Introduction: Two Kinds of Proportion
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Symmetry As an Aesthetic Factor
Comp. & Maths. with Appls, Vol. 12B, Nos. I/2, pp. 77-82. 1986 0886-9561/86 $3,1)0+ .00 Printed in Great Britain. © 1986 Pergamon Press Ltd. SYMMETRY AS AN AESTHETIC FACTOR HAROLD OSBORNEt Kreutzstrasse 12, 8640 Rappersvill SG, Switzerland Abstract--In classical antiquity symmetry meant commensurability and was believed to constitute a canon of beauty in nature as in art. This intellectualist conception of beauty persisted through the Middle Ages with the addition doctrine that the phenomenal world manifests an imperfect replica of the ideal symmetry of divine Creation. The concept of the Golden Section came to the fore at the Renaissance and has continued as a minority interest both for organic nature and for fine art. The modern idea of symmetry is based more loosely upon the balance of shapes or magnitudes and corresponds to a change from an intellectual to a perceptual attitude towards aesthetic experience. None of these theories of symmetry has turned out to be a principle by following which aesthetically satisfying works of art can be mechanically constructed. In contemporary theory the vaguer notion of organic unity has usurped the prominence formerly enjoyed by that of balanced symmetry. From classical antiquity the idea of symmetry in close conjunction with that of proportion dominated the studio practice of artists and the thinking of theorists. Symmetry was asserted to be the key to perfection in nature as in art. But the traditional concept was radically different from what we understand by symmetry today--so different that "symmetry" can no longer be regarded as a correct translation of the Greek word symmetria from which it derives--and some acquaintance with the historical background of these ideas is essential in order to escape from the imbroglio of confusion which has resulted from the widespread conflation of the two. -
Fibonacci Number
Fibonacci number From Wikipedia, the free encyclopedia • Have questions? Find out how to ask questions and get answers. • • Learn more about citing Wikipedia • Jump to: navigation, search A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, … , are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, ... (Sometimes this sequence is considered to start at F1 = 1, but in this article it is regarded as beginning with F0=0.) The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India. [1] [2] • [edit] Origins The Fibonacci numbers first appeared, under the name mātrāmeru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandah-shāstra, the Art of Prosody, 450 or 200 BC). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well-known text on these. -
De Divino Errore ‘De Divina Proportione’ Was Written by Luca Pacioli and Illustrated by Leonardo Da Vinci
De Divino Errore ‘De Divina Proportione’ was written by Luca Pacioli and illustrated by Leonardo da Vinci. It was one of the most widely read mathematical books. Unfortunately, a strongly emphasized statement in the book claims six summits of pyramids of the stellated icosidodecahedron lay in one plane. This is not so, and yet even extensively annotated editions of this book never noticed this error. Dutchmen Jos Janssens and Rinus Roelofs did so, 500 years later. Fig. 1: About this illustration of Leonardo da Vinci for the Milanese version of the ‘De Divina Proportione’, Pacioli erroneously wrote that the red and green dots lay in a plane. The book ‘De Divina Proportione’, or ‘On the Divine Ratio’, was written by the Franciscan Fra Luca Bartolomeo de Pacioli (1445-1517). His name is sometimes written Paciolo or Paccioli because Italian was not a uniform language in his days, when, moreover, Italy was not a country yet. Labeling Pacioli as a Tuscan, because of his birthplace of Borgo San Sepolcro, may be more correct, but he also studied in Venice and Rome, and spent much of his life in Perugia and Milan. In service of Duke and patron Ludovico Sforza, he would write his masterpiece, in 1497 (although it is more correct to say the work was written between 1496 and 1498, because it contains several parts). It was not his first opus, because in 1494 his ‘Summa de arithmetic, geometrica, proportioni et proportionalita’ had appeared; the ‘Summa’ and ‘Divina’ were not his only books, but surely the most famous ones. For hundreds of years the books were among the most widely read mathematical bestsellers, their fame being only surpassed by the ‘Elements’ of Euclid. -
Thesis Final Copy V11
