Penrose Tiles and Aperiodic Tessellations
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An Artistic and Mathematical Look at the Work of Maurits Cornelis Escher
University of Northern Iowa UNI ScholarWorks Honors Program Theses Honors Program 2016 Tessellations: An artistic and mathematical look at the work of Maurits Cornelis Escher Emily E. Bachmeier University of Northern Iowa Let us know how access to this document benefits ouy Copyright ©2016 Emily E. Bachmeier Follow this and additional works at: https://scholarworks.uni.edu/hpt Part of the Harmonic Analysis and Representation Commons Recommended Citation Bachmeier, Emily E., "Tessellations: An artistic and mathematical look at the work of Maurits Cornelis Escher" (2016). Honors Program Theses. 204. https://scholarworks.uni.edu/hpt/204 This Open Access Honors Program Thesis is brought to you for free and open access by the Honors Program at UNI ScholarWorks. It has been accepted for inclusion in Honors Program Theses by an authorized administrator of UNI ScholarWorks. For more information, please contact [email protected]. Running head: TESSELLATIONS: THE WORK OF MAURITS CORNELIS ESCHER TESSELLATIONS: AN ARTISTIC AND MATHEMATICAL LOOK AT THE WORK OF MAURITS CORNELIS ESCHER A Thesis Submitted in Partial Fulfillment of the Requirements for the Designation University Honors Emily E. Bachmeier University of Northern Iowa May 2016 TESSELLATIONS : THE WORK OF MAURITS CORNELIS ESCHER This Study by: Emily Bachmeier Entitled: Tessellations: An Artistic and Mathematical Look at the Work of Maurits Cornelis Escher has been approved as meeting the thesis or project requirements for the Designation University Honors. ___________ ______________________________________________________________ Date Dr. Catherine Miller, Honors Thesis Advisor, Math Department ___________ ______________________________________________________________ Date Dr. Jessica Moon, Director, University Honors Program TESSELLATIONS : THE WORK OF MAURITS CORNELIS ESCHER 1 Introduction I first became interested in tessellations when my fifth grade mathematics teacher placed multiple shapes that would tessellate at the front of the room and we were allowed to pick one to use to create a tessellation. -
Modelling of Virtual Compressed Structures Through Physical Simulation
MODELLING OF VIRTUAL COMPRESSED STRUCTURES THROUGH PHYSICAL SIMULATION P.Brivio1, G.Femia1, M.Macchi1, M.Lo Prete2, M.Tarini1;3 1Dipartimento di Informatica e Comunicazione, Universita` degli Studi dell’Insubria, Varese, Italy - [email protected] 2DiAP - Dipartimento di Architettura e Pianificazione, Politecnico Di Milano, Milano, Italy 3Visual Computing Group, Istituto Scienza e Tecnologie dell’Informazione, C.N.R., Pisa, Italy KEY WORDS: (according to ACM CCS): I.3.5 [Computer Graphics]: Physically based modeling I.3.8 [Computer Graphics]: Applications ABSTRACT This paper presents a simple specific software tool to aid architectural heuristic design of domes, coverings and other types of complex structures. The tool aims to support the architect during the initial phases of the project, when the structure form has yet to be defined, introducing a structural element very early into the morpho-genesis of the building shape (in contrast to traditional design practices, where the structural properties are taken into full consideration only much later in the design process). Specifically, the tool takes a 3D surface as input, representing a first approximation of the intended shape of a dome or a similar architectural structure, and starts by re-tessellating it to meet user’s need, according to a recipe selected in a small number of possibilities, reflecting different common architectural gridshell styles (e.g. with different orientations, con- nectivity values, with or without diagonal elements, etc). Alternatively, the application can import the gridshell structure verbatim, directly as defined by the connectivity of an input 3D mesh. In any case, at this point the 3D model represents the structure with a set of beams connecting junctions. -
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. -
Real-Time Rendering Techniques with Hardware Tessellation
