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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. -
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. -
Perspective As a Convention of Spatial Objects Representation
TECHNICAL TRANSACTIONS CZASOPISMO TECHNICZNE ARCHITECTURE ARCHITEKTURA 11-A/2015 ANNA KULIG*, KRYSTYNA ROMANIAK** PERSPECTIVE AS A CONVENTION OF SPATIAL OBJECTS REPRESENTATION PERSPEKTYWA JAKO KONWENCJA PRZEDSTAWIANIA OBIEKTÓW PRZESTRZENNYCH Abstract In this study perspective was presented as one of the conventions for recording 3D objects on a 2D plane. Being the easiest type of projection to understand (for a recipient) perspective, in its historical context, was presented from ancient times until today1. Issues of practice and theory were addressed and examples of precise compliance with the rules and their negation in the 20th century were presented. Contemporary practical applications in architecture were reviewed. Keywords: linear perspective, computer perspective visualization Streszczenie W pracy przedstawiono perspektywę jako jedną z konwencji notowania trójwymiarowych obiektów na dwuwymiarowej płaszczyźnie. Perspektywę, jako najbardziej zrozumiały dla odbiorcy rodzaj rzutowania, zaprezentowano w ujęciu historycznym (od antyku do czasów współczesnych). Odniesiono się do praktyki i teorii, przykładów ścisłego stosowania reguł i ne- gowania ich w XX wieku. Przywołano współczesne zastosowania praktyczne w architekturze. Słowa kluczowe: perspektywa liniowa, komputerowe wizualizacje perspektywiczne * Ph.D. Arch. Anna Kulig, Assoc. Prof. D.Sc. Ph.D. Krystyna Romaniak, Faculty of Architecture, Cracow University of Technology; [email protected], [email protected]. 1 In his work [21] Michał Sufczyński presents the development of the recording methods in architectural design (Chapter XI, p. 78). The analysis based on works [4] and [11] covers from earliest times up to today. 20 1. Introduction Spatial objects are observed in three dimensions – we view them from different sides and directions. „For each look, new shapes of the dimensions sizes and lines are brought. -
Robert Zwijnenberg Introduction Leon Battista Alberti's Treatise De Pictura
WHY DID ALBERTI NOT ILLUSTRATE HIS DE PICTURA? Robert Zwijnenberg Introduction Leon Battista Alberti's treatise De Pictura (1435-36) is best understood as an attempt to elevate painting from its lowly position as a craft, which it still had in Italy at the beginning of the fifteenth century, and endow it with the status of a liberal art. Alberti sought to restore the glory which painting enjoyed in Antiquity. To accomplish this goal, it was necessary to invent a theoretical foundation for it, includ ing a specific technical vocabulary for discussing painting as a lib eral art. The liberal arts differ from the mechanical arts precisely in that they are based on general theoretical principles. For such prin ciples and vocabulary Alberti relied on two disciplines that were available to him in particular: mathematics, which he deployed for discussing the quantifiable aspects of painting, and rhetoric, by which he could elaborate subjects associated with form, content and com position, as well as with the relation between paintings and their viewers. Specifically, Alberti tried to achieve his aim by concentrat ing his discourse in De Pictura on linear perspective and its use of optical and geometrical principles; on an analysis of pictorial com position that is couched in rhetorical terms; and on the connection between painting and poetry with respect to the proper selection and representation of subjects as well as regarding advice on the proper education and attitude of the painter. On account of his decidedly theoretical focus, he succeeded not only in giving painting the sta tus of a liberal art, but also in making painting an appropriate sub ject for civilized, humanist thought and discussion.* The author himself repeatedly emphasized the novel character of his project. -
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. -
Kepler's Cosmos
science and image Kepler’s cosmos Copernicus’s system of the Universe was revolutionary but his method of representing it on paper was anything but. It was left to Kepler to apply Renaissance techniques of spatial visualization to make the theory come alive. Martin Kemp icolaus Copernicus’s programme to 8 purge the Ptolemaic model of the Uni- Nverse of its growing burden of disfig- uring complications was driven not least by what we would call aesthetic considerations. The architecture of his heliocentric system reinstated the geometrical integrity that the Greek astronomers had sought: “At rest... in the middle of everything is the sun. For in this most beautiful temple, who would place this lamp in another or better position than that from which it can light up the whole thing at the same time?” Although Copernicus’s “temple” obeyed the principles of harmonically unified design advocated in the Renaissance, his conventionally diagrammatic representa- tion of his scheme in 1543 as a flat series of concentric circles did not avail itself of the new spatial vision inherent in the buildings and paintings of the Renaissance masters. It was more than 50 years later that Johannes Kepler, fervent Copernican and Platonist, allied the new visual forms with the new astronomical vision. Among the many testimonies to Kepler’s extraordinary powers of spatial visualiza- tion, none is more remarkable than the great cosmological model he illustrated in a fold- out plate in his Mysterium Cosmographicum of 1596. We know how the scheme came to be envisaged. He tells how, when he was teach- ing his “students the way Great Conjunc- tions jump eight signs at a time”, he drew Kepler’s “Model of the Orbits of the Planets” “many triangles, or quasi triangles, in the from Mysterium Cosmographicum, 1596 (above). -
The Deep Space Mission That Might Trigger a Multidisciplinary Scientific Revolution by Judy Hodgkiss
