction, allegedly imposed by Plato (427–347 BC), that geometers use no instruments besides the compass and straightedge. This restriction requires further explanation. For that, we turn to Euclid (fl. 300 BC), who collected and systematized much of the plane geometry of the Greeks in his Elements. Euclid’s goal was to develop geometry in a deductive manner from as few basic assumptions as possible. The first three postulates in the Elements are (in modernized form): 1. Between any two points, there exists a unique straight line. 2. A straight line may be extended indefinitely. 3. Given any point and any length, a circle may be constructed centered at the point with radius equal to the given length.
    [Show full text]
  • History of Mathematics the Early Years
    The History of Mathematics From the Egyptians to Archimedes Michael Flicker OLLI Winter 2011 1 Egyptian and Babylonian • 3000 BC to 260 AD • Essentially empirical • Early number systems • Simple arithmetic, practical geometry • Egyptian papyri & Babylonian cuneiform tablets – Mathematical tables – Collections of mathematical problems 2 3 Hieroglyphic & Hieratic Numerals 4 Hieratic & Hieroglyphic 5 Egyptian Arithmetic • The Egyptians could add, subtract, multiply and divide • Multiplication of whole numbers used the method of doubling Example: 57 x 117 1+ 117 2 234 4 468 8+ 936 16+ 1872 32+ 3744 57 6669 6 Primary Egyptian Sources • Rhind Mathematical Papyrus (RMP) – About 1650 BC from writings made 200 years earlier (18 ft x 13 in) – The Recto Table and 84 (87) mathematical problems • Moscow Mathematical Papyrus – – 1850 BC, (18 ft x 1.5 to 3 in) – 25 problems 7 The Recto Table • The division of 2 by the odd numbers 3 to 101 with the answers expressed as the sum of unit fractions • The entrance into the knowledge of all existing things and all obscure secrets. This book was copied in the year 33, in the 4th month of the inundation season, under the majesty of the king of Upper and Lower Egypt, A-user-Re (Aweserre Apopi), endowed with life, in likeness to writings of old made in the time of Upper and Lower Egypt, Ne-ma-et-Re (Nemare Ammenemes III). It is the scribe Ah-mose who copies this writing. 8 Egyptian Fractions & Algebra • Except for 2/3 and possibly 3/4 the Egyptian arithmetic notation only permitted fractions with unity in the numerator • 2/n equivalents in the RMP Recto table 5: 3 15 7: 4 28 9: 6 18 11: 6 66 13: 8 52 104 15: 10 30 17: 12 51 68 19: 12 76 114 21: 14 42 23: 12 276 25: 15 75 … 97: 56 679 776 … • Could multiply and divide mixed numbers • Simple Algebra – A quantity and its seventh, added together give 19.
    [Show full text]
  • On the Existence of Poncelet-Morley Points in Euclidean Geometry
    Journal of Mathematics Research; Vol. 9, No. 3; June 2017 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education On the Existence of Poncelet-Morley Points in Euclidean Geometry 1 Joseph Dongho1, Nguimbous F. L.1 & Teyou M. R.1 1 Department of Mathematics and Computer Science, University of Maroua Correspondence: Joseph Dongho, Department of Mathematics and Computer Sciences, Faculty of Science , University of Maroua, Po-Box 163 Maroua-Domayo, Cameroon. E-mail: [email protected] Received: December 16, 2016 Accepted: March 30, 2017 Online Published: May 25, 2017 doi:10.5539/jmr.v9n3p68 URL: https://doi.org/10.5539/jmr.v9n3p68 Abstract In this paper, we prove the existence of the Poncelet-Morley point for a given elliptic configuration. The paper ends with an application of such a point in angle trisection problem. Keywords: Poncelet theorem, Morley Theorem, Poncelet-Morpey point, trisection, Galois theory, congruent figure Msc200: 51M04, 51M05, 11R32, 12F10 and 97G99 1. Introduction Trisection is a classic problem of compass and straightedge constructions of ancient Greek mathematics. It concerns the construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge, and a compass. The problem as stated is generally impossible to solve, as shown by Pierre Wantzel in (Arnaudiès J. M. & Fraysse H., 1990). Note that the fact that there is no way to trisect an angle in general with just a compass and a straightedge does not mean that there is no trisecting angle: for example, it is relatively straightforward to trisect a right angle, that is, to construct an angle of measure 30 degrees.
