Looking at Islamic Patterns I: the Perception of Order
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The Sasanian Tradition in ʽabbāsid Art: Squinch Fragmentation As The
The Sasanian Tradition in ʽAbbāsid Art: squinch fragmentation as The structural origin of the muqarnas La tradición sasánida en el arte ʿabbāssí: la fragmentación de la trompa de esquina como origen estructural de la decoración de muqarnas A tradição sassânida na arte abássida: a fragmentação do arco de canto como origem estrutural da decoração das Muqarnas Alicia CARRILLO1 Abstract: Islamic architecture presents a three-dimensional decoration system known as muqarnas. An original system created in the Near East between the second/eighth and the fourth/tenth centuries due to the fragmentation of the squinche, but it was in the fourth/eleventh century when it turned into a basic element, not only all along the Islamic territory but also in the Islamic vocabulary. However, the origin and shape of muqarnas has not been thoroughly considered by Historiography. This research tries to prove the importance of Sasanian Art in the aesthetics creation of muqarnas. Keywords: Islamic architecture – Tripartite squinches – Muqarnas –Sasanian – Middle Ages – ʽAbbāsid Caliphate. Resumen: La arquitectura islámica presenta un mecanismo de decoración tridimensional conocido como decoración de muqarnas. Un sistema novedoso creado en el Próximo Oriente entre los siglos II/VIII y IV/X a partir de la fragmentación de la trompa de esquina, y que en el siglo XI se extendió por toda la geografía del Islam para formar parte del vocabulario del arte islámico. A pesar de su importancia y amplio desarrollo, la historiografía no se ha detenido especialmente en el origen formal de la decoración de muqarnas y por ello, este estudio pone de manifiesto la influencia del arte sasánida en su concepción estética durante el Califato ʿabbāssí. -
Islamic Geometric Patterns Jay Bonner
Islamic Geometric Patterns Jay Bonner Islamic Geometric Patterns Their Historical Development and Traditional Methods of Construction with a chapter on the use of computer algorithms to generate Islamic geometric patterns by Craig Kaplan Jay Bonner Bonner Design Consultancy Santa Fe, New Mexico, USA With contributions by Craig Kaplan University of Waterloo Waterloo, Ontario, Canada ISBN 978-1-4419-0216-0 ISBN 978-1-4419-0217-7 (eBook) DOI 10.1007/978-1-4419-0217-7 Library of Congress Control Number: 2017936979 # Jay Bonner 2017 Chapter 4 is published with kind permission of # Craig Kaplan 2017. All Rights Reserved. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. -
Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
Geometric Algorithms for Optimal Airspace Design and Air Traffic
Geometric Algorithms for Optimal Airspace Design and Air Traffic Controller Workload Balancing AMITABH BASU Carnegie Mellon University JOSEPH S. B. MITCHELL Stony Brook University and GIRISHKUMAR SABHNANI Stony Brook University The National Airspace System (NAS) is designed to accommodate a large number of flights over North America. For purposes of workload limitations for air traffic controllers, the airspace is partitioned into approximately 600 sectors; each sector is observed by one or more controllers. In order to satisfy workload limitations for controllers, it is important that sectors be designed carefully according to the traffic patterns of flights, so that no sector becomes overloaded. We formulate and study the airspace sectorization problem from an algorithmic point of view, model- ing the problem of optimal sectorization as a geometric partition problem with constraints. The novelty of the problem is that it partitions data consisting of trajectories of moving points, rather than static point set partitioning that is commonly studied. First, we formulate and solve the 1D version of the problem, showing how to partition a line into “sectors” (intervals) according to historical trajectory data. Then, we apply the 1D solution framework to design a 2D sectoriza- tion heuristic based on binary space partitions. We also devise partitions based on balanced “pie partitions” of a convex polygon. We evaluate our 2D algorithms experimentally, applying our algorithms to actual historical flight track data for the NAS. We compare the workload balance of our methods to that of the existing set of sectors for the NAS and find that our resectorization yields competitive and improved workload balancing. -
Coloring Copoints of a Planar Point Set
Coloring Copoints of a Planar Point Set Walter Morris Department of Mathematical Sciences George Mason University 4400 University Drive, Fairfax, VA 22030, USA Abstract To a set of n points in the plane, one can associate a graph that has less than n2 vertices and has the property that k-cliques in the graph correspond vertex sets of convex k-gons in the point set. We prove an upper bound of 2k¡1 on the size of a planar point set for which the graph has chromatic number k, matching the bound con- jectured by Szekeres for the clique number. Constructions of Erd}os and Szekeres are shown to yield graphs that have very low chromatic number. The constructions are carried out in the context of pseudoline arrangements. 1 Introduction Let X be a ¯nite set of points in IR2. We will assume that X is in general position, that is, no three points of X are on a line. A subset C of X is called closed if C = K \ X for some convex subset K of IR2. The set C of closed subsets of X, partially ordered by inclusion, is a lattice. If x 2 X and A is a closed subset of X that does not contain x and is inclusion-maximal among all closed subsets of X that do not contain x, then A is called a copoint of X attached to x. In the more general context of antimatroids, or convex geometries, co- points have been studied in [1], [2] and [3]. It is shown in these references that every copoint is attached to a unique point of X, and that the copoints are the meet-irreducible elements of the lattice of closed sets. -
