In the paper it is considered a construction of an egg curve
obtained by continual joining of the circular arcs with three different
centers. These three centers form sharp or flat triangles of the
pentagon (sharp and flat triangles are the basic building shapes of
Penrose tiling). They have many relationships with both the Fibonacci numbers and Phi. In such an egg curve, it is possible to inscribe two juxtaposed pentagons. We give also another construction which employs several, even infinite number of the pentagons. Those pentagons are forming a sequence related to Fibonacci sequence. The construction using the flat triangles will provide a curve which ratio between length of the perimeter and the sum of the length of its minor and major axes will give the coefficient Φ.http://mongeometrija.com/media/mongeometrija/2010/moNGeometrija%202010%20-%20PAGINACIJA.pd

Obradović, Marija

Petrović, Maja

GraFar - Repository of the Faculty of Civil Engineering

   2 UNDER THE AUSPICES OF: Republic of Serbia – Ministry of Science and Technological Development Faculty of Architecture in Belgrade Faculty of Mechanical Engineering in Belgrade Faculty of Civil Engineering in Belgrade Faculty of Forestry in Belgrade Faculty of Transport and Traffic Engineering in Belgrade Faculty of Applied Arts in Belgrade  3 25th National and 2nd International Scientific Conference moNGeometrija 2010  Publisher | Izdavač Faculty of Architecture in Belgrade Arhitektonski fakultet u Beogradu Serbian Society for Geometry and Graphics Srpsko udruženje za geometriju i grafiku  SUGIG Title of Publication PROCEEDINGS | BILTEN Reviewers | Recezenti Ph. D.  Miodrag Nestorović  Ph. D.  Aleksandar Čučaković Ph. D.  Marija Obradović Ph. D.  Branislav Popkonstantinović M. Sc.  Magdalena Dimitrijević  M. Sc. Slobodan Mišić Editor-in-Chief | Glavni urednik Ph.D. Miodrag Nestorović   Co-Editor | Zamenik urednika M. Sc.  Magdalena Dimitrijević     Text formatting Miljan Radunović, Ph. D. Candidate     ISBN 978-86-7924-040-8  Numbers of copies printed | Tiraž: 100  Printing | Štampa:  All rights reserved. No parts of this publication may be reprinted without either the prior written permission of the publisher.  Izdavač zadržava sva prava. Reprodukcija pojedinih delova ili celine nije dozvoljena.    4 Conference Organizers    Main Organizers Faculty of Architecture, Belgrade  Faculty of Mechanical Engineering, Belgrade    Co-organizers Faculty of Civil Engineering, Belgrade  Faculty of Forestry, Belgrade  Faculty of Transport and Traffic Engineering, Belgrade Faculty of Applied Arts, Belgrade    Conference Committees  Executive Committee  Ph.D. Miodrag Nestorović, Full professor, Faculty of Architecture, Belgrade Ph.D. Branislav Popkonstantinović, Associate professor, Faculty of Mechanical Engineering, Belgrade  Ph.D. Biserka Marković, Full professor, Faculty of Civil Engineering and Architecture, Niš Ph.D. Aleksandar Čučaković, Associate professor, Faculty of Civil Engineering, Belgrade  Ph.D. Radovan Štulić, Full professor, Faculty of Technical Sciences, Novi Sad Ph.D. Ljubica Velimirović, Full professor, Faculty of Science and Mathematics, Niš  5 Supervisory Committee  Ph.D. Miodrag Stoimenov, Associate professor, Faculty of Mechanical Engineering, Belgrade  M.Sc. Zorana Jeli, Lecturer Assistant, Faculty of Mechanical Engineering, Belgrade  Ph.D. Ratko Obradović, Associate professor, Faculty of Technical Sciences, Novi Sad Ph.D. Jelisava Kalezić, Associate professor, Faculty of Civil Engineering, Podgorica, Montenegro M.Sc. Marija Jevrić, Lecturer Assistant, Faculty of Civil Engineering, Podgorica, Montenegro  Ph.D. Ljiljana Petruševski, Associate professor, Faculty of Architecture, Belgrade Ph.D. Dragan Petrović, Associate professor, Faculty of Mechanical Engineering, Belgrade Dr Mirjana Devetaković Radojević, Docent, Faculty of Architecture, Belgrade M.Sc. Gordana Đukanović, Lecturer Assistant, Faculty of Forestry, Belgrade Maja Petrović, architect, Faculty of Transport and Traffic Engineering, Belgrade   Organizational Committee  Ph.D. Vladimir Mako, Dean of the Faculty of Architecture, Belgrade  Ph.D. Đorđe Vuksanović, Dean of the Faculty of Civil Engineering, Belgrade  Ph.D. Milorad Milovančević, Dean of the Faculty of