Volume 34 (2015), Number x pp. 0–24 COMPUTER GRAPHICS forum Real-time Rendering Techniques with Hardware Tessellation M. Nießner1 and B. Keinert2 and M. Fisher1 and M. Stamminger2 and C. Loop3 and H. Schäfer2 1Stanford University 2University of Erlangen-Nuremberg 3Microsoft Research Abstract Graphics hardware has been progressively optimized to render more triangles with increasingly flexible shading. For highly detailed geometry, interactive applications restricted themselves to performing transforms on fixed geometry, since they could not incur the cost required to generate and transfer smooth or displaced geometry to the GPU at render time. As a result of recent advances in graphics hardware, in particular the GPU tessellation unit, complex geometry can now be generated on-the-fly within the GPU’s rendering pipeline. This has enabled the generation and displacement of smooth parametric surfaces in real-time applications. However, many well- established approaches in offline rendering are not directly transferable due to the limited tessellation patterns or the parallel execution model of the tessellation stage. In this survey, we provide an overview of recent work and challenges in this topic by summarizing, discussing, and comparing methods for the rendering of smooth and highly-detailed surfaces in real-time. 1. Introduction Hardware tessellation has attained widespread use in computer games for displaying highly-detailed, possibly an- Graphics hardware originated with the goal of efficiently imated, objects. In the animation industry, where displaced rendering geometric surfaces. GPUs achieve high perfor- subdivision surfaces are the typical modeling and rendering mance by using a pipeline where large components are per- primitive, hardware tessellation has also been identified as a formed independently and in parallel. -
ON a PROOF of the TREE CONJECTURE for TRIANGLE TILING BILLIARDS Olga Paris-Romaskevich
ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS Olga Paris-Romaskevich To cite this version: Olga Paris-Romaskevich. ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS. 2019. hal-02169195v1 HAL Id: hal-02169195 https://hal.archives-ouvertes.fr/hal-02169195v1 Preprint submitted on 30 Jun 2019 (v1), last revised 8 Oct 2019 (v3) 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. ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS. OLGA PARIS-ROMASKEVICH To Manya and Katya, to the moments we shared around mathematics on one winter day in Moscow. Abstract. Tiling billiards model a movement of light in heterogeneous medium consisting of homogeneous cells in which the coefficient of refraction between two cells is equal to −1. The dynamics of such billiards depends strongly on the form of an underlying tiling. In this work we consider periodic tilings by triangles (and cyclic quadrilaterals), and define natural foliations associated to tiling billiards in these tilings. By studying these foliations we manage to prove the Tree Conjecture for triangle tiling billiards that was stated in the work by Baird-Smith, Davis, Fromm and Iyer, as well as its generalization that we call Density property. -
Counting the Angels and Devils in Escher's Circle Limit IV
Journal of Humanistic Mathematics Volume 5 | Issue 2 July 2015 Counting the Angels and Devils in Escher's Circle Limit IV John Choi Nicholas Pippenger Harvey Mudd College Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Choi, J. and Pippenger, N. "Counting the Angels and Devils in Escher's Circle Limit IV," Journal of Humanistic Mathematics, Volume 5 Issue 2 (July 2015), pages 51-59. DOI: 10.5642/jhummath.201502.05 . Available at: https://scholarship.claremont.edu/jhm/vol5/iss2/5 ©2015 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Counting the Angels and Devils in Escher's Circle Limit IV Cover Page Footnote The research reported here was supported by Grant CCF 0646682 from the National Science Foundation. This work is available in Journal of Humanistic Mathematics: https://scholarship.claremont.edu/jhm/vol5/iss2/5 Counting the Angels and Devils in Escher's Circle Limit IV John Choi Goyang, Gyeonggi Province, Republic of Korea 412-724 [email protected] Nicholas Pippenger Department of Mathematics, Harvey Mudd College, Claremont CA, USA [email protected] Abstract We derive the rational generating function that enumerates the angels and devils in M. -
Tiles of the Plane: 6.891 Final Project