The Traditional Arts and Crafts of Turnery Or Mashrabiya
THE TRADITIONAL ARTS AND CRAFTS OF TURNERY OR MASHRABIYA BY JEHAN MOHAMED A Capstone submitted to the Graduate School-Camden Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Master of Art Graduate Program in Liberal Studies Written under the direction of Dr. Martin Rosenberg And Approved by ______________________________ Dr. Martin Rosenberg Camden, New Jersey May 2015 CAPSTONE ABSTRACT The Traditional Arts and Crafts of Turnery or Mashrabiya By JEHAN MOHAMED Capstone Director: Dr. Martin Rosenberg For centuries, the mashrabiya as a traditional architectural element has been recognized and used by a broad spectrum of Muslim and non-Muslim nations. In addition to its aesthetic appeal and social component, the element was used to control natural ventilation and light. This paper will analyze the phenomenon of its use socially, historically, artistically and environmentally. The paper will investigate in depth the typology of the screen; how the different techniques, forms and designs affect the function of channeling direct sunlight, generating air flow, increasing humidity, and therefore, regulating or conditioning the internal climate of a space. Also, in relation to cultural values and social norms, one can ask how the craft functioned, and how certain characteristics of the mashrabiya were developed to meet various needs. Finally, the study of its construction will be considered in relation to artistic representation, abstract geometry, as well as other elements of its production. ii Table of Contents Abstract……………………………………………………………………….……….…..ii List of Illustrations………………………………………………………………………..iv Introduction……………………………………………….…………………………….…1 Chapter One: Background 1.1. Etymology………………….……………………………………….……………..3 1.2. Description……………………………………………………………………...…6 1.3. -
Illumination Underfoot the Design Origins of Mamluk Carpets by Peter Samsel
Illumination Underfoot The Design Origins of Mamluk Carpets by Peter Samsel Sophia: The Journal of Traditional Studies 10:2 (2004), pp.135-61. Mamluk carpets, woven in Cairo during the period of Mamluk rule, are widely considered to be the most exquisitely beautiful carpets ever created; they are also perhaps the most enigmatic and mysterious. Characterized by an intricate play of geometrical forms, woven from a limited but shimmering palette of colors, they are utterly unique in design and near perfect in execution.1 The question of the origin of their design and occasion of their manufacture has been a source of considerable, if inconclusive, speculation among carpet scholars;2 in what follows, we explore the outstanding issues surrounding these carpets, as well as possible sources of inspiration for their design aesthetic. Character and Materials The lustrous wool used in Mamluk carpets is of remarkably high quality, and is distinct from that of other known Egyptian textiles, whether earlier Coptic textiles or garments woven of wool from the Fayyum.3 The manner in which the wool is spun, however – “S” spun, rather than “Z” spun – is unique to Egypt and certain parts of North Africa.4,5 The carpets are knotted using the asymmetrical Persian knot, rather than the symmetrical Turkish knot or the Spanish single warp knot.6,7,8 The technical consistency and quality of weaving is exceptionally high, more so perhaps than any carpet group prior to mechanized carpet production. In particular, the knot counts in the warp and weft directions maintain a 1:1 proportion with a high degree of regularity, enabling the formation of polygonal shapes that are the most characteristic basis of Mamluk carpet design.9 The red dye used in Mamluk carpets is also unusual: lac, an insect dye most likely imported from India, is employed, rather than the madder dye used in most other carpet groups.10,11 Despite the high degree of technical sophistication, most Mamluks are woven from just three colors: crimson, medium blue and emerald green. -
30 ARRANGEMENTS Dan Halperin and Micha Sharir
30 ARRANGEMENTS Dan Halperin and Micha Sharir INTRODUCTION Given a finite collection of geometric objects such as hyperplanes or spheres in Rd, the arrangement ( S) is the decomposition of Rd into connected open cells of dimensions 0, 1,...,d AinducedS by . Besides being interesting in their own right, ar- rangements of hyperplanes have servedS as a unifying structure for many problems in discrete and computational geometry. With the recent advances in the study of arrangements of curved (algebraic) surfaces, arrangements have emerged as the underlying structure of geometric problems in a variety of “physical world” applica- tion domains such as robot motion planning and computer vision. This chapter is devoted to arrangements of hyperplanes and of curved surfaces in low-dimensional Euclidean space, with an emphasis on combinatorics and algorithms. In the first section we introduce basic terminology and combinatorics of ar- rangements. In Section 30.2 we describe substructures in arrangements and their combinatorial complexity. Section 30.3 deals with data structures for representing arrangements and with special refinements of arrangements. The following two sections focus on algorithms: algorithms for constructing full arrangements are described in Section 30.4, and algorithms for constructing substructures in Sec- tion 30.5. In Section 30.6 we discuss the relation between arrangements and other structures. Several applications of arrangements are reviewed in Section 30.7. Sec- tion 30.8 deals with robustness issues when implementing algorithms and data structures for arrangements and Section 30.9 surveys software implementations. We conclude in Section 30.10 with a brief review of Davenport-Schinzel sequences, a combinatorial structure that plays an important role in the analysis of arrange- ments. -