Mechanical Engineering, Belgrade Ph.D. Milan Medarević, Dean of the Faculty of Forestry, Belgrade  Ph.D. Slobodan Gvozdenović, Dean of the Faculty of Transport and Traffic Engineering, Belgrade  Ph.D. Vladimir Kostić, Dean of the Faculty of Applied Arts, Belgrade  Ph.D. Aleksandar Čučaković, Associate professor, Faculty of Civil Engineering, Belgrade  Ph.D. Branislav Popkonstantinović, Associate professor, Faculty of Mechanical Engineering, Belgrade  Ph.D. Marija Obradović, Docent, Faculty of Civil Engineering, Belgrade  M.Sc. Magdalena Dimitrijević, Lecturer assistant, Faculty of Civil Engineering, Belgrade  M.Sc. Slobodan Mišić, Lecturer assistant, Faculty of Civil Engineering, Belgrade  M.Sc. Biserka Nestorović, Lecturer assistant, Faculty of Forestry, Belgrade  M.Sc. Biljana Jović, Lecturer assistant, Faculty of Forestry, Belgrade   6 Scientific Review Committee    Professor Đorđe Zloković, Member of Serbian Academy of Sciences and Arts, Department of Technical Sciences, Belgrade Ph.D. Miodrag Nestorović, Full professor, Faculty of Architecture, Belgrade Ph.D. Hellmuth Stachel, University of Technology Vienna, Austria  Ph.D. Gunter Weiss, Technical University of Dresden, Germany  Ph.D. Emil Molnar, Technical University of Budapest, Hungary Ph.D. Aleksander Dvoretsky, Kiev National University of Building and Architecture, Ukraine Ph.D. Milena Stavrić, TU Graz, Institut für Architektur und Medien, Austria Ph.D. Aleksandar Veg, Full professor, Faculty of Mechanical Engineering, Belgrade  Ph.D. Aleksandar Čučaković, Associate professor, Faculty of Civil Engineering, Belgrade  Ph.D. Natasa Danilova, Faculty of Civil engineering, Architecture and Geodesy, Sofia, Bulgaria Ph.D. Carmen Marza, Technical University of Cluj-Napoca, Romania Ph.D. Sonja Gorjanc, Faculty of Civil Engineering, Zagreb, Croatia, Hrvatska Ph.D. Marija Obradović, docent, Faculty of Civil Engineering, Belgrade M.Sc. Branko Pavić, Full professor, Faculty of Architecture, Belgrade  Ph.D. Ivana Marcikić, Full professor, Faculty of Applied Arts, Belgrade  Ph.D. Hranislav Anđelković, Retired full professor, Faculty of Civil Engineering and Architecture, Niš Ph.D. Miroslav Marković, Retired full professor, Faculty of Civil Engineering and Architecture, Niš   Ph.D. Lazar Dovniković, Retired full professor, Faculty of Technical Sciences, Novi Sad  7 Topics Theoretical geometry, exposed by synthetical or analytical methodology:    * Descriptive and constructive geometry    * Projective geometry    * Central projection, Perspective and Restitution    * Cartography    * Theory of Polyhedra    * Fractal geometry Geometry and Graphics applied in Engineering and Architecture:    * Engineering graphics    * Computational geometry (algorithms, computer modeling of abstract geometrical objects, structures, procedures and operations)    * Computer Aided Design and Drafting; Geometric and Solid Modeling; Product Modeling; Image Synthesis; Pattern Recognition; Digital Image Processing; Graphics Standards; Scientific and Technical Visualization    * Kinematics Geometry and Mechanisms    * Applications of Polyhedra theory    * Fractals    * Computational restitution    * Stereoscopy and Stereography    * Virtual reality Geometry applied in Visual Arts and Design:    * Theory and application of Visual Aesthetics    * Geometrical and mathematical criteria of Aesthetic values    * Perception and meaning of colors    * Geometrical forms applied in Visual Arts    * Optical illusions and its applications History of Geometry:    * Famous scientist and their contribution     * Origin, derivation and development of particular geometrical branches     * History of geometrical education Education and didactics:    * Descriptive Geometry and Graphics Education, including the Reform of Education           * Education Technology Research    * Multimedia Educational Software Development    * Virtual Reality Educational Systems    * Educational Software Development Tools Research and so on  532 CONSTRUCTING THE EGG CURVES USING THE GOLDEN RATIO OF PENTAGON  Maja Petrović117 Marija Obradović118   RESUME  In the paper it is considered a construction of an egg curve obtained by continual joining of the circular arcs with three different centers. These