<<

International Journal of Trends and Technology (IJMTT) – 50 5 October 2017

The Duplication Solution (A - () Construction) Kimuya .M. Alex#1, Josephine Mutembei#2 1 Department of Physical Sciences, Faculty of , Meru University of Science and Technology, Kenya 2 Department of Mathematics, Meru University of Science and Technology, Kenya

Abstract problem (doubling the volume of a given cube) to a This paper objectively presents a provable realizable accuracy of , in reference to the construction of generating a of magnitude; numerical value from the scientific calculator, using , as the geometrical only a ruler and compass. Through the ages, solution for the ancient classical problem of doubling mathematicians and other practitioners have wrestled the volume of a cube. Cube duplication is believed to the problem of doubling a cube, but no geometrical be impossible under the stated restrictions of solution has been found. The deeper need upon taking Euclidean , because the Delian constant up this research project was inspired by the objectives; is classified as an , which was how could the length of a given edge of a cube be stated to be geometrically irreducible (Pierre Laurent geometrically increased to produce a new edge, such Wantzel, 1837) [1]. Contrary to the impossibility that ratio of the new edge to the original edge would consideration, the solution for this ancient problem is be . To have the method for , in which an elegant approach is presented, solving the Delian problem at this reasonable as a refute to the cube duplication impossibility accuracy, achieved under the specified conditions of statement. Geogebra software as one of the interactive Euclidean constructions [5, 10]. The presented method geometry software is used to illustrate the accuracy of involves the generating a certain constant length from the obtained results, at higher accuracies which a given face of a cube, which when geometrically cannot be perceived using the idealized platonic added to the original edge would produce the required straightedge and compass construction. results. This application is in harmony with the compass equivalence theorem as provided in Key Words: Doubling a cube, Delian constant, in Book of ‟s Elements which Compass, Straightedge (Ruler), Classical states “any construction via „fixed‟ compass may be construction, Euclidean number, Irrational number, attained with a collapsing compass. That is; it is GeoGebra software possible to construct of equal centered at any on a ”. The elementary idea was Abbreviations based on the consideration that; some objects Two dimensional („atomic‟ angular units in realm of circles) redistribute Three dimensional in a given plane to produce some significant figures Three significant figures and objects. This is in harmony with the classical geometric division of a given straight segment into a Five significant figures number of even parts. Therefore, some of these units

(circles and segments), if geometrically identified Notations could help solve the Delian problem apparently to a Used to denotes a straight line (length) meaningful of accuracy. This rise the deeper Used to denote an need for geometrical inspection, for the relationship between and lines in a given plane. I. Introduction Definition involves the geometrical definition of a There are three influential problems in geometry posed cube, and definition brings out the uncertainty in by the ancient Greek mathematicians; The problem of the presented impossibility proof of cube duplication. doubling a cube, which require the construction of a and involves the geometrical cube whose volume is twice that of a given cube, the problem of squaring a , in which a square of construction of the fractions and equal to that of a given circle has to be constructed, respectively, as the deductive possibility of and the problem of which involves constructing the factor . The desire to construct constructing one third of a given angle or the these ratios was motivated by the possibility of construction of an angle whose size is three times a constructing some algebraically irrational (in given angle. However, a high profile contemporaries algebraic language) such as and using ruler have closed the door in solving these problems by compass construction. Theorem presents an assuming them impossible for ruler-compass construction [1, 3, and 4]. This paper is focused on the algorithm for solving . Geometric transformation exposition of a geometrical solution for the Delian relation of enlargement (resizing objects) was used to