    [Show full text]
  • The Angle Trisection Solution (A Compass-Straightedge (Ruler) Construction) Kimuya .M
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE I S S N 2 3provided 4 7 - by1921 KHALSA PUBLICATIONS Volume 13 Number 4 Journal of Advance in Mathematics The Angle Trisection Solution (A Compass-Straightedge (Ruler) Construction) Kimuya .M. Alex1, Josephine Mutembei2 1Meru University of Science and Technology, Kenya, Department of Physical Sciences, Faculty of Physics 2Meru University of Science and Technology, Kenya, Department of Mathematics 1Cell; +254 704600418 2Cell; +254 721567967 [email protected] [email protected] ABSTRACT This paper is devoted to exposition of a provable classical solution for the ancient Greek’s classical geometric problem of angle trisection [3]. (Pierre Laurent Wantzel, 1837), presented an algebraic proof based on ideas from Galois theory showing that, the angle trisection solution correspond to an implicit solution of the cubic equation; , which he stated as geometrically irreducible [23]. The primary objective of this novel work is to show the possibility to solve the ages old problem of trisecting an arbitrary angle using the traditional Greek’s tools of geometry (classical compass and straightedge), and refute the presented proof of angle trisection impossibility statement. The exposed proof of the solution is theorem , which is based on the classical rules of Euclidean geometry, contrary to the Archim edes proposition of using a marked straightedge construction [4], [11]. Key Words Angle trisection; An arbitrary angle; A particular angle; Compass; Ruler (Straightedge);
    [Show full text]
  • Constructibility of Regular Polygons
    Constructibility of Regular Polygons Iowa State University MATH 599 Creative Component Eric T. Eekhoff Constructibility of Regular Polygons 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272 The above list of 38 numbers appears as the concluding statement of the greatest work written by one of the elite mathematicians of all time. Carl Friedrich Gauss ended his masterpiece Disquisitiones Arithmeticae with this sequence. It represents the numbers of sides (under 300) of regular polygons that can be constructed by traditional straight-edge and compass techniques. While the idea of constructing regular polygons had been studied by the early Greeks, along with many others, it was Gauss that proved with finality which regular polygons could be constructed, and which ones could not. In this paper, I will discuss what was known on this topic in the time of Euclid, and relate it to what Gauss was able to prove in what could be described as the discovery in which he took the most pride. (mathcircle.berkeley.edu) The construction of regular polygons is a topic that has been studied for many years. Going back to the early Greek mathematicians, we can see that by using specific tools with well defined rules for their use, the techniques to construct certain regular polygons were widely known. It was the Greek mathematician Euclid who chronicled what was known at that time in what was regarded as the greatest compilation on the subject of Geometry for over two millennia, Euclid’s Elements.
