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Islamic Geometric Ornaments in Istanbul
►SKETCH 2 CONSTRUCTIONS OF REGULAR POLYGONS Regular polygons are the base elements for constructing the majority of Islamic geometric ornaments. Of course, in Islamic art there are geometric ornaments that may have different genesis, but those that can be created from regular polygons are the most frequently seen in Istanbul. We can also notice that many of the Islamic geometric ornaments can be recreated using rectangular grids like the ornament in our first example. Sometimes methods using rectangular grids are much simpler than those based or regular polygons. Therefore, we should not omit these methods. However, because methods for constructing geometric ornaments based on regular polygons are the most popular, we will spend relatively more time explor- ing them. Before, we start developing some concrete constructions it would be worthwhile to look into a few issues of a general nature. As we have no- ticed while developing construction of the ornament from the floor in the Sultan Ahmed Mosque, these constructions are not always simple, and in order to create them we need some knowledge of elementary geometry. On the other hand, computer programs for geometry or for computer graphics can give us a number of simpler ways to develop geometric fig- ures. Some of them may not require any knowledge of geometry. For ex- ample, we can create a regular polygon with any number of sides by rotat- ing a point around another point by using rotations 360/n degrees. This is a very simple task if we use a computer program and the only knowledge of geometry we need here is that the full angle is 360 degrees. -
Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension. -
Polygon Review and Puzzlers in the Above, Those Are Names to the Polygons: Fill in the Blank Parts. Names: Number of Sides
Polygon review and puzzlers ÆReview to the classification of polygons: Is it a Polygon? Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Polygon Not a Polygon Not a Polygon (straight sides) (has a curve) (open, not closed) Regular polygons have equal length sides and equal interior angles. Polygons are named according to their number of sides. Name of Degree of Degree of triangle total angles regular angles Triangle 180 60 In the above, those are names to the polygons: Quadrilateral 360 90 fill in the blank parts. Pentagon Hexagon Heptagon 900 129 Names: number of sides: Octagon Nonagon hendecagon, 11 dodecagon, _____________ Decagon 1440 144 tetradecagon, 13 hexadecagon, 15 Do you see a pattern in the calculation of the heptadecagon, _____________ total degree of angles of the polygon? octadecagon, _____________ --- (n -2) x 180° enneadecagon, _____________ icosagon 20 pentadecagon, _____________ These summation of angles rules, also apply to the irregular polygons, try it out yourself !!! A point where two or more straight lines meet. Corner. Example: a corner of a polygon (2D) or of a polyhedron (3D) as shown. The plural of vertex is "vertices” Test them out yourself, by drawing diagonals on the polygons. Here are some fun polygon riddles; could you come up with the answer? Geometry polygon riddles I: My first is in shape and also in space; My second is in line and also in place; My third is in point and also in line; My fourth in operation but not in sign; My fifth is in angle but not in degree; My sixth is in glide but not symmetry; Geometry polygon riddles II: I am a polygon all my angles have the same measure all my five sides have the same measure, what general shape am I? Geometry polygon riddles III: I am a polygon. -
Formulas Involving Polygons - Lesson 7-3
you are here > Class Notes – Chapter 7 – Lesson 7-3 Formulas Involving Polygons - Lesson 7-3 Here’s today’s warmup…don’t forget to “phone home!” B Given: BD bisects ∠PBQ PD ⊥ PB QD ⊥ QB M Prove: BD is ⊥ bis. of PQ P Q D Statements Reasons Honors Geometry Notes Today, we started by learning how polygons are classified by their number of sides...you should already know a lot of these - just make sure to memorize the ones you don't know!! Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon 13 Tridecagon 14 Tetradecagon 15 Pentadecagon 16 Hexadecagon 17 Heptadecagon 18 Octadecagon 19 Enneadecagon 20 Icosagon n n-gon Baroody Page 2 of 6 Honors Geometry Notes Next, let’s look at the diagonals of polygons with different numbers of sides. By drawing as many diagonals as we could from one diagonal, you should be able to see a pattern...we can make n-2 triangles in a n-sided polygon. Given this information and the fact that the sum of the interior angles of a polygon is 180°, we can come up with a theorem that helps us to figure out the sum of the measures of the interior angles of any n-sided polygon! Baroody Page 3 of 6 Honors Geometry Notes Next, let’s look at exterior angles in a polygon. First, consider the exterior angles of a pentagon as shown below: Note that the sum of the exterior angles is 360°. -