three centers form sharp or flat triangles of the pentagon (sharp and flat triangles are the basic building shapes of Penrose tiling). They have many relationships with both the Fibonacci numbers and Phi. In such an egg curve, it is possible to inscribe two juxtaposed pentagons. We give also another construction which employs several, even infinite number of the pentagons. Those pentagons are forming a sequence related to Fibonacci sequence. The construction using the flat triangles will provide a curve which ratio between length of the perimeter and the sum of the length of its minor and major axes will give the coefficient Φ. Key words: pentagon, egg curve, golden ratio, circular arc                                                    117 Assistant Trainee, Faculty of Transport and Traffic Engineering, Belgrade, Serbia 118 Associate Professor, PhD, Faculty of Civil Engineering, Belgrade, Serbia  533 1. INTRODUCTION Forming a planar curve using the method of continuation of circular arcs does not require a complicated geometric construction, or exact mathematical formula. This construction has a practical application in the designing.    The construction of egg-shaped curve by method of 45º angle (Figure 1): At the first semicircle of radius r with center C1, there are attached, symmetrically on the both sides of it, the two eighth of circle twice the larger radii (2r) with centers C2 and C3 which are located at the endpoints of the semicircle. Then there is attached another quarter circle arc, with center C4, set in the intersection of the radii of the previous arcs lined at an angle of 45º respectively to the axis of the curve, with radius a=2r-r√2.  Figure 1: Construction Of The Egg Curve By The Method Of 45o Angle The egg-shaped plane figure formed with continuous circular arcs centered on the vertices of the Pythagorean triangle AOB - the method 345 - is shown in the Figure 2: The first semi-circle with center A has the radius AN = 5 (Pythagorean hypotenuse of a triangle), the other one is centered at the point B twice the larger radius, while the third circular arc with center at the point O has the radius equal to the sum of the laterals of the  534 Pythagorean triangle (3 +4 = 7).   Figure 2: Construction Of The Egg Curve By The Method „345“  1.1 The Construction Of Egg-Shaped Curves With 3, 4 Or 5 Points In the book Mathographics (Mathographics - How to draw a real Chicken Egg), the mathematician Robert Dixon shows the constructions of seven egg-shaped curves, using circular arcs continuing to each other, and he called such an egg - the Euclidean egg (Egg Euclidean) which is shown in the figure 3.  Figure 3: The Egg Curves Obtained By Robert Dixon Construction  This paper deals with one of the possible construction of the golden egg using the described methods.  535  1.2  Connection Of The Golden Section And The Construction Of Egg-Shaped Curve – The Method Of 45º Angle The figure 4 presents the previously described method of 45º angle, associated with the golden section. For the specific value of the initial semicircle radius, we may observe more easily the construction of the golden egg.  If the first arc, semicircle, would have the radius r = Φ with the center at the point O, the second (and the third) arc, centered at the point H (and G) of the radius twice as large - would value 2Φ,then the fourth arc, quarter of the circle centered at the point I would have the radius p.   Figure 4: The Golden Egg Curve Construction By The Method Of 45o Angle 2. THE EGG CURVE AND THE GOLDEN SECTION 2.1. Constructing The Egg Shaped Curve Using The Regular Pentagon  536 Forming the egg-shaped curve using two inserted pentagons, by the method “345”of continuous circular arcs can produce the golden egg (Fig. 5).  The circular arc l1, with the center at the point C1 has the radius of 1, the circular arc l2 centered at the point C2 has the radius 1+ Φ, while the circular arc l3 with the center at the point C3 has the radius of Φ.  Figure 5: The Golden Egg And The Pentagon For the previously described method, it is possible to obtain, by iteration, a series of egg shaped curves, which will be described in three, four, five, ... pentagons (Fig. 6).  