Generating a Fractal Tree
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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. -
Joint Session: Measuring Fractals Diversity in Mathematics, 2019
Joint Session: Measuring Fractals Diversity in Mathematics, 2019 Tongou Yang The University of British Columbia, Vancouver [email protected] July 2019 Tongou Yang (UBC) Fractal Lectures July 2019 1 / 12 Let's Talk Geography Which country has the longest coastline? What is the longest river on earth? Tongou Yang (UBC) Fractal Lectures July 2019 2 / 12 (a) Map of Canada (b) Map of Nunavut Which country has the longest coastline? List of countries by length of coastline Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12 (b) Map of Nunavut Which country has the longest coastline? List of countries by length of coastline (a) Map of Canada Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12 Which country has the longest coastline? List of countries by length of coastline (a) Map of Canada (b) Map of Nunavut Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12 What's the Longest River on Earth? A Youtube Video Tongou Yang (UBC) Fractal Lectures July 2019 4 / 12 Measuring a Smooth Curve Example: use a rope Figure: Boundary between CA and US on the Great Lakes Tongou Yang (UBC) Fractal Lectures July 2019 5 / 12 Measuring a Rugged Coastline Figure: Coast of Nova Scotia Tongou Yang (UBC) Fractal Lectures July 2019 6 / 12 Covering by Grids 4 3 2 1 0 0 1 2 3 4 Tongou Yang (UBC) Fractal Lectures July 2019 7 / 12 Counting Grids Number of Grids=15 Side Length ≈ 125km Coastline ≈ 15 × 125 = 1845km. Tongou Yang (UBC) Fractal Lectures July 2019 8 / 12 Covering by Finer Grids 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 Tongou Yang (UBC) Fractal Lectures July 2019 9 / 12 Counting Grids Number of Grids=34 Side Length ≈ 62km Coastline ≈ 34 × 62 = 2108km. -
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 -
Grade 6 Math Circles Fractals Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 6 Math Circles March 10/11 2020 Fractals Introduction Fractals are mathematical objects that are self-similar, meaning that they have the same structure when you zoom in on them. For example, this is a famous fractal called the Mandelbrot set: Image by Arnaud Cheritat, retrieved from: https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/File:MilMand_1000iter.png 1 The main shape, which looks a bit like a snowman, is shaded in the image below: As an exercise, shade in some of the smaller snowmen that are all around the largest one in the image above. This is an example of self-similarity, and this pattern continues into infinity. Watch what happens when you zoom deeper and deeper into the Mandelbrot Set here: https://www.youtube.com/watch?v=pCpLWbHVNhk. Drawing Fractals Fractals like the Mandelbrot Set are generated using formulas and computers, but there are a number of simple fractals that we can draw ourselves. 2 Sierpinski Triangle Image by Beojan Stanislaus, retrieved from: https://commons.wikimedia.org/wiki/File:Sierpinski_triangle.svg Use the template and the steps below to draw this fractal: 1. Start with an equilateral triangle. 2. Connect the midpoints of all three sides. This will create one upside down triangle and three right-side up triangles. 3. Repeat from step 1 for each of the three right-side up triangles. 3 Apollonian Gasket Retrieved from: https://mathlesstraveled.com/2016/04/20/post-without-words-6/ Use the template and the steps below to draw this fractal: 1. -
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. -
Arm Streamline Tutorial Analyzing an Example Streamline Report with Arm
Arm Streamline Tutorial Revision: NA Analyzing an example Streamline report with Arm DS Non-Confidential Issue 0.0 Copyright © 2021 Arm Limited (or its affiliates). NA All rights reserved. Arm Streamline Tutorial Analyzing an example Streamline NA report with Arm DS Issue 0.0 Arm Streamline Tutorial Analyzing an example Streamline report with Arm DS Copyright © 2021 Arm Limited (or its affiliates). All rights reserved. Release information Document history Issue Date Confidentiality Change 0.0 30th of Non- First version March 2021 Confidential Confidential Proprietary Notice This document is CONFIDENTIAL and any use by you is subject to the terms of the agreement between you and Arm or the terms of the agreement between you and the party authorized by Arm to disclose this document to you. This document is protected by copyright and other related rights and the practice or implementation of the information contained in this document may be protected by one or more patents or pending patent applications. No part of this document may be reproduced in any form by any means without the express prior written permission of Arm. No license, express or implied, by estoppel or otherwise to any intellectual property rights is granted by this document unless specifically stated. Your access to the information in this document is conditional upon your acceptance that you will not use or permit others to use the information: (i) for the purposes of determining whether implementations infringe any third party patents; (ii) for developing technology or products which avoid any of Arm's intellectual property; or (iii) as a reference for modifying existing patents or patent applications or creating any continuation, continuation in part, or extension of existing patents or patent applications; or (iv) for generating data for publication or disclosure to third parties, which compares the performance or functionality of the Arm technology described in this document with any other products created by you or a third party, without obtaining Arm's prior written consent. -
FOR IMMEDIATE RELEASE August 18, 2015