ISSN: 2231-5373 http://www.ijmttjournal.org Page 307 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 50 Number 5 October 2017 justify the geometrical accuracy of the generated are algebraically solvable. He draws the same method, based on results obtained using an interactive conclusion for the both problems of cube duplication computer software. By following the exposed and the trisection of an angle [7]. It is not methodology, it was possible to geometrically figure geometrically clear the connection between the two out the geometrical rationality of the Delian constant problems from his work. Consider the follows as with precision. This work interpreted version of the cube duplication is contained in the formal rules of classical Euclidean impossibility proof. geometry. Theorem 1.0: Given a cube of volume it is Definition 1.1: Geometrical Construction of a not geometrically possible to construct a cube of Cube volume , using only a straightedge (ruler) and a Geometrically, a cube is any three-dimensional solid compass. object bounded by six square faces, facets or sides, with three of the edges meeting at each [6]. The Proof: Let the initial cube be of a unit length. Assume cube is the only regular hexahedron and is one of the that one of the sides of the cube is the line between the five Platonic solids. It has 6 faces, 12 edges, and 8 coordinates ) and . The volume of such a vertices as shown in figure 1: cube would be , so that constructing a cube of volume would correspond to constructing some point , such that . Let be the smallest containing and . The minimum of over is . This is a degree three polynomial and therefore, based on dimensionality we have: (1) Based on application of in plane geometry, the point is constructible from the configuration , if is a power of two. However, as seen from ( ), is a power of three and not two. It can therefore be concluded that the relation is not reducible, and thus the delian constant is not geometrically constructible.

From this discussion one can deduce the following points: 1. The cube duplication impossibility proof does not disproof a geometrical construction aimed at a solution to the cube duplication problem. 2. The proof involves use of coordinates, which is basically an analytical geometric approach, and not on the classical geometric rigor of analysis. Fig. 1 Geometric View of a Cube 3. The cube duplication impossibility statement show a degree of misconception in the sense that it does not As observed from figure (1), each face of the cube take into account the difference between the classical represent a two dimensional plane. Logically, the geometric problems, and the analytical geometric genetic construction of a cubical structure would problems. For instance, the difference between the involve constructing each of the six facets of the cube, classical and the analytic one after another and joining the planes at some geometry is that, Euclidean plane geometry is a common endpoints to produce the eight corners at the in that; it proceeds logically from vertices. to without use of the coordinate Definition 1.2 The Mistake In Pierre Laurent system, while the analytical geometry completely Wantzel‟s Proof Of Cube Duplication Impossibility accomplish the use of coordinate systems. In the Statement (1837) ancient Greek‟s geometry, the only numbers were This section begins by presenting an interpreted (positive) . was represented version of the presented cube duplication impossibility by a ratio of integers. Any other quantity was statement, to bring out the geometrical represented as a geometrical magnitude. The term misconstruction involved in defining the impossibility irrationality did not exist. This point of consideration statement, based on the algebraic consideration. In his is presented by (Descartes, 1637). In Book III of La proof, Wantzel used a series of quadratic to Géométrie [30], Descartes considers with show that all the geometrically constructible problems coefficients. If there is an integer root, that

ISSN: 2231-5373 http://www.ijmttjournal.org Page 308 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 50 Number 5 October 2017

gives a numerical solution to the problem. But if there are no integral roots, the solutions has be constructed geometrically. A quadratic equation gives rise to a plane problem whose solution can be constructed with ruler and compass. Cubic and quartic equations are solid problems that require the of conics for their solution. The root of the equation is a certain constructed geometrically, not a number. Up to this point, it is clear that any form of an algebraic proof taming the cube duplication problem as an impossible problem is not a classical geometric solution. It is thus not geometrically valid to assume the cube duplication problem as an impossible problem, since as stated in this context, all geometric problems has to be sought geometrically. 4. The early mathematicians were able to construct algebraic irrationalities of the form: and (see Annex-1), as employed in the famous , where, the factor has no specific value Fig. 2 An illustration of existence of significant points algebraically. This consideration provide clear along jurisdiction that, all algebraic irrationalities can be geometrically constructed. It should be noted that, the term exactness in geometry is completely defined by III. Materials and Methods solving a specific problem, based on the Materials framework. The use of technology is as well limited in The required mathematical in solving this the sense that, the CAD methods provide just problem for the proposed methods include; approximate solutions, and not exact measurements. This paper is in the view that, any classical geometric - Classical compass solution has to be reasonably exact, and not just an - Ruler(straightedge) eyeballing construction. - Piece of a drawing paper - Pencil In this paper, an attempt is made to show that, the - Rubber factor is geometrically constructible and it is not an - Computer irrational number problem. It is the consideration of - GeoGebra Software installed on the PC. this paper that, the use of algebra in solving a classical geometric problem is not a geometric verification, but In this paper, though all the provided constructions are an analysis. typically compass and straightedge methods, the use of GeoGebra software is preferred, purposely for good II. Hypothesis results visualization. Consider figure (2). The aim of this paper is to provide an algorithm for geometrical solution of the ancient The GeoGebra Software problem of cube duplication, using both the Greek‟s tools of geometry and computer interactive software GeoGebra is an open source dynamic software for (GeoGebra 5.0). From the following figure, the square teaching and learning mathematics with suitable face represents a of the given cube features for topics such as geometry, algebra and . It was developed by Markus Hohenwarter with base . Piont is a reflection of point about and has now been translated into several languages; point . Thus . Also, and (Abdul Saha et al. 2010). It is a software designed as a are equal to . Therefore for a line of magnitude combination of other geometry dynamic software to be correctly constructible, must features such as Cabri Geometry, C.a.R, and geometrically equal . Theorem presents a proof to Geometer‟s Sketchpad. It is found to be more confirm that the generated length , interactive compared to other geometric software. In to an accuracy of . this era, geometry is one of the mathematics contents that is concerned with the integration of technology during the teaching and learning process. Therefore, having such good geometry environments (integrated toolboxes), GeoGebra is a suitable dynamic package to be used as an alternative teaching aid that is based on technology. Besides geometrical learning GeoGebra has also other blocksets related to topics of