II. Scientific Revolutions Will Shape the Future The Deep Space Mission that Might Trigger a Multidisciplinary Scientific Revolution by Judy Hodgkiss President Trump insists he wants the U.S. to lead Is There a Fundamental Principle of the world in scientific and technological develop- Development in the Universe? ment. He instinctively understands that the key to In 1921, Albert Einstein went beyond his relativity doing that is resuscitating and upgrading the U.S. theory, and put forward a theory of the cosmos, what he space program. called the concept of a “finite but unbounded” universe. As an example of the fundamentals of physical In a lecture that year before the Prussian Academy of economy, Lyndon LaRouche frequently pointed out Sciences, he asked the audience to consider a new met- how John F. Kennedy’s Apollo Project space mission aphor: He described the physical universe as a finite had provided the impetus for explosive growth of the number of shadows projected from different kinds of “Silicon Valley,” the previously higher-order geometries. He chal- bucolic rural area near San Jose, lenged the audience to not only California, which became Ameri- imagine the unseen geometries, ca’s high-tech center in the 1960s. but to imagine novel yardsticks LaRouche always insisted that that could measure them. Such a space exploration must be the universe would be bound only by point-on-the-spear in terms of the principles of development that scientific R&D. If the nation’s characterized it.1 frontier missions in space are ap- Einstein would be delighted propriately defined and fully- today to see examples of scien- funded, they not only provide tists in such fields as physics, spin-off benefits in innovative chemistry, or electronics, discov- ALMA technologies in obviously related Proto-planetary disc surrounding the young ering, to their amazement, that fields, but, even more impor- stellar object, Elias 2-27, 450 light-years away. -
Alberti on Perspective: Calling His Bluff
Online Journal of Art and Design volume 3, issue 2, 2015 Alberti on Perspective: Calling His Bluff Donald Richardson [email protected] ABSTRACT There are lacunae in the paragraphs of De Pictura in which Leon Battista Alberti describes his method of perspective construction. Although this has been a matter of considerable discussion among Alberti scholars in recent decades, none has deconstructed the text enough to state unequivocally that the theory is both defective and circular – which it is: • Alberti does not provide sufficient information to enable the construction to be drawn with any degree of mathematical precision, in spite of his basing its validity on mathematical proof; • when the construction is drawn (which can only be achieved after making certain assumptions), it does not produce an array of continuous and contiguous foreshortened squares, as Alberti maintains it does; • the ‘distance point’ of the viewer is irrelevant; and • the construction is less practically useful and less technically advanced than the method or methods used by Alberti’s great artist contemporaries - Ghiberti, Masaccio, Mantegna, Donatello and Piero della Francesca - rather than the reverse. This paper maintains that Alberti does not deserve his reputation as the originator of perspective theory and that the Western academic community stands condemned for allowing this situation to occur and to continue. Keywords: Alberti, perspective, quattrocento, painting, writers, academics It is generally agreed that Leon Battista Alberti wrote the Latin version of his treatise on painting, De Pictura, in 1435, not long after his arrival in Florence for his second and most extended period of residence there (1434-46). -
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. -
On a Human Scale. Drawing and Proportion of the Vitruvian Figure Veronica Riavis
7 / 2020 On a Human Scale. Drawing and Proportion of the Vitruvian Figure Veronica Riavis Abstract Among the images that describe the proportions of the human body, Leonardo da Vinci’s one is certainly the most effective, despite the fact that the iconic drawing does not faithfully follow the measurements indicated by Vitruvius. This research concerned the geometric analysis of the interpretations of the Vitruvian man proposed in the Renaissance editions of De Architectura, carried out after the aniconic editio princeps by Sulpicio da Veroli. Giovanni Battista da Sangallo drew the Vitruvian figure directly on his Sulpician copy, very similar to the images by Albrecht Dürer in The Symmetry of the Human Bodies [Dürer 1591]. Fra Giocondo proposes in 1511 two engravings of homo ad quadratum and ad circulum in the first Latin illustrated edition of De Architectura, while the man by Cesare Cesariano, author of the first version in vernacular of 1521, has a deformed body extension to adapt a geometric grid. Francesco di Giorgio Martini and Giacomo Andrea da Ferrara also propose significant versions believed to be the origin of Leonardo’s figuration due to the friendship that bound them. The man inscribed in the circle and square in the partial translation of Francesco di Giorgio’s De Architectura anticipates the da Vinci’s solution although it does not have explicit metric references, while the drawing by Giacomo Andrea da Ferrara reproduces a figure similar to Leonardo’s one. The comparison between the measures expressed by Vitruvius to proportion the man and the various graphic descriptions allows us to understand the complex story of the exegesis of the Roman treatise. -
Il Movimento Illusorio, L'illusione Del Movimento
Il movimento illusorio, l’illusione del movimento : sull’impossibilità della resa pittorica del moto negli scritti di Leon Battista Alberti e Leonardo da Vinci Elena Paroli To cite this version: Elena Paroli. Il movimento illusorio, l’illusione del movimento : sull’impossibilità della resa pittorica del moto negli scritti di Leon Battista Alberti e Leonardo da Vinci. Italies, Centre aixois d’études romanes, 2021, 2020 (24), pp.17-29. halshs-03151603 HAL Id: halshs-03151603 https://halshs.archives-ouvertes.fr/halshs-03151603 Submitted on 7 Mar 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License Il movimento illusorio, l’illusione del movimento Sull’impossibilità della resa pittorica del moto negli scritti di Leon Battista Alberti e Leonardo da Vinci Elena Paroli ENS Lyon, France Résumé : L’article porte sur la dimension illusoire de la représentation picturale du mouvement au sein de la théorie artistique de la Renaissance, notamment par le biais de l’analyse du De pictura de Leon Battista Alberti et du Traité de la peinture de Léonard de Vinci. Si, d’une part, les théories d’Alberti se trouvent scindées entre la nécessité de la mise en mouvement de la istoria picturale et la recherche d’une harmonie esthétique figée, d’autre part, le système imitatif conçu par le dernier Léonard constitue une véritable renonciation à la représentation naturaliste du mouvement.