FOR IMMEDIATE RELEASE August 18, 2015 MEDIA CONTACT Emily Kowalski | (919) 664-6795 | [email protected] North Carolina Museum of Art Presents M. C. Escher, Leonardo da Vinci Exhibitions and Related Events Raleigh, N.C.—The North Carolina Museum of Art (NCMA) presents two exhibitions opening in October 2015: The Worlds of M. C. Escher: Nature, Science, and Imagination and Leonardo da Vinci’s Codex Leicester and the Creative Mind. The Worlds of M. C. Escher features over 130 works (some never before exhibited) and will be the most comprehensive Escher exhibition ever presented in the United States. The Codex Leicester is a 500-year-old notebook handwritten and illustrated by inventor, scientist, and artist Leonardo da Vinci—the only manuscript by Leonardo in North America—that offers a glimpse into one of the greatest minds in history. “This is going to be an exciting fall at the Museum—an incredibly rare opportunity for our visitors to see not only centuries-old writings and sketches by Leonardo da Vinci, but also the work of M. C. Escher, another observer of nature and a perfect modern counterpart to Leonardo,” says NCMA Director Lawrence J. Wheeler. “These exhibitions will thrill art lovers and science lovers alike, and we hope that all visitors leave with a piqued curiosity, an ignited imagination, and a desire to more closely observe the world around them.” The Worlds of M. C. Escher: Nature, Science, and Imagination October 17, 2015−January 17, 2016 Comprising over 130 woodcuts, lithographs, wood engravings, and mezzotints, as well as numerous drawings, watercolors, wood blocks, and lithographic stones never before exhibited, The Worlds of M. -
Symmetry in Perspective
Symmetry in Perspective Stefan Carlsson Numerical Analysis and Computing Science, Royal Institute of Technology, (KTH), S-100 44 Stockholm, Sweden, [email protected], http://www.nada.kth.se/,-,stefanc Abstract. When a symmetric object in 3D is projected to an image, the symmetry properties are lost except for specific relative viewpoints. For sufficiently complex objects however, it is still possible to infer sym- metry in 3D from a single image of an uncalibrated camera. In this paper we give a general computational procedure for computing 3D structure and finding image symmetry constraints from single view projections of symmetric 3D objects. For uncalibrated cameras these constraints take the form of polynomials in brackets (determinants) and by using pro- jectively invariant shape constraints relating 3D and image structure, the symmetry constraints can be derived easily by considering the ef- fects of the corresponding symmetry transformation group in 3D. The constraints can also be given a useful geometric interpretation in terms of projective incidence using the Grassmann-Cayley algebra formalism. We demonstrate that in specific situations geometric intuition and the use of GC-algebra is a very simple and effective tool for finding image symmetry constraints. 1 Introduction The detection and analysis of symmetry is a problem that has many aspects and levels of complexity depending on e.g. the geometry of the imaging process and the shape of the objects being imaged. An object that is symmetric in 3D will only give rise to a symmetric image for specific viewing positions. It is however in general still possible to decide whether an image is a view of a symmetric object. -
Working with Fractals a Resource for Practitioners of Biophilic Design
WORKING WITH FRACTALS A RESOURCE FOR PRACTITIONERS OF BIOPHILIC DESIGN A PROJECT OF THE EUROPEAN ‘COST RESTORE ACTION’ PREPARED BY RITA TROMBIN IN COLLABORATION WITH ACKNOWLEDGEMENTS This toolkit is the result of a joint effort between Terrapin Bright Green, Cost RESTORE Action (REthinking Sustainability TOwards a Regenerative Economy), EURAC Research, International Living Future Institute, Living Future Europe and many other partners, including industry professionals and academics. AUTHOR Rita Trombin SUPERVISOR & EDITOR Catherine O. Ryan CONTRIBUTORS Belal Abboushi, Pacific Northwest National Laboratory (PNNL) Luca Baraldo, COOKFOX Architects DCP Bethany Borel, COOKFOX Architects DCP Bill Browning, Terrapin Bright Green Judith Heerwagen, University of Washington Giammarco Nalin, Goethe Universität Kari Pei, Interface Nikos Salingaros, University of Texas at San Antonio Catherine Stolarski, Catherine Stolarski Design Richard Taylor, University of Oregon Dakota Walker, Terrapin Bright Green Emily Winer, International Well Building Institute CITATION Rita Trombin, ‘Working with fractals: a resource companion for practitioners of biophilic design’. Report, Terrapin Bright Green: New York, 31 December 2020. Revised June 2021 © 2020 Terrapin Bright Green LLC For inquiries: [email protected] or [email protected] COVER DESIGN: Catherine O. Ryan COVER IMAGES: Ice crystals (snow-603675) by Quartzla from Pixabay; fractal gasket snowflake by Catherine Stolarski Design. 2 Working with Fractals: A Toolkit © 2020 Terrapin Bright Green LLC -
Exploring Fractal Dimension
MATH2916 A Working Seminar University of Sydney School of Mathematics and Statistics Exploring Fractal Dimension Dibyendu Roy SID:480344308 June 2, 2019 MATH2916 Exploring Fractal Dimension Dibyendu Roy 1 Motivations We all have an intuitive understanding of roughness, and can easily tell that a jagged rock is rougher than a marble. We can similarly understand that the Koch curve is "rougher" than a straight line. To make this definition of "roughness" or "fractal-ness" more rigorous, we introduce the idea of fractal dimension. What would such a dimension be defined as? It makes sense to attempt to define such a "dimen- sion" to fill the gaps in the standard integers that we use to define dimension conventionally. We know that an object in one dimension is a line and an example of a two dimensional object is a square. But what would an example of a 1.26 dimensional object look like? In this text we will explore the nature of fractal dimension culminating in a rigorous definition that can be applied to any object. 2 Compass dimension 2.1 The coastline paradox Let us imagine that we are are a cartographer trying to find the length of the coastline of Britain. As the coastline is not a regular mathematically defined shape it would make sense for us to try and find a series of approximations that would converge to the true value after a large number of iterations. Let us consider the following method for evaluating the perimeter of a shape. We take a compass (or more appropriately a divider) and open it up to a fixed distance s.