ISSN: 2231-5373 http://www.ijmttjournal.org Page 309 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 50 Number 5 October 2017 algebra and calculus. As depicted in figure 3, Methodology GeoGebra is dynamic geometry software fitted with Theorem 2.0: Approximate Construction of the various characteristics that allow users to construct fraction form the edge of a given cube object such as points, segments, lines, circles, , The following method would help the construction of and other dynamic functions. However, like a line of magnitude , given a cube of side one any other software, GeoGebra has its own limitations unit. in the sense that, it does not give the exact position of a quantity on the graphics pad, but, just 1. Draw a straight line through two points approximation. This shows the reason why it is so . much difficult to get an exact measurement in classical 2. Construct square with as its base. geometry, as most people are concerned with exactness of quantities, a property that never exist in 3. Locate a point at the center of the square face science. As stated earlier, this will specifically be by constructing its . used for results visualization and analysis, and not as 4. Construct the of line at . part of the provided proofs. Consider figure 3. 5. Further construct the bisection of at . 6. By placing the pair of compass at , trace the length to cut the edge at . 7. Place the compass at and make an arc of

length , to cut line externally at . 8. Join point to point using a straight line. Fig. 3: The GeoGebra Software Interface and Functions Figure (4) represents the results of this construction.

ISSN: 2231-5373 http://www.ijmttjournal.org Page 310 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 50 Number 5 October 2017

2. Construct the bisection of line at a point . 3. Further, construct the bisection of at point . 4. Reflect point about point to get point as shown in diagram 5.

The edge .

Fig. 5 The construction of the fraction

Proof From the construction results it is evident that, point Fig. 4 Construction of the fraction is at . (8) This implies that . Proof Mapping point to means adding the length to The purpose of this proof is to show that the fraction line . Therefore, is geometrically constructible. . (9) Let the above unit length square be the face of Taking , and substituting for and a cube whose volume is to be doubled. in equation (9) we get; Let , and . (2) . (10) The magnitude of can be given by Equation (10) shows that the fraction is applying the expression . geometrically constructible. Thus . (3) The largest fraction of the diagonal would Theorem 4.0: It is geometrically possible to therefore be construct the ratio , . (4) for c ompass and ruler (straightedge) construction

Considering (Right angled at ), the Under this theorem, the objective is to proof that the . (5) factor is constructible using the The magnitude of can be found using the classical construction tools. Consider the following theorem: steps of construction;

. (6) 1. Given a cube of unit length, construct square face Therefore, the magnitude of would be given of sides . by . (7) 2. Using the radius of the constructed square, Substituting for we get the value of place the compass at point and make an arc . through vertices and . Reflecting point about point to produce point 3. Without adjusting the compass, position the implies adding the length to the edge of the compass spike at point and make another arc square face. Thus the constructed length is from point through . Label the point of of the two arc inside the square. Theorem 3.0: It is possible to construct the4. Join point to using a straight line, and construct its fraction from the side of a given cube bisection at point , to cut at a point . The fraction can be geometrically solved by5. Using , place the compass at and mark a point following the presented steps of construction: on the extended edge . 6. Again using , place the compass at and 1. Draw a straight line through two points , at a mark another arc at a point on the extended line. equal to the side of the given cube.