“VIENS A LA MAISON" MOROCCAN HOSPITALITY, A CONTEMPORARY VIEW by Anita Schwartz A Thesis Submitted to the Faculty of The Dorothy F. Schmidt College of Arts & Letters in Partial Fulfillment of the Requirements for the Degree of Master of Art in Teaching Art Florida Atlantic University Boca Raton, Florida May 2011 "VIENS A LA MAlSO " MOROCCAN HOSPITALITY, A CONTEMPORARY VIEW by Anita Schwartz This thesis was prepared under the direction of the candidate's thesis advisor, Angela Dieosola, Department of Visual Arts and Art History, and has been approved by the members of her supervisory committee. It was submitted to the faculty ofthc Dorothy F. Schmidt College of Arts and Letters and was accepted in partial fulfillment of the requirements for the degree ofMaster ofArts in Teaching Art. SUPERVISORY COMMIITEE: • ~~ Angela Dicosola, M.F.A. Thesis Advisor 13nw..Le~ Bonnie Seeman, M.F.A. !lu.oa.twJ4..,;" ffi.wrv Susannah Louise Brown, Ph.D. Linda Johnson, M.F.A. Chair, Department of Visual Arts and Art History .-dJh; -ZLQ_~ Manjunath Pendakur, Ph.D. Dean, Dorothy F. Schmidt College ofArts & Letters 4"jz.v" 'ZP// Date Dean. Graduate Collcj;Ze ii ACKNOWLEDGEMENTS I would like to thank the members of my committee, Professor John McCoy, Dr. Susannah Louise Brown, Professor Bonnie Seeman, and a special thanks to my committee chair, Professor Angela Dicosola. Your tireless support and wise counsel was invaluable in the realization of this thesis documentation. Thank you for your guidance, inspiration, motivation, support, and friendship throughout this process. To Karen Feller, Dr. Stephen E. Thompson, Helena Levine and my colleagues at Donna Klein Jewish Academy High School for providing support, encouragement and for always inspiring me to be the best art teacher I could be. -
Greek Architecture
2nd Year Architecture 2018/2019 second Semester History of Architecture I Lecture (6 and 7) : Classic Greek Architecture by : SEEMA K. ALFARIS 1 Lecture ’s information Course name History of Architecture I Lecturer Seema k. Alfaris Course ’s information This course traces the history of Architecture from the early developments in the Paleolithic Age (Early Stone Age) to the Rome (16th century).. The objective 1. Understanding the Greek Architecture , and the factors which shape this Architecture. 2. Understanding The Main Types of buildings that Greek famous with . 2 The Historical Timeline of Architecture 3 FACTORS INFLUENCING ARCHITECTURE 4 Greek and Rome Architecture CLASSICAL PERIODS – GREEK AND ROMAN Classical antiquity (also the classical era, classical period or classical age) • is the period of cultural history between the 8th century BC and the 5th or 6th century AD centered around the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the Greco-Roman world. • It is the period in which Greek and Roman society flourished and wielded great influence throughout Europe, North Africa and Western Asia. 5 Greek Architecture 1. Geographical factor: • Greek civilization started from group of islands in the Mediterranean Sea . • Greece has a broken coast line with about 3000 islands, which made the Greeks into a sea-faring people. • Greek civilization occurred in the area around the Greek mainland, on a peninsula that extends into the Mediterranean Sea . • Greek civilization expanded by The colonization of neighboring lands such as the Dorian colonies of Sicily & the Ionian colonies of Asia minor , and that’s the reason of the separation of Greek civilization. -
Columns & Construction
ABPL90267 Development of Western Architecture columns & construction COMMONWEALTH OF AUSTRALIA Copyright Regulations 1969 Warning This material has been reproduced and communicated to you by or on behalf of the University of Melbourne pursuant to Part VB of the Copyright Act 1968 (the Act). The material in this communication may be subject to copyright under the Act. Any further copying or communication of this material by you may be the subject of copyright protection under the Act. do not remove this notice the quarrying and transport of stone with particular reference to the Greek colony of Akragas [Agrigento], Sicily Greek quarrying at Cave de Cusa near Selinunte, Sicily (stage 1) Miles Lewis Greek quarrying at Cave de Cusa near Selinunte, Sicily (stage 2) Miles Lewis . ·++ Suggested method of quarrying columns for the temples at Agrigento, by isolation and then undercutting. Pietro Arancio [translated Pamela Crichton], Agrigento: History and Ancient Monuments (no place or date [Agrigento (Sicily) 1973), fig 17 column drum from the Temple of Hercules, Agrigento; diagram Miles Lewis; J G Landels, Engineering in the Ancient World (Berkeley [California] 1978), p 184 suggested method of transporting a block from the quarries of Agrigento Arancio, Agrigento, fig 17 surmised means of moving stone blocks as devised by Metagenes J G Landels, Engineering in the Ancient World (Berkeley [California] 1978), p 18 the method of Paconius Landels, Engineering in the Ancient World, p 184 the raising & placing of stone earth ramps cranes & pulleys lifting -
