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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. -
Golden Ratio: a Subtle Regulator in Our Body and Cardiovascular System?
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/306051060 Golden Ratio: A subtle regulator in our body and cardiovascular system? Article in International journal of cardiology · August 2016 DOI: 10.1016/j.ijcard.2016.08.147 CITATIONS READS 8 266 3 authors, including: Selcuk Ozturk Ertan Yetkin Ankara University Istinye University, LIV Hospital 56 PUBLICATIONS 121 CITATIONS 227 PUBLICATIONS 3,259 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: microbiology View project golden ratio View project All content following this page was uploaded by Ertan Yetkin on 23 August 2019. The user has requested enhancement of the downloaded file. International Journal of Cardiology 223 (2016) 143–145 Contents lists available at ScienceDirect International Journal of Cardiology journal homepage: www.elsevier.com/locate/ijcard Review Golden ratio: A subtle regulator in our body and cardiovascular system? Selcuk Ozturk a, Kenan Yalta b, Ertan Yetkin c,⁎ a Abant Izzet Baysal University, Faculty of Medicine, Department of Cardiology, Bolu, Turkey b Trakya University, Faculty of Medicine, Department of Cardiology, Edirne, Turkey c Yenisehir Hospital, Division of Cardiology, Mersin, Turkey article info abstract Article history: Golden ratio, which is an irrational number and also named as the Greek letter Phi (φ), is defined as the ratio be- Received 13 July 2016 tween two lines of unequal length, where the ratio of the lengths of the shorter to the longer is the same as the Accepted 7 August 2016 ratio between the lengths of the longer and the sum of the lengths. -
Simple Rules for Incorporating Design Art Into Penrose and Fractal Tiles
Bridges 2012: Mathematics, Music, Art, Architecture, Culture Simple Rules for Incorporating Design Art into Penrose and Fractal Tiles San Le SLFFEA.com [email protected] Abstract Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable M.C. Escher-like image that works esthetically as well as functionally requires resolving incongruencies at a tile’s edge while constrained by its shape. Escher was the most well known practitioner in this style of mathematical visualization, but there are significant mathematical objects to which he never applied his artistry including Penrose Tilings and fractals. In this paper, we show that the rules of creating a traditional tile extend to these objects as well. To illustrate the versatility of tiling art, images were created with multiple figures and negative space leading to patterns distinct from the work of others. 1 1 Introduction M.C. Escher was the most prominent artist working with tessellations and space filling. Forty years after his death, his creations are still foremost in people’s minds in the field of tiling art. One of the reasons Escher continues to hold such a monopoly in this specialty are the unique challenges that come with creating Escher type designs inside a tessellation[15]. When an image is drawn into a tile and extends to the tile’s edge, it introduces incongruencies which are resolved by continuously aligning and refining the image. This is particularly true when the image consists of the lizards, fish, angels, etc. which populated Escher’s tilings because they do not have the 4-fold rotational symmetry that would make it possible to arbitrarily rotate the image ± 90, 180 degrees and have all the pieces fit[9]. -
Leonardo and the Whale
Biology Faculty Publications Biology 6-17-2019 Leonardo and the Whale Kay Etheridge Gettysburg College Follow this and additional works at: https://cupola.gettysburg.edu/biofac Part of the Ancient, Medieval, Renaissance and Baroque Art and Architecture Commons, Biology Commons, Ecology and Evolutionary Biology Commons, and the Marine Biology Commons Share feedback about the accessibility of this item. Recommended Citation Etheridge, Kay. "Leonardo and the Whale." In Leonardo da Vinci – Nature and Architecture, edited by C. Moffat and S.Taglialagamba, 89-106. Leiden: Brill, 2019. This is the author's version of the work. This publication appears in Gettysburg College's institutional repository by permission of the copyright owner for personal use, not for redistribution. Cupola permanent link: https://cupola.gettysburg.edu/biofac/81 This open access book chapter is brought to you by The Cupola: Scholarship at Gettysburg College. It