ISSN: 2231-5373 http://www.ijmttjournal.org Page 311 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 50 Number 5 October 2017

7. Construct the bisection of at as shown in figure 10. Draw a straight line from point thorough point (6). to meet line at point as shown in figure Line is the required length. (7). 11. Reflect point about point to get point . 12. Draw a straight line from point through point to meet at point . 13. Join point to point using a straight line.

Fig. 6 How to locate point

Verification of the Method From figure (6), it is clear that the points and are

very close that for one to construct the bisection at ,

the concept of constructing a though a point in a straight line has to be employed. This Fig. 7 Illustrative proof for constructing a line of challenge calls for the use of a geometry software for magnitude clarification purpose. GeoGebra software is preferred Proof for use in this paper because of its good geometry From figure (7), several properties are evident: Using tools which makes it more reliable for plane geometry the compass equivalence theorem, the purpose of the constructions, as stated earlier. The following algorithm was used to generate figure (7). green and the blue circles with centres and respectively is to show that the radii 1. Given a cube of unit length, construct square face , to accuracy of of side . . The radii and are inherently generate 2. Using the radius of the constructed square, place from the construction. Since the of the blue the compass at point and make an arc through circle with center passes through point (center of vertices and . the green circle), and the vertex , it is therefore clear 3. Without adjusting the compass, position the compass that , and is the initial radius. It spike at point and make another arc from point can therefore be deduced that, triangle is through . Label the point of intersection of the two equilateral (all sides equal), implying, arc inside the square. . Let 4. Join point to using a straight line, and construct its , and side of the original cube be . bisection at point , to cut curve at a point . This proof is to show that; for . 5. Using , place the compass at and mark a point Where is the length of the given cube. Using on the extended edge . and following the above deductions, the 6. Again using chord , place the compass at and cosine rule can be applied to triangle as follows: mark another arc at a point on the extended line. 7. Construct the bisection of at . 8. Further, construct the bisection of at . (11)

9. Using , place the compass at point , and make ) an arc to cut curve at a point outside the square. (12) Substituting for the lines , and we have:

ISSN: 2231-5373 http://www.ijmttjournal.org Page 312 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 50 Number 5 October 2017

, and . construction using the GeoGebra software: Further, equation (12) can be rewritten as: 1. Given a cube of unit length, construct square face (13) of sides . (14) 2. Using the radius of the constructed square, Applying the Menelaus‟ Theorem to triangle , place the compass at point and make an arc with , we have; through vertices and . . 3. Without adjusting the compass, position the Since , we have compass at point and make another arc from . This equation reduces to; point through . Label the point of . (15) intersection of the two arc inside the square. Equations (14) and (15) can be solved simultaneously 4. Join point to using a straight line, and as follows; construct its bisection at point , to cut curve at a point . 5. Using , place the compass at and mark a point on the extended edge . 6. Again using chord , place the compass at The equation reduces to . Thus the factor and mark another arc at a point on the extended line. is constructed. In this proof, an attempt has 7. Construct the bisection of at . been made to correctly map lengths and onto 8. Use to locate the vertices and . and respectively to produce line as illustrated in figure (7). This implies to an accuracy of decimal places.