131 865- EDRS PRICE Technclogy,Liedhanical
..DOCUMENT RASH 131 865- JC 760 420 TITLE- Training Program for Teachers of Technical Mathematics in Two-Year Cutricnla. INSTITUTION Queensborough Community Coll., Bayside, N Y. SPONS AGENCY New York State Education Dept., Albany. D v.of Occupational Education Instruction. PUB DATE Jul 76 GRANT VEA-78-2-132 NOTE 180p. EDRS PRICE MP-$0.83 HC-S10.03 Plus Postage. DESCRIPTORS *College Mathematics; Community Colleges; Curriculum Development; Engineering Technology; *Junior Colleges Mathematics Curriculum; *Mathematics Instruction; *Physics Instruction; Practical ,Mathematics Technical Education; *Technical Mathematics Technology ABSTRACT -This handbook 1 is designed'to.aS ist teachers of techhical'mathematics in:developing practically-oriented-curricula -for their students. The upderlying.assumption is that, while technology students are nOt a breed apartr their: needs-and orientation are_to the concrete, rather than the abstract. It describes the:nature, scope,and dontett,of curricula.. in Electrical, TeChnClogy,liedhanical Technology4 Design DraftingTechnology, and: Technical Physics', ..with-particularreterence to:the mathematical skills which are_important for the students,both.incollege.andot -tle job. SaMple mathematical problemsr-derivationSeand.theories tei be stressed it..each of_these.durricula-are presented, as- ate additional materials from the physics an4 mathematics. areas. A frame ok-reference-is.provided through diScussiotS ofthe\careers,for.which-. technology students-are heing- trained'. There.iS alsO §ion deioted to -
Chaos Based RFID Authentication Protocol
Chaos Based RFID Authentication Protocol Hau Leung Harold Chung A thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the degree of MASTER OF APPLIED SCIENCE in Electrical and Computer Engineering School of Electrical Engineering and Computer Science Faculty of Engineering University of Ottawa c Hau Leung Harold Chung, Ottawa, Canada, 2013 Abstract Chaotic systems have been studied for the past few decades because of its complex be- haviour given simple governing ordinary differential equations. In the field of cryptology, several methods have been proposed for the use of chaos in cryptosystems. In this work, a method for harnessing the beneficial behaviour of chaos was proposed for use in RFID authentication and encryption. In order to make an accurate estimation of necessary hardware resources required, a complete hardware implementation was designed using a Xilinx Virtex 6 FPGA. The results showed that only 470 Xilinx Virtex slices were required, which is significantly less than other RFID authentication methods based on AES block cipher. The total number of clock cycles required per encryption of a 288-bit plaintext was 57 clock cycles. This efficiency level is many times higher than other AES methods for RFID application. Based on a carrier frequency of 13:56Mhz, which is the standard frequency of common encryption enabled passive RFID tags such as ISO-15693, a data throughput of 5:538Kb=s was achieved. As the strength of the proposed RFID authentication and encryption scheme is based on the problem of predicting chaotic systems, it was important to ensure that chaotic behaviour is maintained in this discretized version of Lorenz dynamical system. -
900.00 Modelability
986.310 Strategic Use of Min-max Cosmic System Limits 986.311 The maximum limit set of identical facets into which any system can be divided consists of 120 similar spherical right triangles ACB whose three corners are 60 degrees at A, 90 degrees at C, and 36 degrees at B. Sixty of these right spherical triangles are positive (active), and 60 are negative (passive). (See Sec. 901.) 986.312 These 120 right spherical surface triangles are described by three different central angles of 37.37736814 degrees for arc AB, 31.71747441 degrees for arc BC, and 20.90515745 degrees for arc AC__which three central-angle arcs total exactly 90 degrees. These 120 spherical right triangles are self-patterned into producing 30 identical spherical diamond groups bounded by the same central angles and having corresponding flat-faceted diamond groups consisting of four of the 120 angularly identical (60 positive, 60 negative) triangles. Their three surface corners are 90 degrees at C, 31.71747441 degrees at B, and 58.2825256 degrees at A. (See Fig. 986.502.) 986.313 These diamonds, like all diamonds, are rhombic forms. The 30- symmetrical- diamond system is called the rhombic triacontahedron: its 30 mid- diamond faces (right- angle cross points) are approximately tangent to the unit- vector-radius sphere when the volume of the rhombic triacontahedron is exactly tetravolume-5. (See Fig. 986.314.) 986.314 I therefore asked Robert Grip and Chris Kitrick to prepare a graphic comparison of the various radii and their respective polyhedral profiles of all the symmetric polyhedra of tetravolume 5 (or close to 5) existing within the primitive cosmic hierarchy (Sec. -
