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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. -
Cosmic Architecture Kozmička Arhitektura
je značio nešto poput ‘univerzuma, reda i ornamenta’. [...] Za stare Grke riječ ‘Kosmos’ stavljena je u opreku s riječi ‘Chaos’. Kaos je prethodio nastanku svijeta kakvog poznajemo, ali ga Kozmička je naslijedio Kozmos koji je simbolizirao apsolutni red svi- jeta i ukupnost njegovih prirodnih fenomena. [...] Stari grčki ‘Kosmeo’ znači ‘rasporediti, urediti i ukrasiti’, a osoba Kosmése arhitektura (ukrašava) sebe kako bi svoj Kozmos učinila vidljivim.2 kako ¶ Ornament ima gramatiku. Ornament bi trebao posje- dovati prikladnost, proporcije, sklad čiji je rezultat mir... onaj mir koji um osjeća kada su oko, intelekt i naklonosti zadovo- Cosmic ljeni.3 ¶ Vjerujem, kao što sam rekao, da se može projektirati izvrsna i lijepa zgrada koju neće krasiti nikakvi ornamenti; ali jednako čvrsto vjerujem da se ukrašenu građevinu, skladno zamišljenu, dobro promišljenu, ne može lišiti njezinog sustava architecture ornamenata, a da se ne uništi njezina individualnost.4 ¶ Tipičan postupak drevne arhitekture je dodavanje idealnih aspekata ili idealnih struktura površini zgrade. [...] Cijepanje ili klizanje stvarne površine zida u izražajnu površinu je čin transfor- macije. ¶ Govori li nepravilna evolucija kamena o nevjerojat- noj gotičkoj priči o ljudskom životu? Ili je to usputna pojava nevažnih činjenica iscrpljenih kamenoloma i klesara? Ili je to pustolovina vremena? 5 gdje ¶ Ornament je svjesna zanatska intervencija u proi- napisao fotografije Arhiva / Archive Alberto Alessi (aaa) zvodnji polugotovih proizvoda, prije nego što budu montirani written by photographs by Arhiva / Archive Alinari (aa) na gradilištu. Ornament stvara sidrenu točku protiv homo- Ruskin Library, University of Lancaster (rl) genizacije i uniformnosti suvremene građevinske produkcije. Ornamentacija omogućuje izravan odgovor na lokalne uvjete proizvodnje, na geografske ili kulturne osobitosti. -
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. -
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. -
Optimization of the Manufacturing Process for Sheet Metal Panels Considering Shape, Tessellation and Structural Stability Optimi
Institute of Construction Informatics, Faculty of Civil Engineering Optimization of the manufacturing process for sheet metal panels considering shape, tessellation and structural stability Optimierung des Herstellungsprozesses von dünnwandigen, profilierten Aussenwandpaneelen by Fasih Mohiuddin Syed from Mysore, India A Master thesis submitted to the Institute of Construction Informatics, Faculty of Civil Engineering, University of Technology Dresden in partial fulfilment of the requirements for the degree of Master of Science Responsible Professor : Prof. Dr.-Ing. habil. Karsten Menzel Second Examiner : Prof. Dr.-Ing. Raimar Scherer Advisor : Dipl.-Ing. Johannes Frank Schüler Dresden, 14th April, 2020 Task Sheet II Task Sheet Declaration III Declaration I confirm that this assignment is my own work and that I have not sought or used the inadmissible help of third parties to produce this work. I have fully referenced and used inverted commas for all text directly quoted from a source. Any indirect quotations have been duly marked as such. This work has not yet been submitted to another examination institution – neither in Germany nor outside Germany – neither in the same nor in a similar way and has not yet been published. Dresden, Place, Date (Signature) Acknowledgement IV Acknowledgement First, I would like to express my sincere gratitude to Prof. Dr.-Ing. habil. Karsten Menzel, Chair of the "Institute of Construction Informatics" for giving me this opportunity to work on my master thesis and believing in my work. I am very grateful to Dipl.-Ing. Johannes Frank Schüler for his encouragement, guidance and support during the course of work. I would like to thank all the staff from "Institute of Construction Informatics " for their valuable support. -
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]. -
Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture
REPORTS 21. Materials and methods are available as supporting 27. N. Panagia et al., Astrophys. J. 459, L17 (1996). Supporting Online Material material on Science Online. 28. The authors would like to thank L. Nelson for providing www.sciencemag.org/cgi/content/full/315/5815/1103/DC1 22. A. Heger, N. Langer, Astron. Astrophys. 334, 210 (1998). access to the Bishop/Sherbrooke Beowulf cluster (Elix3) Materials and Methods 23. A. P. Crotts, S. R. Heathcote, Nature 350, 683 (1991). which was used to perform the interacting winds SOM Text 24. J. Xu, A. Crotts, W. Kunkel, Astrophys. J. 451, 806 (1995). calculations. The binary merger calculations were Tables S1 and S2 25. B. Sugerman, A. Crotts, W. Kunkel, S. Heathcote, performed on the UK Astrophysical Fluids Facility. References S. Lawrence, Astrophys. J. 627, 888 (2005). T.M. acknowledges support from the Research Training Movies S1 and S2 26. N. Soker, Astrophys. J., in press; preprint available online Network “Gamma-Ray Bursts: An Enigma and a Tool” 16 October 2006; accepted 15 January 2007 (http://xxx.lanl.gov/abs/astro-ph/0610655) during part of this work. 10.1126/science.1136351 be drawn using the direct strapwork method Decagonal and Quasi-Crystalline (Fig. 1, A to D). However, an alternative geometric construction can generate the same pattern (Fig. 1E, right). At the intersections Tilings in Medieval Islamic Architecture between all pairs of line segments not within a 10/3 star, bisecting the larger 108° angle yields 1 2 Peter J. Lu * and Paul J. Steinhardt line segments (dotted red in the figure) that, when extended until they intersect, form three distinct The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in polygons: the decagon decorated with a 10/3 star medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, line pattern, an elongated hexagon decorated where the lines were drafted directly with a straightedge and a compass. -