has been accepted for inclusion by an authorized administrator of The Cupola. For more information, please contact [email protected]. Leonardo and the Whale Abstract Around 1480, when he was 28 years old, Leonardo da Vinci recorded what may have been a seminal event in his life. In writing of his travels to view nature he recounted an experience in a cave in the Tuscan countryside: Having wandered for some distance among overhanging rocks, I can to the entrance of a great cavern... [and after some hesitation I entered] drawn by a desire to see whether there might be any marvelous thing within..." [excerpt] Keywords Leonardo da Vinci, fossils, whale Disciplines Ancient, Medieval, Renaissance and Baroque Art and Architecture | Biology | Ecology and Evolutionary Biology | Marine Biology Comments Please note that that this is pre-print version of the article and has not yet been peer-reviewed. -
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. -
Works Issued As Multipart Monographs Works Issued
LC-PCC: Policy Statements for Chapter 6: Identifying Works and Expressions Policy Statements for Chapter 6: Identifying Works and Expressions LC-PCC PS FOR 6.1.3.1 WORKS ISSUED AS MULTIPART MONOGRAPHS LC practice/PCC practice: If a work embodied in a multipart monograph is identified by a creator based on the first or earliest volume received, and subsequent volumes indicate additional creators are involved, do not change the authorized access point for the work, but record additional creators when considered important. [2013-01] LC-PCC PS FOR 6.1.3.2 WORKS ISSUED AS SERIALS Expressions When Preferred Title of Work Changes LC practice/PCC practice: When there are different language expressions of a serial work and the preferred title of the work (as determined according to RDA 6.2.2 ) changes, create a new description for each different expression of that work even if the title proper of the manifestation of the specific language expression did not change. 130 0# $a Inzhenernyĭ zhurnal. Mekhanika tverdogo tela. $l English. 245 10 $a Mechanics of solids. 785 00 $a $t Izvestiíà. Mekhanika tverdogo tela. English. Mechanics of solids 130 0# $a Izvestiíà. Mekhanika tverdogo tela. $l English. 245 10 $a Mechanics of solids. 780 00 $t Inzhenernyĭ zhurnal. Mekhanika tverdogo tela. English. Mechanics of solids "Mechanics of solids" is the title proper of an English-language expression of a work in Russian. Although the English title proper did not change, a new description is necessary because the preferred title of the work in Russian changed. LC Policy & Standards Division and PCC Standing Committee on Standards LCPCC6-1 LC-PCC: Policy Statements for Chapter 6: Identifying Works and Expressions Subseries and the Omission/Addition of Main Series PCC practice: When either of the situations below occurs, create a new series authority record (SAR) and link the two authorized access points via MARC 5XX fields . -
The Mathematics of Art: an Untold Story Project Script Maria Deamude Math 2590 Nov 9, 2010
The Mathematics of Art: An Untold Story Project Script Maria Deamude Math 2590 Nov 9, 2010 Art is a creative venue that can, and has, been used for many purposes throughout time. Religion, politics, propaganda and self-expression have all used art as a vehicle for conveying various messages, without many references to the technical aspects. Art and mathematics are two terms that are not always thought of as being interconnected. Yet they most definitely are; for art is a highly technical process. Following the histories of maths and sciences – if not instigating them – art practices and techniques have developed and evolved since man has been on earth. Many mathematical developments occurred during the Italian and Northern Renaissances. I will describe some of the math involved in art-making, most specifically in architectural and painting practices. Through the medieval era and into the Renaissance, 1100AD through to 1600AD, certain significant mathematical theories used to enhance aesthetics developed. Understandings of line, and placement and scale of shapes upon a flat surface developed to the point of creating illusions of reality and real, three dimensional space. One can look at medieval frescos and altarpieces and witness a very specific flatness which does not create an illusion of real space. Frescos are like murals where paint and plaster have been mixed upon a wall to create the image – Michelangelo’s work in the Sistine Chapel and Leonardo’s The Last Supper are both famous examples of frescos. The beginning of the creation of the appearance of real space can be seen in Giotto’s frescos in the late 1200s and early 1300s. -