Syddansk Universitet Review of "Henning, Herbert: La Divina
Syddansk Universitet Review of "Henning, Herbert: La divina proportione und die Faszination des Schönen oder das Schöne in der Mathematik" Robering, Klaus Published in: Mathematical Reviews Publication date: 2014 Document version Accepted author manuscript Citation for pulished version (APA): Robering, K. (2014). Review of "Henning, Herbert: La divina proportione und die Faszination des Schönen oder das Schöne in der Mathematik". Mathematical Reviews. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 14. Feb. 2017 MR3059375 Henning, Herbert La divina proportione und die Faszination des Sch¨onenoder das Sch¨onein der Mathematik. (German) [The divine proportion and the fascination of beauty or beauty in mathematics] Mitt. Math. Ges. Hamburg 32 (2012), 49{62 00A66 (11B39 51M04) The author points out that there is a special relationship between mathematics and the arts in that both aim at the discovery and presentation of truth and beauty. -
Leonardo Universal
Leonardo Universal DE DIVINA PROPORTIONE Pacioli, legendary mathematician, introduced the linear perspective and the mixture of colors, representing the human body and its proportions and extrapolating this knowledge to architecture. Luca Pacioli demonstrating one of Euclid’s theorems (Jacobo de’Barbari, 1495) D e Divina Proportione is a holy expression commonly outstanding work and icon of the Italian Renaissance. used in the past to refer to what we nowadays call Leonardo, who was deeply interested in nature and art the golden section, which is the mathematic module mathematics, worked with Pacioli, the author of the through which any amount can be divided in two text, and was a determined spreader of perspectives uneven parts, so that the ratio between the smallest and proportions, including Phi in many of his works, part and the largest one is the same as that between such as The Last Supper, created at the same time as the largest and the full amount. It is divine for its the illustrations of the present manuscript, the Mona being unique, and triune, as it links three elements. Lisa, whose face hides a perfect golden rectangle and The fusion of art and science, and the completion of the Uomo Vitruviano, a deep study on the human 60 full-page illustrations by the preeminent genius figure where da Vinci proves that all the main body of the time, Leonardo da Vinci, make it the most parts were related to the golden ratio. Luca Pacioli credits that Leonardo da Vinci made the illustrations of the geometric bodies with quill, ink and watercolor. -
FACTS Standards
FACTS Standards INTERNATIONAL STANDARD GUIDE FRM-400 Addopted-1997 Using The Goldenmean For Proportional Divisions In Decorative Standards-1998 Matting Revised-1999 Revised-2000 FACTS publishes this document as a public service. Its use is voluntary, and all results obtained by its use must be entirely the responsibility of the user. This document is subject to revision, change and/or withdrawal at any time. © FACTS 2000 1.00 Design (defined) To plan out in systematic, usually graphic form: To create or contrive for a particular purpose or effect: To create or execute in an artistic or highly skilled manner. 1.01 Everything can be called a design, but to achieve good design the particular purpose must be clear. 1.02 When designing for the framing of artwork art the purpose is " create a surrounding to complement with out distraction". 1.03 When designing for wall decor the latitude is greater, the purpose may be to create a greater interest, make a color or size statement, brighten a drab corner or create something everyone will ask about. 1.04 The Golden Mean provides a formula for the pleasing division of space. 2.00 The Golden Mean (divine proportion) 2.01 In about 300 BC a mathematician by the name of Euclid established a formula that also occurs in nature, when applied to an area that area can be divided into visually balanced unequal proportions that were visually pleasing to the eye. 2.02 In the 1st century BC Vitruvius, a Roman architect of c.90-c.20 BC, wrote De Architectura or the "Ten Books" documenting Roman and Greek architectural history and building practices. -
Towards Van Der Laan's Plastic Number in the Plane