    [Show full text]
  • NUMBER the Language of Science
    Dantzig_FM.qxd 2/18/05 9:29 AM Page i Dantzig_FM.qxd 2/18/05 9:29 AM Page ii Dantzig_FM.qxd 2/18/05 9:29 AM Page iii Tobias Dantzig NUMBER The Language of Science Edited by Joseph Mazur Foreword by Barry Mazur The Masterpiece Science Edition Pi Press New York Dantzig_FM.qxd 2/18/05 9:29 AM Page iv PI PRESS An imprint of Pearson Education, Inc. 1185 Avenue of the Americas, New York, New York 10036 Foreword, Notes, Afterword and Further Readings © 2005 by Pearson Education, Inc.© 1930, 1933, 1939, and 1954 by the Macmillan Company This edition is a republication of the 4th edition of Number, originally published by Scribner, an Imprint of Simon & Schuster Inc. Pi Press offers discounts for bulk purchases. For more information, please contact U.S. Corporate and Government Sales, 1-800-382-3419, [email protected]. For sales outside the U.S., please contact International Sales at [email protected]. Company and product names mentioned herein are the trademarks or registered trademarks of their respective owners. Printed in the United States of America First Printing: March, 2005 Library of Congress Number: 2004113654 Pi Press books are listed at www.pipress.net. ISBN 0-13-185627-8 Pearson Education LTD. Pearson Education Australia PTY, Limited. Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia, Ltd. Pearson Education Canada, Ltd. Pearson Educatión de Mexico, S.A. de C.V. Pearson Education—Japan Pearson Education Malaysia, Pte. Ltd. Dantzig_FM.qxd 2/18/05 9:29 AM Page v Contents Foreword vii Editor's Note xiv Preface to the Fourth Edition xv Preface to the First Edition xvii 1.
    [Show full text]
  • Arxiv:1807.08416V3
    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 200 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were written down. Since [431] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context” we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The number of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable” theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For litera- ture on general mathematics, see [158, 154, 26, 190, 204, 478, 330, 114], for history [179, 484, 304, 62, 43, 170, 306, 296, 535, 95, 477, 68, 208, 275], for popular, beautiful or elegant things [10, 406, 163, 149, 15, 518, 519, 41, 166, 155, 198, 352, 475, 240, 163, 2, 104, 121, 105, 389]. For comprehensive overviews in large parts of mathematics, [63, 138, 139, 47, 458] or predictions on developments [44]. For reflections about mathematics in general [120, 358, 42, 242, 350, 87, 435]. Encyclopedic source examples are [153, 547, 516, 88, 157, 127, 182, 156, 93, 489].
    [Show full text]
  • The Possibility of Angle Trisection (A Compass-Straightedge Construction) Kimuya M Alex
    Journal of Mathematics and System Science 7 (2017) 25-42 doi: 10.17265/2159-5291/2017.01.003 D DAVID PUBLISHING The Possibility of Angle Trisection (A Compass-Straightedge Construction) Kimuya M Alex Kimuya M Alex Department of Mathematics, Faculty of Physics, Meru University of Science and Technology, Kenya. Received: September 19, 2016 / Accepted: October 16, 2016 / Published: January 25, 2017. Abstract: The objective of this paper is to provide a provable solution of the ancient Greek problem of trisecting an arbitrary angle employing only compass and straightedge (ruler). (Pierre Laurent Wantzel, 1837) obscurely presented a proof based on ideas from Galois field showing that, the solution of angle trisection corresponds to solution of the cubic equation; 3 3 1 = 0, which is geometrically irreducible [1]. The focus of this work is to show the possibility to solve the trisection of an angle by correcting some − − flawed methods meant for general construction of angles, and exemplify why the stated trisection impossible proof is not geometrically valid. The revealed proof is based on a concept from the Archimedes proposition of straightedge construction [2, 3]. Key Words: Angle trisection; Compass; Ruler (Straightedge); Classical Construction; GeoGebra Software; Greek’s geometry; Cubic equation; Plane geometry; Solid geometry Notations sought under the Greek’s rules of Geometry’ [1, 2]. Three algebraic constraints exists which state; a length Denotes an angle Denotes a straight line and a length can only be constructible if and only if it represents a ∠ 2 Two Dimensional constructible number, an angle is constructible if and ∩����∀� 3 Three Dimensional only if its cosine is a constructible number, and, a 1.Introduction number is constructible if and only if it can be written in the four basic arithmetic operations and the The early Greek mathematicians were unable to extraction of the square roots but not on higher roots solve three problems of compass-ruler (straightedge) [4].
    [Show full text]