Tiles of the Plane: 6.891 Final Project Luis Blackaller Kyle Buza Justin Mazzola Paluska May 14, 2007 1 Introduction Our 6.891 project explores ways of generating shapes that tile the plane. A plane tiling is a covering of the plane without holes or overlaps. Figure 1 shows a common tiling from a building in a small town in Mexico. Traditionally, mathematicians approach the tiling problem by starting with a known group of symme- tries that will generate a set of tiles that maintain with them. It turns out that all periodic 2D tilings must have a fundamental region determined by 2 translations as described by one of the 17 Crystallographic Symmetry Groups [1]. As shown in Figure 2, to generate a set of tiles, one merely needs to choose which of the 17 groups is needed and create the desired tiles using the symmetries in that group. On the other hand, there are aperiodic tilings, such as Penrose Tiles [2], that use different sets of generation rules, and a completely different approach that relies on heavy math like optimization theory and probability theory. We are interested in the inverse problem: given a set of shapes, we would like to see what kinds of tilings can we create. This problem is akin to what a layperson might encounter when tiling his bathroom — the set of tiles is fixed, but the ways in which the tiles can be combined is unconstrained. In general, this is an impossible problem: Robert Berger proved that the problem of deciding whether or not an arbitrary finite set of simply connected tiles admits a tiling of the plane is undecidable [3]. -
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]. -
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. -
Examples of the Golden Ratio You Can Find in Nature
Examples Of The Golden Ratio You Can Find In Nature It is often said that math contains the answers to most of universe’s questions. Math manifests itself everywhere. One such example is the Golden Ratio. This famous Fibonacci sequence has fascinated mathematicians, scientist and artists for many hundreds of years. The Golden Ratio manifests itself in many places across the universe, including right here on Earth, it is part of Earth’s nature and it is part of us. In mathematics, the Fibonacci sequence is the ordering of numbers in the following integer sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… and so on forever. Each number is the sum of the two numbers that precede it. The Fibonacci sequence is seen all around us. Let’s explore how your body and various items, like seashells and flowers, demonstrate the sequence in the real world. As you probably know by now, the Fibonacci sequence shows up in the most unexpected places. Here are some of them: 1. Flower petals number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55. For example, the lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy has often 34 or 55 petals, etc. 2. Faces Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. -
The Golden Relationships: an Exploration of Fibonacci Numbers and Phi
The Golden Relationships: An Exploration of Fibonacci Numbers and Phi Anthony Rayvon Watson Faculty Advisor: Paul Manos Duke University Biology Department April, 2017 This project was submitted in partial fulfillment of the requirements for the degree of Master of Arts in the Graduate Liberal Studies Program in the Graduate School of Duke University. Copyright by Anthony Rayvon Watson 2017 Abstract The Greek letter Ø (Phi), represents one of the most mysterious numbers (1.618…) known to humankind. Historical reverence for Ø led to the monikers “The Golden Number” or “The Devine Proportion”. This simple, yet enigmatic number, is inseparably linked to the recursive mathematical sequence that produces Fibonacci numbers. Fibonacci numbers have fascinated and perplexed scholars, scientists, and the general public since they were first identified by Leonardo Fibonacci in his seminal work Liber Abacci in 1202. These transcendent numbers which are inextricably bound to the Golden Number, seemingly touch every aspect of plant, animal, and human existence. The most puzzling aspect of these numbers resides in their universal nature and our inability to explain their pervasiveness. An understanding of these numbers is often clouded by those who seemingly find Fibonacci or Golden Number associations in everything that exists. Indeed, undeniable relationships do exist; however, some represent aspirant thinking from the observer’s perspective. My work explores a number of cases where these relationships appear to exist and offers scholarly sources that either support or refute the claims. By analyzing research relating to biology, art, architecture, and other contrasting subject areas, I paint a broad picture illustrating the extensive nature of these numbers.