537  Figure 6: The Golden Egg and the Series of Pentagons These curves may create interesting rosettes (Fig.10) which can be used in designing and arts.  2.2 The Construction Of The Egg Curve Using The Penrose    Tiling Triangle The golden Egg curve is primary obtained using the triangle formed by Penrose tiling119, so that the vertices of the triangle are the centers of circular arcs C1, C2 and C3 (Fig. 7).                                              119 Kepler (Johannes Kepler, 1571-1630) was the one of the first scientists who investigated the paving (tiling) and symmetry of the pentagon (5-fold symmetry, Harmonices Mundi, Book II), and who also inspired Roger Penrose (Roger Penrose), to discover in 1973. and 1974. the three new sets of aperiodic pentagonal tiling with symmetry, that are used as models in geometry of quasicrystal structures.  538  Figure 7: Sharp And Flat Triangles Of The Pentagon120 When the egg-shaped curve is formed by the sharp triangle of the Penrose tiling - the triangle with angles of 36 º - 72 º - 72 º (Figure 8) then we will get the same curve as in the Figure 5.  Figure 8: The Golden Egg And The Sharp Triangle (36º - 72º - 72º)                                             120 The picture is taken from: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi2DGeomTrig.html   539 For a blunt (flat) triangle of the Penrose tiling, with the angles of 36º - 36º - 108º (Figure 9) there will be obtained  the golden egg, ie. the curve which ratio between its volume and the sum of major and minor axis equals Φ.  Figure 9:  The Golden Egg And The Flat Triangle (36º - 36º - 108º)  3. CONCLUSIONS   540 Construction of the egg shaped curves produced by the method of continuous circular arcs, so that the centers of these circles are vertices of the sharp or the flat triangles, and the construction of the curve using the pentagon were not yet presented. Thus, the resulting curve can be inspiring and can be used for further research:  • to create interesting rosettes (Fig. 10)  • to create aperiodic tiling, and so on.   Figure 10: Rosettes formed by rotation of golden egg shaped curves REFERENCES:  1. Johannes Kepler (1571 - 1630) : New Astronomy, Translated by William H. Donahue. Cambridge, U.K., 1992. Translation of Astronomia Nova Aitiologetos, sev Physica Coelestis (1609). 2. Dixon, R.: The Drawing Out of an Egg, New Sci., 290-295, 1982. 3. Dixon, R.: Mathographics, New York, Dover, p. 3-11,75-78,159,  1991.  4. Gardner М.: The Last Recreations, Hydras, Eggs, and Other Math.Mystifications, Springer,  New York, 45-50, 1997. 5. Gardner M.: Extraordinary nonperiodic tiling that enriches the  theory of tiles, Mathematical Games, Scientific American, January,  1977, p. 110-121 6. Grunbaum B. and Shephard G. C.: Tilings and Patterns, Freeman, 1987.  7. Cundy H. and Rollett A. : Mathematical Models, 3rd ed. Stradbroke, England, Tarquin Pub., p. 19-21, 35, 1989.  541 8. Pritchard C.: The Changing Shape of Geometry : Celebrating a Century of Geometry and Geometry Teaching, Cambridge, 2003 9. Atanassova K., Atanassova V., Shannon A. and Turner J.: New Visual Perspectives on Fibonacci Numbers, World Scientific, 2002 10.Evan A. A.: Practical Sheet And Plate Metal Work, Chapter XVI. Hip And Sponge Baths Egg-Shaped Oval,  Wellhausen Press, 2008. 11.http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi2DGeomTrig.html 12.Rojas C. C.: Geometry of parabola according to the golden number, vismath - Visual Mathematics,  Electronic journal,  volume 4, No. 3, 2002, ISSN 1821-1437 http://vismath7.tripod.com/rojas/index.html 13.Carlos Calvimontes Rojas: The golden triangle  http://urbtecto.zzl.org/GOLDEN.html http://urbtecto.zzl.org/PENTACAEDRO.html http://exapenta.zxq.net/CALVIMONT.html 14.http://numerodeoro.wordpress.com/aplicaciones-de-phimatrix/ 15.http://intendo.net/penrose/info.html 16.http://local.wasp.uwa.edu.au/~pbourke/texture_colour/nonperiodic/ 17.EggMath:    http://chickscope.itg.uiuc.edu/explore/eggmath/ 18.Köller J. 2000.:   http://www.mathematische-basteleien.de/eggcurves.htm 

Constructing the Egg Curves Using the Golden Ratio of Pentagon

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Constructing the Egg Curves Using the Golden Ratio of Pentagon

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