Journal for Geometry and Graphics Volume 13 (2009), No. 2, 163–175. Towards van der Laan’s Plastic Number in the Plane Vera W. de Spinadel1, Antonia Redondo Buitrago2 1Laboratorio de Matem´atica y Dise˜no, Universidad de Buenos Aires Buenos Aires, Argentina email: vspinadel@fibertel.com.ar 2Departamento de Matem´atica, I.E.S. Bachiller Sabuco Avenida de Espa˜na 9, 02002 Albacete, Espa˜na email: [email protected] Abstract. In 1960 D.H. van der Laan, architect and member of the Benedic- tine order, introduced what he calls the “Plastic Number” ψ, as an ideal ratio for a geometric scale of spatial objects. It is the real solution of the cubic equation x3 x 1 = 0. This equation may be seen as example of a family of trinomials − − xn x 1=0, n =2, 3,... Considering the real positive roots of these equations we− define− these roots as members of a “Plastic Numbers Family” (PNF) com- prising the well known Golden Mean φ = 1, 618..., the most prominent member of the Metallic Means Family [12] and van der Laan’s Number ψ = 1, 324... Similar to the occurrence of φ in art and nature one can use ψ for defining special 2D- and 3D-objects (rectangles, trapezoids, ellipses, ovals, ovoids, spirals and even 3D-boxes) and look for natural representations of this special number. Laan’s Number ψ and the Golden Number φ are the only “Morphic Numbers” in the sense of Aarts et al. [1], who define such a number as the common solution of two somehow dual trinomials. -
De Allereerste Nederlandse Vertaling Van (De) 'Divina Proportione'
Grootveld, Roelo s, Huylebrouck De allereerste Nederlandse vertaling van (De) ‘Divina Proportione’, mét talloze kritische voetnoten. D. Huylebrouck: vertaling van ‘Divina Proportione’ Inleidende vraag (Boekenbeurs%): 15de eeuw: uitvinding boekdrukkunst) wat was het meest gedrukte en verspreide boek* De Bijbel Nog ‘bestsellers’ uit de 15 de eeuw* Euclides: ‘Elementen’. Luca Pacioli: - ‘Summa de arit metica, ...’ - ‘Goddelijke Ver ouding’. Terloops: wat is wereldwi,d het meest gedrukte en verspreide boek vandaag* De catalogus van Ikea. D. Huylebrouck: vertaling van ‘Divina Proportione’ Luca Pacioli (1445-1517), broeder en wiskundige. In 14,- ontmoette ij Leonardo da Vinci Ludovico (145. - 1519 ). Sforza Luca Leonardo (Hertog Pacioli maakte de Milaan illustraties: /ijn enige offici1le publicaties. -iniatuur uit het manuscript van Gen.ve D. Huylebrouck: vertaling van ‘Divina Proportione’ Twee manuscripten nu bewaard in Gen4ve en in 5ilaan (14,8). 7 een gedrukte versie (150,). Boek: succes door de tekeningen van Leonardo. D. Huylebrouck: vertaling van ‘Divina Proportione’ Boek Pacioli: versc illende ‘delen’, waarvan vertaald werden: Einde Divina Proportione 0egin De 1rchitectura Beide delen niet gesc eiden door een titelblad. 1) ‘Divina Proportione’ .) ‘De Arc itectura’ D. Huylebrouck: vertaling van ‘Divina Proportione’ ‘De Arc itectura’; Er is een /ogenaamd ‘3 de deel’; Libellus: niet in de vertaling. Pacioli /egt: deel 3 = Piero della Francesca, ik publiceerde et als een gedrukt wederdienst in… Geen plagiaat; … eindigt met ‘Finis’. Dus: be oort niet tot Pacioli’s tractaat. D. Huylebrouck: vertaling van ‘Divina Proportione’ Later werd (de titel van) et boek veel geciteerd door adepten van de ‘myt e van de gulden snede’ Door gebrek aan een goede vertaling? Duits: ‘De Leer van de Gulden Snede’. -
Leonardo's Vitruvian Man Drawing: a New Interpretation Looking at Leonardo's Geometric Constructions
Nexus Netw J (2015) 17:507–524 DOI 10.1007/s00004-015-0247-7 RESEARCH Leonardo’s Vitruvian Man Drawing: A New Interpretation Looking at Leonardo’s Geometric Constructions Vitor Murtinho1 Published online: 9 April 2015 Ó Kim Williams Books, Turin 2015 Abstract Generally speaking, today’s scientific community considers that the famous figure drawn by Leonardo da Vinci at the end of the fifteenth century was made using the Golden ratio. More specifically, the relationship established between the circle’s diameter and the side of the square is a consequence of the geometric relationship probably discovered by Euclid, but made famous by Luca Pacioli in his De Divina proportione. Aware of the close working relationship between Leonardo and Pacioli, namely in the writing of this last book, the theory that establishes a close relationship between these two figures, making use of this remarkable mathematical relationship, has gained credibility. In fact, the use of the Divina proporzione, despite being a very stimulating construction on an intellectual level, presents too great a margin of error, especially for such a competent geometrician as Leonardo da Vinci was. For that reason, the relationship between these two figures (square and circle) is grounded on a much simpler geometric relationship than the one found at the base of the definition, for instance, of Le Corbusier’s Modulor. Keywords Leonardo da Vinci Á Vitruvian man Á Vesica piscis Á Golden number Á Proportion Introduction No human investigation may claim to be a true science if it has not passed through mathematical demonstrations, and if you say that the sciences that begin and end in the mind exhibit truth, this cannot be allowed, but must be & Vitor Murtinho [email protected] 1 CES-Department of Architecture, University of Coimbra, Colegio das Artes, Largo D.