Justification of the proof This section uses the transformation relation of object similarities and enlargement to justify the correctness of the proposed method for doubling a cube. Consider figure (8) generated after performing the following

ISSN: 2231-5373 http://www.ijmttjournal.org Page 313 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 50 Number 5 October 2017

respect to the traditional rigor of Greek‟s geometry, Fig. 8 Analytical Justification of the proposed method the constant ; (the geometrical magnitude of the Delian constant) has been approximated to the accuracy; s to , and is Considering and , the scale geometrically achievable following the Euclidian factor mapping onto can be found. From construction rules according to the present proof. This the GeoGebra window (Algebra view), it is observed, paper presents a geometrically possible method under the points and have the co-ordinates the set restrictions of Euclidean geometry (in the sense and that, all presented constructions have been reduced to respectively. Considering these two points, let the the Euclidean postulates of practical geometry), by the scale factor of enlargement be . Taking the point construction of the relation as depicted in to be the center of enlargement. the justification section. Therefore, for the : (16) V. Conclusion We get the scale factor as . In this paper, an elementary proof for solving the ancient problem of doubling the volume of a given Therefore the scale factor mapping point onto is cube, to a certain correctness, is presented. The , which is approximately equal to obtained results indicated that, algebraic irrationalities . The same value would be obtained if should not be extended to plane geometric calculated for the . constructions, since subject to application, the desired degree of precision could be possible for compass- IV. Results and Discussion straightedge construction. Through the presented The problem of doubling a cube is a well-known discussion, it can be concluded that the Wantzel‟s millenary problem about which mathematicians stated statement of impossibility is not geometrically valid, as impossible to geometrically resolve because the since it does not give the geometrical relationship factor is classified as an irrational number. The between the quadratic and the cubic extensions incomprehensible concerned employed in the proof of cube duplication showing that the solution of the constant impossibility, with respect to the formal framework of corresponds to the solution of the ; classical geometric constructions. The impossibility , which is not reducible, and thus geometrically proof simply justify a statement and not a concept. unsolvable. Irrational numbers are mathematically The focus of theorem was to convert the problem defined as being not a finite solution from a division. from the complex consideration as presented in the However, this is not a fashionable definition, as most impossibility proof, into a simpler problem, and its of number division are an open loop operation that can solution found following the formal Greek‟s rules of never be ended [19]. The impossibility proof of geometry. For centuries, the problem of doubling a doubling a cube was based on three dimensional cubic cube has been a subject to pseudo mathematical extensions in , an approach which approaches, which do not reach the set limits of entirely shifted the problem to from its accuracy [25]-[29]. The present method has presented Greek‟s definition in plane geometry, and therefore a justified geometrical rationality (since all the the algebraic statement of impossibility has no presented constructions have been reduced to the geometrical validity. Genetically, Euclidean plane Euclidean postulates of practical geometry) of the geometry obliges the solution of all the compass-ruler constant as to an accuracy of in constructions to be carried out in a plane (according to reference to the numerical value from the calculator. It the first thee postulates of of the thirteen books can therefore be affirmed that, by following the of Euclid elements). This is evident from the fact that revealed approach, it is geometrically possible to solve no two facets of a cube can share all their four vertices the factors: and other from two different planes. Thus according to this ratios in that order with a meaningful precision, where paper, the impossible imprecise classification should is the magnitude of the given cube. not be extended to geometry so that the irrationality definition was stated as “algebraic irrationality is not a Acknowledgements of the geometry”. The numerical We greatly acknowledge the constructive suggestions value of the Delian constant from the numerical by the friends and reviewers who took part in the calculator settle at . This value lie evaluation of the developed theorems. in the range , and these two fractions are geometrically constructible as VI. References demonstrated earlier. The existence of these two [1] Wantzel, P.L., “Recherches sur les moyens de reconnaitre si un problème de géométrie peut se résoudre avec la règle et le fractions implies the possibility to solve the rationality compass”, Journal de Mathematiques pures et appliques, 1837. of the constant to a meaningful precision. In [2] Tom Davis, “Classical Geometric Construction”, , December 4, 2002.

