Golden Ratio: a Subtle Regulator in Our Body and Cardiovascular System?
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Linguistic Presentation of Objects
Zygmunt Ryznar dr emeritus Cracow Poland [email protected] Linguistic presentation of objects (types,structures,relations) Abstract Paper presents phrases for object specification in terms of structure, relations and dynamics. An applied notation provides a very concise (less words more contents) description of object. Keywords object relations, role, object type, object dynamics, geometric presentation Introduction This paper is based on OSL (Object Specification Language) [3] dedicated to present the various structures of objects, their relations and dynamics (events, actions and processes). Jay W.Forrester author of fundamental work "Industrial Dynamics”[4] considers the importance of relations: “the structure of interconnections and the interactions are often far more important than the parts of system”. Notation <!...> comment < > container of (phrase,name...) ≡> link to something external (outside of definition)) <def > </def> start-end of definition <spec> </spec> start-end of specification <beg> <end> start-end of section [..] or {..} 1 list of assigned keywords :[ or :{ structure ^<name> optional item (..) list of items xxxx(..) name of list = value assignment @ mark of special attribute,feature,property @dark unknown, to obtain, to discover :: belongs to : equivalent name (e.g.shortname) # number of |name| executive/operational object ppppXxxx name of item Xxxx with prefix ‘pppp ‘ XXXX basic object UUUU.xxxx xxxx object belonged to UUUU object class & / conjunctions ‘and’ ‘or’ 1 If using Latex editor we suggest [..] brackets -
Theon of Alexandria and Hypatia
CREATIVE MATH. 12 (2003), 111 - 115 Theon of Alexandria and Hypatia Michael Lambrou Abstract. In this paper we present the story of the most famous ancient female math- ematician, Hypatia, and her father Theon of Alexandria. The mathematician and philosopher Hypatia flourished in Alexandria from the second part of the 4th century until her violent death incurred by a mob in 415. She was the daughter of Theon of Alexandria, a math- ematician and astronomer, who flourished in Alexandria during the second part of the fourth century. Information on Theon’s life is only brief, coming mainly from a note in the Suda (Suida’s Lexicon, written about 1000 AD) stating that he lived in Alexandria in the times of Theodosius I (who reigned AD 379-395) and taught at the Museum. He is, in fact, the Museum’s last attested member. Descriptions of two eclipses he observed in Alexandria included in his commentary to Ptolemy’s Mathematical Syntaxis (Almagest) and elsewhere have been dated as the eclipses that occurred in AD 364, which is consistent with Suda. Although originality in Theon’s works cannot be claimed, he was certainly immensely influential in the preservation, dissemination and editing of clas- sic texts of previous generations. Indeed, with the exception of Vaticanus Graecus 190 all surviving Greek manuscripts of Euclid’s Elements stem from Theon’s edition. A comparison to Vaticanus Graecus 190 reveals that Theon did not actually change the mathematical content of the Elements except in minor points, but rather re-wrote it in Koini and in a form more suitable for the students he taught (some manuscripts refer to Theon’s sinousiai). -
Alexandrea Ad Aegyptvm the Legacy of Multiculturalism in Antiquity
Alexandrea ad aegyptvm the legacy of multiculturalism in antiquity editors rogério sousa maria do céu fialho mona haggag nuno simões rodrigues Título: Alexandrea ad Aegyptum – The Legacy of Multiculturalism in Antiquity Coord.: Rogério Sousa, Maria do Céu Fialho, Mona Haggag e Nuno Simões Rodrigues Design gráfico: Helena Lobo Design | www.hldesign.pt Revisão: Paula Montes Leal Inês Nemésio Obra sujeita a revisão científica Comissão científica: Alberto Bernabé, Universidade Complutense de Madrid; André Chevitarese, Universidade Federal, Rio de Janeiro; Aurélio Pérez Jiménez, Universidade de Málaga; Carmen Leal Soares, Universidade de Coimbra; Fábio Souza Lessa, Universidade Federal, Rio de Janeiro; José Augusto Ramos, Universidade de Lisboa; José Luís Brandão, Universidade de Coimbra; Natália Bebiano Providência e Costa, Universidade de Coimbra; Richard McKirahan, Pomona College, Claremont Co-edição: CITCEM – Centro de Investigação Transdisciplinar «Cultura, Espaço e Memória» Via Panorâmica, s/n | 4150-564 Porto | www.citcem.org | [email protected] CECH – Centro de Estudos Clássicos e Humanísticos | Largo da Porta Férrea, Universidade de Coimbra Alexandria University | Cornice Avenue, Shabty, Alexandria Edições Afrontamento , Lda. | Rua Costa Cabral, 859 | 4200-225 Porto www.edicoesafrontamento.pt | [email protected] N.