( ) Methods for Construction of Odd Number Pointed
Daniel DOBRE METHODS FOR CONSTRUCTION OF ODD NUMBER POINTED POLYGONS Abstract: The purpose of this paper is to present methods for constructing of polygons with an odd number of sides, although some of them may not be built only with compass and straightedge. Odd pointed polygons are difficult to construct accurately, though there are relatively simple ways of making a good approximation which are accurate within the normal working tolerances of design practitioners. The paper illustrates rather complicated constructions to produce unconstructible polygons with an odd number of sides, constructions that are particularly helpful for engineers, architects and designers. All methods presented in this paper provide practice in geometric representations. Key words: regular polygon, pentagon, heptagon, nonagon, hendecagon, pentadecagon, heptadecagon. 1. INTRODUCTION n – number of sides; Ln – length of a side; For a specialist inured to engineering graphics, plane an – apothem (radius of inscribed circle); and solid geometries exert a special fascination. Relying R – radius of circumscribed circle. on theorems and relations between linear and angular sizes, geometry develops students' spatial imagination so 2. THREE - POINT GEOMETRY necessary for understanding other graphic discipline, descriptive geometry, underlying representations in The construction for three point geometry is shown engineering graphics. in fig. 1. Given a circle with center O, draw the diameter Construction of regular polygons with odd number of AD. From the point D as center and, with a radius in sides, just by straightedge and compass, has preoccupied compass equal to the radius of circle, describe an arc that many mathematicians who have found ingenious intersects the circle at points B and C. -
Τα Cellular Automata Στο Σχεδιασμό
τα cellular automata στο σχεδιασμό μια προσέγγιση στις αναδρομικές σχεδιαστικές διαδικασίες Ηρώ Δημητρίου επιβλέπων Σωκράτης Γιαννούδης 2 Πολυτεχνείο Κρήτης Τμήμα Αρχιτεκτόνων Μηχανικών ερευνητική εργασία Ηρώ Δημητρίου επιβλέπων καθηγητής Σωκράτης Γιαννούδης Τα Cellular Automata στο σχεδιασμό μια προσέγγιση στις αναδρομικές σχεδιαστικές διαδικασίες Χανιά, Μάιος 2013 Chaos and Order - Carlo Allarde περιεχόμενα 0001. εισαγωγή 7 0010. χάος και πολυπλοκότητα 13 a. μια ιστορική αναδρομή στο χάος: Henri Poincare - Edward Lorenz 17 b. το χάος 22 c. η πολυπλοκότητα 23 d. αυτοοργάνωση και emergence 29 0011. cellular automata 31 0100. τα cellular automata στο σχεδιασμό 39 a. τα CA στην στην αρχιτεκτονική: Paul Coates 45 b. η φιλοσοφική προσέγγιση της διεπιστημονικότητας του σχεδιασμού 57 c. προσομοίωση της αστικής ανάπτυξης μέσω CA 61 d. η περίπτωση της Changsha 63 0101. συμπεράσματα 71 βιβλιογραφία 77 1. Metamorphosis II - M.C.Escher 6 0001. εισαγωγή Η επιστήμη εξακολουθεί να είναι η εξ αποκαλύψεως προφητική περιγραφή του κόσμου, όπως αυτός φαίνεται από ένα θεϊκό ή δαιμονικό σημείο αναφοράς. Ilya Prigogine 7 0001. 8 0001. Στοιχεία της τρέχουσας αρχιτεκτονικής θεωρίας και μεθοδολογίας προτείνουν μια εναλλακτική λύση στις πάγιες αρχιτεκτονικές μεθοδολογίες και σε ορισμένες περιπτώσεις υιοθετούν πτυχές του νέου τρόπου της κατανόησής μας για την επιστήμη. Αυτά τα στοιχεία εμπίπτουν σε τρεις κατηγορίες. Πρώτον, μεθοδολογίες που προτείνουν μια εναλλακτική λύση για τη γραμμικότητα και την αιτιοκρατία της παραδοσιακής αρχιτεκτονικής σχεδιαστικής διαδικασίας και θίγουν τον κεντρικό έλεγχο του αρχιτέκτονα, δεύτερον, η πρόταση μιας μεθοδολογίας με βάση την προσομοίωση της αυτο-οργάνωσης στην ανάπτυξη και εξέλιξη των φυσικών συστημάτων και τρίτον, σε ορισμένες περιπτώσεις, συναρτήσει των δύο προηγούμενων, είναι μεθοδολογίες οι οποίες πειραματίζονται με την αναδυόμενη μορφή σε εικονικά περιβάλλοντα. -
Volume 2 Shape and Space
