Interlocking Star Polygons in Persian Architecture: the Special Case Of
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And Twelve-Pointed Star Polygon Design of the Tashkent Scrolls
Bridges 2011: Mathematics, Music, Art, Architecture, Culture A Nine- and Twelve-Pointed Star Polygon Design of the Tashkent Scrolls B. Lynn Bodner Mathematics Department Cedar Avenue Monmouth University West Long Branch, New Jersey, 07764, USA E-mail: [email protected] Abstract In this paper we will explore one of the Tashkent Scrolls’ repeat units, that, when replicated using symmetry operations, creates an overall pattern consisting of “nearly regular” nine-pointed, regular twelve-pointed and irregularly-shaped pentagonal star polygons. We seek to determine how the original designer of this pattern may have determined, without mensuration, the proportion and placement of the star polygons comprising the design. We will do this by proposing a plausible Euclidean “point-joining” compass-and-straightedge reconstruction. Introduction The Tashkent Scrolls (so named because they are housed in Tashkent at the Institute of Oriental Studies at the Academy of Sciences) consist of fragments of architectural sketches attributed to an Uzbek master builder or a guild of architects practicing in 16 th century Bukhara [1, p. 7]. The sketch from the Tashkent Scrolls that we will explore shows only a small portion, or the repeat unit , of a nine- and twelve-pointed star polygon design. It is contained within a rectangle and must be reflected across the boundaries of this rectangle to achieve the entire pattern. This drawing, which for the remainder of this paper we will refer to as “T9N12,” is similar to many of the 114 Islamic architectural and ornamental design sketches found in the much larger, older and better preserved Topkapı Scroll, a 96-foot-long architectural scroll housed at the Topkapı Palace Museum Library in Istanbul. -
Semiology Study of Shrine Geometric Patterns of Damavand City of Tehran Province1
Special Issue INTERNATIONAL JOURNAL OF HUMANITIES AND December 2015 CULTURAL STUDIES ISSN 2356-5926 Semiology Study of Shrine Geometric patterns of Damavand City of Tehran Province1 Atieh Youzbashi Masterof visual communication, Faculty of Art, Shahed University, Tehran, Iran [email protected] Seyed Nezam oldin Emamifar )Corresponding author) Assistant Professor of Faculty of Art, Shahed University, Tehran city, Iran [email protected] Abstract Remained works of decorative Arts in Islamic buildings, especially in religious places such as shrines, possess especial sprits and visual depth. Damavand city having very beautiful architectural works has been converted to a valuable treasury of Islamic architectural visual motifs. Getting to know shrines and their visual motifs features is leaded to know Typology, in Typology, Denotation and Connotation are the concept of truth. This research is based on descriptive and analytical nature and the collection of the data is in a mixture way. Sampling is in the form of non-random (optional) and there are 4 samples of geometric motifs of Damavand city of Tehran province and the analysis of information is qualitatively too. In this research after study of geometric designs used in this city shrines, the amount of this motifs confusion are known by semiotic concepts and denotation and connotation meaning is stated as well. At first the basic articles related to typology and geometric motifs are discussed. Discovering the meaning of these motifs requires a necessary deep study about geometric motifs treasury of believe and religious roots and symbolic meaning of this motifs. Geometric patterns with the centrality of the circle In drawing, the incidence abstractly and creating new combination is based on uniformly covering surfaces in order not to attract attention to designs independently creating an empty space also recalls “the principle of unity in diversity” and “diversity in unity”. -
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. -
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. -
Applying the Polygon Angle
POLYGONS 8.1.1 – 8.1.5 After studying triangles and quadrilaterals, students now extend their study to all polygons. A polygon is a closed, two-dimensional figure made of three or more non- intersecting straight line segments connected end-to-end. Using the fact that the sum of the measures of the angles in a triangle is 180°, students learn a method to determine the sum of the measures of the interior angles of any polygon. Next they explore the sum of the measures of the exterior angles of a polygon. Finally they use the information about the angles of polygons along with their Triangle Toolkits to find the areas of regular polygons. See the Math Notes boxes in Lessons 8.1.1, 8.1.5, and 8.3.1. Example 1 4x + 7 3x + 1 x + 1 The figure at right is a hexagon. What is the sum of the measures of the interior angles of a hexagon? Explain how you know. Then write an equation and solve for x. 2x 3x – 5 5x – 4 One way to find the sum of the interior angles of the 9 hexagon is to divide the figure into triangles. There are 11 several different ways to do this, but keep in mind that we 8 are trying to add the interior angles at the vertices. One 6 12 way to divide the hexagon into triangles is to draw in all of 10 the diagonals from a single vertex, as shown at right. 7 Doing this forms four triangles, each with angle measures 5 4 3 1 summing to 180°. -
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. -
Properties of N-Sided Regular Polygons