º edição: 1152 ISBN: 978-972-36-1336-0 (Edições Afrontamento) ISBN: 978-989-8351-25-8 (CITCEM) ISBN: 978-989-721-53-2 (CECH) Depósito legal: 366115/13 Impressão e acabamento: Rainho & Neves Lda. | Santa Maria da Feira [email protected] Distribuição: Companhia das Artes – Livros e Distribuição, Lda. [email protected] Este trabalho é financiado por Fundos Nacionais através da FCT – Fundação para a Ciência e Tecnologia no âmbito do projecto PEst-OE/HIS/UI4059/2011 HYPATIA AND THE IDIOSCYNCRASIES OF CHRISTIANITY IN EGYPT – A STUDY OF THE EVENTS OCCURRED AT EASTER 415 A.D. -
Thales of Miletus Sources and Interpretations Miletli Thales Kaynaklar Ve Yorumlar
Thales of Miletus Sources and Interpretations Miletli Thales Kaynaklar ve Yorumlar David Pierce October , Matematics Department Mimar Sinan Fine Arts University Istanbul http://mat.msgsu.edu.tr/~dpierce/ Preface Here are notes of what I have been able to find or figure out about Thales of Miletus. They may be useful for anybody interested in Thales. They are not an essay, though they may lead to one. I focus mainly on the ancient sources that we have, and on the mathematics of Thales. I began this work in preparation to give one of several - minute talks at the Thales Meeting (Thales Buluşması) at the ruins of Miletus, now Milet, September , . The talks were in Turkish; the audience were from the general popu- lation. I chose for my title “Thales as the originator of the concept of proof” (Kanıt kavramının öncüsü olarak Thales). An English draft is in an appendix. The Thales Meeting was arranged by the Tourism Research Society (Turizm Araştırmaları Derneği, TURAD) and the office of the mayor of Didim. Part of Aydın province, the district of Didim encompasses the ancient cities of Priene and Miletus, along with the temple of Didyma. The temple was linked to Miletus, and Herodotus refers to it under the name of the family of priests, the Branchidae. I first visited Priene, Didyma, and Miletus in , when teaching at the Nesin Mathematics Village in Şirince, Selçuk, İzmir. The district of Selçuk contains also the ruins of Eph- esus, home town of Heraclitus. In , I drafted my Miletus talk in the Math Village. Since then, I have edited and added to these notes. -
The Golden Ratio and the Diagonal of the Square
Bridges Finland Conference Proceedings The Golden Ratio and the Diagonal of the Square Gabriele Gelatti Genoa, Italy [email protected] www.mosaicidiciottoli.it Abstract An elegant geometric 4-step construction of the Golden Ratio from the diagonals of the square has inspired the pattern for an artwork applying a general property of nested rotated squares to the Golden Ratio. A 4-step Construction of the Golden Ratio from the Diagonals of the Square For convenience, we work with the reciprocal of the Golden Ratio that we define as: φ = √(5/4) – (1/2). Let ABCD be a unit square, O being the intersection of its diagonals. We obtain O' by symmetry, reflecting O on the line segment CD. Let C' be the point on BD such that |C'D| = |CD|. We now consider the circle centred at O' and having radius |C'O'|. Let C" denote the intersection of this circle with the line segment AD. We claim that C" cuts AD in the Golden Ratio. B C' C' O O' O' A C'' C'' E Figure 1: Construction of φ from the diagonals of the square and demonstration. Demonstration In Figure 1 since |CD| = 1, we have |C'D| = 1 and |O'D| = √(1/2). By the Pythagorean Theorem: |C'O'| = √(3/2) = |C''O'|, and |O'E| = 1/2 = |ED|, so that |DC''| = √(5/4) – (1/2) = φ. Golden Ratio Pattern from the Diagonals of Nested Squares The construction of the Golden Ratio from the diagonal of the square has inspired the research of a pattern of squares where the Golden Ratio is generated only by the diagonals. -
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. -
Linguistic Presentation of Object Structure and Relations in OSL Language