ISSN: 2231-5373 http://www.ijmttjournal.org Page 314 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 50 Number 5 October 2017

[3] University of Toronto, “The Three Impossible Constructions of Geometry”, , 14 Aug, 1997. Annex-1: Algebraic Proof of the Impossibility to [4] B. Bold, “The Delian problem, in: Famous Problems of Geometry and How to Solve Them”, New York: Dover, 1982. construct [5] Burton, David M., “A : An Introduction, 4th Ed”., McGraw-Hill, p. 116, 1999. Consider the following section of proof: [6] English cube from Old French < Latin cubus < Greek κύβος (kubos) meaning "a cube, a , vertebra". In turn Suppose that is rational, then, can be written as from PIE*keu(b)-, "to bend, turn" [7] Wantzel, P.L., “Recherches sur les moyens de reconnaitre si un , where both and are whole numbers problème de géométrie peut se résoudre avec la règle et le with no common factor. It follows that; compass”, Journal de Mathematiques pures et appliques, pp. 366-372, 1837. [8] C.A.Hart and Daniel D Feldman, “Plane and Solid Geometry”, Therefore, American Book Company, Chicago, 1912, 2013, p.96, p.171. [9]Rich Cochrane, et.al, “Euclidean Geometry”, Fine Art Maths Clearly, following the algebraic fact that; Centre, October 2015. [10] W, Dunham,”The Mathematical ”, John Wiley & 1. The product of two odd numbers is odd Sons, Inc., New York, 1994. 2. The product between an odd number and an even [11] Francois Rivest and Stephane Zafirov, “Duplication of the number is even cube”, , Zafiroff, 3. The product between two even numbers is even 1998. [12] H. Dorrie, “The Delian cube-doubling problem, in: 100 Great Problems of : Their History and It can be deduced that; if both and have no Solutions”, New York: Dover, 1965. common factor, then the relation pose a [13] . A. Janes,S.Morris, and K.Pearson,Abstruct “Algebra and Famous Impossibilities”, Springer-verlag, New York, 1991. contradiction, since the product is even, [14] W. R. Knorr, “The Ancient Tradition of Geometric Problems”, Boston, 1986. because of the number This implies that, itself [15]Weisstein,w.Eric,“CubeDuplication”,,1999. must be even. Therefore, it is not possible to construct [16]J.H. Conway and R.K Guy, “The Book of Numbers”. New the factor , which is classified as irrational. York, Springer-Verlag, 1996. [17] I.Thomas, “Selections Illustrating the History of ”, William Heinemann, London, and Harvard University Press, Cambridge, MA. 1939, 1941. [18]F. Klein, "The Delian Problem and the Trisection of the Angle." Ch. 2 in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, 1980, pp. 13-15. [19] Nguyên Tân Tài, “Doubling a Cube”, , 2011. [20] University of Wisconsin, “Duplicating the Cube”, , Wisconsin-Green Bay, 6 September 2001. [21]M. Kac, S. M. Ulam, “Mathematics and Logic”, Dover Publications, 1992. [22]C.S.Ogilvy, “Excursions in Geometry”, Dover, New York, Fig. 9: Geometrical construction of the irrational 1990, pp.135-138. [23]R. Courant, H. Robbins, I. Stewart, “What Is Mathematics?”, number Oxford University Press, 1996. [24] D. E.Smith, “History of Mathematics”, Dover Publications, Annex-1.2: Geometric construct of New York. vol. 1, 1923. [25]J.O.Connorand E.F.Robertson, “”, From figure 9, it is evident that, the factor is , April 1999. cannot be algebraically determined, the geometric [26]D.Wells, “The Penguin Dictionary of Curious and Interesting of this case is that, the factor is a Geometry”, London: Penguin, 1991, pp. 49-50. [27]Kristóf Fenyvesi et.al, “Two Solutions to an Unsolvable multiplicative factor, which propagate along any such Problem”, Connecting and GeoGebra in a Serbian two dimensional figure of the kind shown in figure 9. High School, Proceedings of Bridges 2014. Thus in this case, the impossibility proof presented in [28] Demaine, Erik D. - O'Rourke, Joseph, “Geometric Folding annex-1 correspond to definition 1.2 in the sense that, Algorithms, Linkages, Origami, Polyhedra. Cambridge”, Cambridge University Press (2007). it provides proof to a statement, and not a [29] Boakes, Norma, “Origami-Mathematics Lessons, Paper Folding construction. as a Teaching Tool”, Mathitudes 1(1) (2008), 1-9. [30] R. DESCARTES, “La Géométrie”, C. Angot, Paris, 1664 (first published in 1637).

ISSN: 2231-5373 http://www.ijmttjournal.org Page 315