Arrangements of Approaching Pseudo-Lines
Arrangements of Approaching Pseudo-Lines Stefan Felsner∗1, Alexander Pilzy2, and Patrick Schnider3 1Institut f¨urMathematik, Technische Universit¨atBerlin, [email protected] 2Institute of SoftwareTechnology, Graz University of Technology, Austria, [email protected] 3Department of Computer Science, ETH Z¨urich, Switzerland, [email protected] January 24, 2020 Abstract We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line `i is represented by a bi-infinite connected x-monotone curve fi(x), x 2 R, s.t. for any two pseudo-lines `i and `j with i < j, the function x 7! fj (x) − fi(x) is monotonically decreasing and surjective (i.e., the pseudo-lines approach each other until they cross, and then move away from each other). We show that such arrangements of approaching pseudo-lines, under some aspects, behave similar to arrangements of lines, while for other aspects, they share the freedom of general pseudo-line arrangements. For the former, we prove: • There are arrangements of pseudo-lines that are not realizable with approaching pseudo-lines. • Every arrangement of approaching pseudo-lines has a dual generalized configura- tion of points with an underlying arrangement of approaching pseudo-lines. For the latter, we show: 2 • There are 2Θ(n ) isomorphism classes of arrangements of approaching pseudo-lines (while there are only 2Θ(n log n) isomorphism classes of line arrangements). • It can be decided in polynomial time whether an allowable sequence is realizable by an arrangement of approaching pseudo-lines. Furthermore, arrangements of approaching pseudo-lines can be transformed into each arXiv:2001.08419v1 [cs.CG] 23 Jan 2020 other by flipping triangular cells, i.e., they have a connected flip graph, and every bichromatic arrangement of this type contains a bichromatic triangular cell. -
The Rinceau Design, the Minor Arts and the St. Louis Psalter
The Rinceau Design, the Minor Arts and the St. Louis Psalter Suzanne C. Walsh A thesis submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Master of Arts in the Department of Art History. Chapel Hill 2011 Approved by: Dr. Jaroslav Folda Dr. Eduardo Douglas Dr. Dorothy Verkerk Abstract Suzanne C. Walsh: The Rinceau Design, the Minor Arts and the St. Louis Psalter (Under the direction of Dr. Jaroslav Folda) The Saint Louis Psalter (Bibliothèque National MS Lat. 10525) is an unusual and intriguing manuscript. Created between 1250 and 1270, it is a prayer book designed for the private devotions of King Louis IX of France and features 78 illustrations of Old Testament scenes set in an ornate architectural setting. Surrounding these elements is a heavy, multicolored border that uses a repeating pattern of a leaf encircled by vines, called a rinceau. When compared to the complete corpus of mid-13th century art, the Saint Louis Psalter's rinceau design has its origin outside the manuscript tradition, from architectural decoration and metalwork and not other manuscripts. This research aims to enhance our understanding of Gothic art and the interrelationship between various media of art and the creation of the complete artistic experience in the High Gothic period. ii For my parents. iii Table of Contents List of Illustrations....................................................................................................v Chapter I. Introduction.................................................................................................1 -
DTP102288.Pdf
Contents 1. SELECTED HISTORICAL BACKGROUND ................................................................... 2 2. PROJECT OBJECTIVES ................................................................................................ 4 3. PROJECT ACTIVITIES ................................................................................................... 4 3.1 ENABLING WORKS ................................................................................................ 5 3.1.1 Fencing: ............................................................................................................. 5 3.1.2 Scaffolding: ........................................................................................................ 5 3.1.3 Clearance Works: .............................................................................................. 5 3.1.4 Excavations: ...................................................................................................... 5 3.2 CONSOLIDATION WORKS: ................................................................................... 6 3.2.1 Dismantling of previous works and reconstruction of buttresses: ..................... 6 3.2.2 Structural Consolidation: ................................................................................... 6 3.2.3 Construction of timber cover: .......................................................................... 11 3.2.4 External Finishes: ............................................................................................ 11 4. PROJECT DATA .......................................................................................................... -
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.