Theorizing Ornament Estelle Thibault
From Herbal to Grammar : Theorizing Ornament Estelle Thibault To cite this version: Estelle Thibault. From Herbal to Grammar : Theorizing Ornament. Fourth International Conference of the European Architectural History Network, Jun 2016, Dublin, Ireland. pp.384-394. hal-01635839 HAL Id: hal-01635839 https://hal.archives-ouvertes.fr/hal-01635839 Submitted on 27 Oct 2019 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. EAHN Dublin 2016 1 PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE OF THE EUROPEAN ARCHITECTURAL HISTORY NETWORK Edited by Kathleen James-Chakraborty EAHN Dublin 2016 2 Published by UCD School of Art History and Cultural Policy University College Dublin, Belfield, Dublin, Ireland. Copyright © UCD School of Art History and Cultural Policy No images in this publication may be reproduced without permission of the copyright holder. ISBN 978-1-5262-0376-2 EAHN Dublin 2016 3 * Indicates full paper included Table of Contents KEYNOTE .................................................................................................................................... -
Reflection Tessellation Is Created, It Is Important to Review a Few Key Mathematical Concepts
1 Before we can understand how a glide reflection tessellation is created, it is important to review a few key mathematical concepts. The first is reflection: this is when an object is flipped across a line—called the line of reflection—without changing its shape or size. By reflecting the object across this line, the original shape is transformed into a mirror image of itself. The reflected shape is also the same distance away from the line of reflection. We see reflected shapes in our everyday lives. For example, reflections in smooth water. See how the bodies of the birds in the photograph above look as if they have been “flipped” upside down in the surface of the water? We can also reflect an object multiple times, over several lines. In the diagram to the right, notice how triangle A is flipped across line of reflection 1 to produce shape B. Then, triangle B—that is, the center green triangle—is flipped across a second line of reflection, resulting in shape C. 2 A glide reflection is a two-fold operation: a “flip” and then a “slide.” The first step is a reflection: as just discussed, the shape is flipped across a line of reflection. The second step is a translation: the shape is “slid” parallel to the line of reflection. In the diagram to the right, you can see the glide reflection broken down into these two steps. However, it actually does not matter which operation—the reflection or the translation—is performed first: translating the shape and then reflecting it will produce the same result, as long as the distance “slid” and the line of reflection are the same. -
Ornamental Design Is… a Kind of Practical Science’ Theories of Ornament at the London School of Design and Department of Science and Art1
‘Ornamental design is… a kind of practical science’ Theories of ornament at the London School of Design and Department of Science and Art1 Elena Chestnova ‘When Semper came to England he fell in with the wrong people’ – wrote Joseph Rykwert in his commentary to Semper’s first public lecture delivered at the Metropolitan School of Design in Marlborough House.2 The ‘wrong people’ implied those associated with the Department of Science and Art: 3 Henry Cole, Richard Redgrave, the closely connected Owen Jones and Matthew Digby Wyatt, William Dyce, Ralph Wornum and other instructors linked with the controversial Design Reform of the early 1850s. Semper had been employed at the school between 1852 and 1855, and while his time in London generally is considered to have been pivotal for his theoretical work, the impact of his associations at the school has been appraised negatively and the importance of his intellectual links with the Cole circle has been downplayed.4 This approach, however, ignores a host of significant issues that formed the context for the development of Semper’s theory of design as well as 1 This article is an outcome of the workshop ‘Gottfried Semper in London 1850–55’ held at the Accademia di architettura, Università della Svizzera italiana (USI), Mendrisio, on February 26, 2014, as part of the research and edition project ‘Architecture and the Globalization of Knowledge in the 19th Century: Gottfried Semper and the Discipline of Architectural History’ headed by Sonja Hildebrand (USI, responsible) and Philip Ursprung (ETH Zurich, co-responsible) with the support of the Swiss National Science Foundation (SNSF).