Zygmunt Ryznar Linguistic presentation of object structure and relations in OSL language Abstract OSL© is a markup descriptive language for the description of any object in terms of structure and behavior. In this paper we present a geometric approach, the kernel and subsets for IT system, business and human-being as a proof of universality. This language was invented as a tool for a free structured modeling and design. Keywords free structured modeling and design, OSL, object specification language, human descriptive language, factual markup, geometric view, object specification, GSO -geometrically structured objects. I. INTRODUCTION Generally, a free structured modeling is a collection of methods, techniques and tools for modeling the variable structure as an opposite to the widely used “well-structured” approach focused on top-down hierarchical decomposition. It is assumed that system boundaries are not finally defined, because the system is never to be completed and components are are ready to be modified at any time. OSL is dedicated to present the various structures of objects, their relations and dynamics (events, actions and processes). This language may be counted among markup languages. The fundamentals of markup languages were created in SGML (Standard Generalized Markup Language) [6] descended from GML (name comes from first letters of surnames of authors: Goldfarb, Mosher, Lorie, later renamed as Generalized Markup Language) developed at IBM [15]. Markups are usually focused on: - documents (tags: title, abstract, keywords, menu, footnote etc.) e.g. latex [14], - objects (e.g. OSL), - internet pages (tags used in html, xml – based on SGML [8] rules), - data (e.g. structured data markup: microdata, microformat, RFDa [13],YAML [12]). -
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 -
The Golden Ratio
Mathematical Puzzle Sessions Cornell University, Spring 2012 1 Φ: The Golden Ratio p 1 + 5 The golden ratio is the number Φ = ≈ 1:618033989. (The greek letter Φ used to 2 represent this number is pronounced \fee".) Where does the number Φ come from? Suppose a line is broken into two pieces, one of length a and the other of length b (so the total length is a + b), and a and b are chosen in a very specific way: a and b are chosen so that the ratio of a + b to a and the ratio of a to b are equal. a b a + b a ! a b a+b a It turns out that if a and b satisfy this property so that a = b then the ratios are equal to the number Φ! It is called the golden ratio because among the ancient Greeks it was thought that this ratio is the most pleasing to the eye. a Try This! You can verify that if the two ratios are equal then b = Φ yourself with a bit of careful algebra. Let a = 1 and use the quadratic equation to find the value of b that makes 1 the two ratios equal. If you successfully worked out the value of b you should find b = Φ. The Golden Rectangle A rectangle is called a golden rectangle if the ratio of the sides of the rectangle is equal to Φ, like the one shown below. 1 Φ p 1 −1+ 5 If the ratio of the sides is Φ = 2 this is also considered a golden rectangle. -
Geometry and Art & Architecture
CHARACTERISTICS AND INTERRELATIONS OF FORMS ARCH 115 Instructor: F. Oya KUYUMCU Mathematics in Architecture • Mathematics can, broadly speaking, be subdivided into to study of quantity - arithmetic structure – algebra space – geometry and change – analysis. In addition to these main concerns, there are also sub-divisions dedicated to exploring links from the heart of mathematics to other field. to logic, to set theory (foundations) to the empirical mathematics of the various sciences to the rigorous study of uncertanity Lecture Topics . Mathematics and Architecture are wholly related with each other. All architectural products from the door to the building cover a place in space. So, all architectural elements have a volume. One of the study field of mathematics, name is geometry is totally concern this kind of various dimensional forms such as two or three or four. If you know geometry well, you can create true shapes and forms for architecture. For this reason, in this course, first we’ll look to the field of geometry especially two and three dimensions. Our second goal will be learning construction (drawing) of this two or three dimensional forms. • Architects intentionally or accidently use ratio and mathematical proportions to shape buildings. So, our third goal will be learning some mathematical concepts like ratio and proportions. • In this course, we explore the many places where the fields of art, architecture and mathematics. For this aim, we’ll try to learn concepts related to these fields like tetractys, magic squares, fractals, penrose tilling, cosmic rythms etc. • We will expose to you to wide range of art covering a long historical period and great variety of styles from the Egyptians up to contemporary world of art and architecture. -
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.