Volume 2 Shape and Space Colin Foster Introduction Teachers are busy people, so I’ll be brief. Let me tell you what this book isn’t. • It isn’t a book you have to make time to read; it’s a book that will save you time. Take it into the classroom and use ideas from it straight away. Anything requiring preparation or equipment (e.g., photocopies, scissors, an overhead projector, etc.) begins with the word “NEED” in bold followed by the details. • It isn’t a scheme of work, and it isn’t even arranged by age or pupil “level”. Many of the ideas can be used equally well with pupils at different ages and stages. Instead the items are simply arranged by topic. (There is, however, an index at the back linking the “key objectives” from the Key Stage 3 Framework to the sections in these three volumes.) The three volumes cover Number and Algebra (1), Shape and Space (2) and Probability, Statistics, Numeracy and ICT (3). • It isn’t a book of exercises or worksheets. Although you’re welcome to photocopy anything you wish, photocopying is expensive and very little here needs to be photocopied for pupils. Most of the material is intended to be presented by the teacher orally or on the board. Answers and comments are given on the right side of most of the pages or sometimes on separate pages as explained. This is a book to make notes in. Cross out anything you don’t like or would never use. Add in your own ideas or references to other resources. -
The Worship of Augustus Caesar
J THE WORSHIP OF AUGUSTUS C^SAR DERIVED FROM A STUDY OF COINS, MONUMENTS, CALENDARS, ^RAS AND ASTRONOMICAL AND ASTROLOGICAL CYCLES, THE WHOLE ESTABLISHING A NEW CHRONOLOGY AND SURVEY OF HISTORY AND RELIGION BY ALEXANDER DEL MAR \ NEW YORK PUBLISHED BY THE CAMBRIDGE ENCYCLOPEDIA CO. 62 Reade Street 1900 (All rights reserrecf) \ \ \ COPYRIGHT BY ALEX. DEL MAR 1899. THE WORSHIP OF AUGUSTUS CAESAR. CHAPTERS. PAGE. Prologue, Preface, ........ Vll. Bibliography, ....... xi. I. —The Cycle of the Eclipses, I — II. The Ancient Year of Ten Months, . 6 III. —The Ludi S^eculares and Olympiads, 17 IV. —Astrology of the Divine Year, 39 V. —The Jovian Cycle and Worship, 43 VI. —Various Years of the Incarnation, 51 VII.—^RAS, 62 — VIII. Cycles, ...... 237 IX. —Chronological Problems and Solutions, 281 X. —Manetho's False Chronology, 287 — XI. Forgeries in Stone, .... 295 — XII. The Roman Messiah, .... 302 Index, ........ 335 Corrigenda, ....... 347 PROLOGUE. THE ABYSS OF MISERY AND DEPRAVITY FROM WHICH CHRISTIANITY REDEEMED THE ROMAN EMPIRE CAN NEVER BE FULLY UNDERSTOOD WITHOUT A KNOWLEDGE OF THE IMPIOUS WoA^P OF EM- PERORS TO WHICH EUROPE ONCE BOWED ITS CREDULOUS AND TERRIFIED HEAD. WHEN THIS OMITTED CHAPTER IS RESTORED TO THE HISTORY OF ROME, CHRISTIANITY WILL SPRING A LIFE FOR INTO NEW AND MORE VIGOROUS ; THEN ONLY WILL IT BE PERCEIVED HOW DEEP AND INERADICABLY ITS ROOTS ARE PLANTED, HOW LOFTY ARE ITS BRANCHES AND HOW DEATH- LESS ARE ITS AIMS. PREFACE. collection of data contained in this work was originally in- " THEtended as a guide to the author's studies of Monetary Sys- tems." It was therefore undertaken with the sole object of estab- lishing with precision the dates of ancient history. -
Neural Network Based Modeling, Characterization and Identification of Chaotic Systems in Nature
Neural Network based Modeling, Characterization and Identification of Chaotic Systems in Nature Thesis submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy by Archana R Reg. No: 3829 Under the guidance of Dr. R Gopikakumari &Dr. A Unnikrishnan Division of Electronics Engineering School of Engineering Cochin University of Science and Technology Kochi- 682022 India March 2015 Neural Network based Modeling, Characterization and Identification of Chaotic Systems in Nature Ph. D thesis under the Faculty of Engineering Author Archana R Research Scholar School of Engineering Cochin University of Science and Technology [email protected] Supervising Guide Dr. R Gopikakumari Professor Division of Electronics Engineering Cochin University of Science and Technology Kochi - 682022 [email protected], [email protected] Co-Guide Dr. A Unnikrishnan Outstanding scientist (Rtd), NPOL, DRDO, Kochi 682021 & Principal Rajagiri School of Engineering & Technology Kochi 682039 [email protected] Division of Electronics Engineering School of Engineering Cochin University of Science and Technology Kochi- 682022 India Certificate This is to certify that the work presented in the thesis entitled “Neural Network based Modeling, Characterization and Identification of Chaotic Systems in Nature” is a bonafide record of the research work carried out by Mrs. Archana R under our supervision and guidance in the Division of Electronics Engineering, School of Engineering, Cochin University of Science and Technology and that no part thereof has been presented for the award of any other degree. Dr. R Gopikakumari Dr. A Unnikrishnan Supervising Guide Co-guide i Declaration I hereby declare that the work presented in the thesis entitled “Neural Network based Modeling, Characterization and Identification of Chaotic Systems in Nature” is based on the original work done by me under the supervision and guidance of Dr.