Leonardo Da Vinci's Study of Light and Optics: a Synthesis of Fields in The
Bitler Leonardo da Vinci’s Study of Light and Optics Leonardo da Vinci’s Study of Light and Optics: A Synthesis of Fields in The Last Supper Nicole Bitler Stanford University Leonardo da Vinci’s Milanese observations of optics and astronomy complicated his understanding of light. Though these complications forced him to reject “tidy” interpretations of light and optics, they ultimately allowed him to excel in the portrayal of reflection, shadow, and luminescence (Kemp, 2006). Leonardo da Vinci’s The Last Supper demonstrates this careful study of light and the relation of light to perspective. In the work, da Vinci delved into the complications of optics and reflections, and its renown guided the artistic study of light by subsequent masters. From da Vinci’s personal manuscripts, accounts from his contemporaries, and present-day art historians, the iterative relationship between Leonardo da Vinci’s study of light and study of optics becomes apparent, as well as how his study of the two fields manifested in his paintings. Upon commencement of courtly service in Milan, da Vinci immersed himself in a range of scholarly pursuits. Da Vinci’s artistic and mathematical interest in perspective led him to the study of optics. Initially, da Vinci “accepted the ancient (and specifically Platonic) idea that the eye functioned by emitting a special type of visual ray” (Kemp, 2006, p. 114). In his early musings on the topic, da Vinci reiterated this notion, stating in his notebooks that, “the eye transmits through the atmosphere its own image to all the objects that are in front of it and receives them into itself” (Suh, 2005, p. -
Computer Vision Today Introductions Introductions Computer Vision Why
Today • Course overview • Requirements, logistics Computer Vision • Image formation Thursday, August 30 Introductions Introductions • Instructor: Prof. Kristen Grauman grauman @ cs TAY 4.118, Thurs 2-4 pm • TA: Sudheendra Vijayanarasimhan svnaras @ cs ENS 31 NQ, Mon/Wed 1-2 pm • Class page: Check for updates to schedule, assignments, etc. http://www.cs.utexas.edu/~grauman/courses/378/main.htm Computer vision Why vision? • Automatic understanding of images and • As image sources multiply, so do video applications • Computing properties of the 3D world from – Relieve humans of boring, easy tasks visual data – Enhance human abilities • Algorithms and representations to allow a – Advance human-computer interaction, machine to recognize objects, people, visualization scenes, and activities. – Perception for robotics / autonomous agents • Possible insights into human vision 1 Some applications Some applications Navigation, driver safety Autonomous robots Assistive technology Surveillance Factory – inspection Monitoring for safety (Cognex) (Poseidon) Visualization License plate reading Visualization and tracking Medical Visual effects imaging (the Matrix) Some applications Why is vision difficult? • Ill-posed problem: real world much more complex than what we can measure in Situated search images Multi-modal interfaces –3D Æ 2D • Impossible to literally “invert” image formation process Image and video databases - CBIR Tracking, activity recognition Challenges: robustness Challenges: context and human experience Illumination Object pose Clutter -
Gerald Holton: Worlds Within Worlds by Barbara Delman Wolfson
NATIONAL ENDOWMENT FOR THE HUMANITIES • VOLUME 2 NUMBER 2 • APRIL 1981 Humanities Gerald Holton: Worlds within worlds by b a r b a r a d e l m a n w o l f s o n PROLOGUE: It is January 1934 in the city of Par working nature, of the style and life of the sci is. A husband and wife are at work in a university entist, and of the power of the human mind." laboratory. They are exposing a piece of ordinary alu Hundreds of thousands of students in this minum to a stream of tiny charged bits of matter country in secondary schools and colleges have called alpha particles. Stated so simply, this hardly used the course, now in the third edition since sounds like an important event. But look more Close its commercial publication in 1970, and millions ly, for it is important indeed. Later you will look at more around the world have used the materials the technical details, but for now they will not get in in French, Arabic, Japanese, Hebrew, Italian the way of the story. and other language adaptations. Although few The story is something of a family affair. The will ever become scientists, they will have a husband and wife are the French physicists Frederic chance to "see physics as the wonderfully Joliot and Irene Curie. The alpha particles they are many-sided human activity that it really is." using in their experiment are shooting from a piece of Throughout his career as physicist, histori naturally radioactive metal. This metal is polonium, an, editor, and educator, Holton has been a lu first identified 36 years before by Irene's parents, cid interpreter of the complexity of the scientific Pierre and Marie Curie, the discoverers of radium. -
Johannes Gutenberg Zim:///A/Johannes Gutenberg.Html
People David Livingstone p2 Henry Morton Stanley p12 Johann Gutenberg p16 Leonardo da Vinci p24 http://cd3wd.com wikipedia-for-schools http://gutenberg.org page no: 1 of 41 David Livingstone zim:///A/David_Livingstone.html David Livingstone 2008/9 Schools Wikipedia Selection. Related subjects: British History 1750-1900; Geographers and explorers David Livingstone ( 19 March 1813 – 1 May 1873) was a British Congregationalist pioneer medical missionary with the London Missionary Society and explorer in central Africa. He was the first European David Livingstone to see Mosi-oa-Tunya (Victoria Falls), to which he gave the English name in honour of his monarch, Queen Victoria. He is the subject of the meeting with H. M. Stanley, which gave rise to the popular quotation, " Dr Livingstone, I presume?" Perhaps one of the most popular national heroes of the late-nineteenth century in Victorian Britain, Livingstone had a mythic status, which operated on a number of interconnected levels: that of Protestant missionary martyr, that of working-class "rags to riches" inspirational story, that of scientific investigator and explorer, that of imperial reformer, anti-slavery crusader and advocate of commercial empire. His fame as an explorer helped drive forward the obsession with discovering the sources of the Nile River that formed the culmination of the classic period of European geographical discovery and colonial penetration of the African continent. At the same time his missionary travels, "disappearance" and death in Africa, and subsequent glorification as posthumous national hero in 1874 led to the founding of several major central African Christian missionary initiatives carried forward in the era of the European "Scramble for Africa." Early life Born 19 March 1813 David Livingstone was born on March 19, 1813 in the mill town of Blantyre, Lanarkshire, Scotland, into Blantyre, United Kingdom a Protestant family believed to be descended from the highland Livingstones, a clan that had been Died 4 May 1873 (aged 60) previously known as the Clan MacLea. -
Linking Math with Art Through the Elements of Design
2007 Asilomar Mathematics Conference Linking Math With Art Through The Elements of Design Presented by Renée Goularte Thermalito Union School District • Oroville, California www.share2learn.com [email protected] The Elements of Design ~ An Overview The elements of design are the components which make up any visual design or work or art. They can be isolated and defined, but all works of visual art will contain most, if not all, the elements. Point: A point is essentially a dot. By definition, it has no height or width, but in art a point is a small, dot-like pencil mark or short brush stroke. Line: A line can be made by a series of points, a pencil or brush stroke, or can be implied by the edge of an object. Shape and Form: Shapes are defined by lines or edges. They can be geometric or organic, predictably regular or free-form. Form is an illusion of three- dimensionality given to a flat shape. Texture: Texture can be tactile or visual. Tactile texture is something you can feel; visual texture relies on the eyes to help the viewer imagine how something might feel. Texture is closely related to Pattern. Pattern: Patterns rely on repetition or organization of shapes, colors, or other qualities. The illusion of movement in a composition depends on placement of subject matter. Pattern is closely related to Texture and is not always included in a list of the elements of design. Color and Value: Color, also known as hue, is one of the most powerful elements. It can evoke emotion and mood, enhance texture, and help create the illusion of distance.