PROPERTIES OF N-SIDED REGULAR POLYGONS When students are first exposed to regular polygons in middle school, they learn their properties by looking at individual examples such as the equilateral triangles(n=3), squares(n=4), and hexagons(n=6). A generalization is usually not given, although it would be straight forward to do so with just a min imum of trigonometry and algebra. It also would help students by showing how one obtains generalization in mathematics. We show here how to carry out such a generalization for regular polynomials of side length s. Our starting point is the following schematic of an n sided polygon- We see from the figure that any regular n sided polygon can be constructed by looking at n isosceles triangles whose base angles are θ=(1-2/n)(π/2) since the vertex angle of the triangle is just ψ=2π/n, when expressed in radians. The area of the grey triangle in the above figure is- 2 2 ATr=sh/2=(s/2) tan(θ)=(s/2) tan[(1-2/n)(π/2)] so that the total area of any n sided regular convex polygon will be nATr, , with s again being the side-length. With this generalized form we can construct the following table for some of the better known regular polygons- Name Number of Base Angle, Non-Dimensional 2 sides, n θ=(π/2)(1-2/n) Area, 4nATr/s =tan(θ) Triangle 3 π/6=30º 1/sqrt(3) Square 4 π/4=45º 1 Pentagon 5 3π/10=54º sqrt(15+20φ) Hexagon 6 π/3=60º sqrt(3) Octagon 8 3π/8=67.5º 1+sqrt(2) Decagon 10 2π/5=72º 10sqrt(3+4φ) Dodecagon 12 5π/12=75º 144[2+sqrt(3)] Icosagon 20 9π/20=81º 20[2φ+sqrt(3+4φ)] Here φ=[1+sqrt(5)]/2=1.618033989… is the well known Golden Ratio. -
The First Treatments of Regular Star Polygons Seem to Date Back to The
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archivio istituzionale della ricerca - Università di Palermo FROM THE FOURTEENTH CENTURY TO CABRÌ: CONVOLUTED CONSTRUCTIONS OF STAR POLYGONS INTRODUCTION The first treatments of regular star polygons seem to date back to the fourteenth century, but a comprehensive theory on the subject was presented only in the nineteenth century by the mathematician Louis Poinsot. After showing how star polygons are closely linked to the concept of prime numbers, I introduce here some constructions, easily reproducible with geometry software that allow us to investigate and see some nice and hidden property obtained by the scholars of the fourteenth century onwards. Regular star polygons and prime numbers Divide a circumference into n equal parts through n points; if we connect all the points in succession, through chords, we get what we recognize as a regular convex polygon. If we choose to connect the points, starting from any one of them in regular steps, two by two, or three by three or, generally, h by h, we get what is called a regular star polygon. It is evident that we are able to create regular star polygons only for certain values of h. Let us divide the circumference, for example, into 15 parts and let's start by connecting the points two by two. In order to close the figure, we return to the starting point after two full turns on the circumference. The polygon that is formed is like the one in Figure 1: a polygon of “order” 15 and “species” two. -
Annual Statistical Bulletin 2013 Annual Statistical Bulletin
2013 OPEC OPEC Annual Statistical Bulletin 2013 Annual Statistical Bulletin OPEC Helferstorferstrasse 17, A-1010 Vienna, Austria Organization of the Petroleum Exporting Countries www.opec.org Team for the preparation of the OPEC Annual Statistical Bulletin 2013 Director, Research Division Editorial Team Omar Abdul-Hamid Head, Public Relations and Information Department Project Leader Angela Agoawike Head, Data Services Department Adedapo Odulaja Editor Alvino-Mario Fantini Coordinator Ramadan Janan Design and Production Coordinator Alaa Al-Saigh Statistics Team Pantelis Christodoulides, Hannes Windholz, Senior Production Assistant Mouhamad Moudassir, Klaus Stöger, Harvir Kalirai, Diana Lavnick Mohammad Sattar, Ksenia Gutman Web and CD Application Dietmar Rudari, Zairul Arifin Questions on data Although comments are welcome, OPEC regrets that it is unable to answer all enquiries concerning the data in the ASB. Data queries: [email protected]. Advertising The OPEC Annual Statistical Bulletin now accepts advertising. For details, please contact the Head, PR and Information Department at the following address: Organization of the Petroleum Exporting Countries Helferstorferstrasse 17, A-1010 Vienna, Austria Tel: +43 1 211 12/0 Fax: +43 1 216 43 20 PR & Information Department fax: +43 1 21112/5081 Advertising: [email protected] Website: www.opec.org Photographs Page 5: Diana Golpashin. Pages 7, 13, 21, 63, 81, 93: Shutterstock. © 2013 Organization of the Petroleum Exporting Countries ISSN 0475-0608 Contents Foreword 5 Tables Page Section 1: -
Compact Design of Modified Pentagon-Shaped Monopole Antenna for UWB Applications
International Journal of Electrical and Electronic Engineering & Telecommunications Vol. 7, No. 2, April 2018 Compact Design of Modified Pentagon-shaped Monopole Antenna for UWB Applications Sanyog Rawat1, Ushaben Keshwala2, and Kanad Ray3 1 Manipal University Jaipur, India. 2 Amity University Uttar Pradesh, India. 3 Amity University Rajasthan, India. Email: [email protected]; [email protected]; [email protected] Abstract—In this work detailed analysis of a modified, introduction, and in Section II the initial antenna pentagon-shaped planar antenna is presented for ultra-wide geometry is presented and discussed. In the preceding bandwidth (UWB) applications. The proposed antenna was section, the modified pentagon-shaped antenna is designed on an FR-4 substrate and has a compact size of 12 elaborated and the results are discussed. × 22 × 1.6 mm3. It achieves an impedance bandwidth of 8.73 GHz (3.8–12.53 GHz) in the UWB range. The design has a II. ANTENNA GEOMETRY uniform gain and stable radiation pattern in the operating bandwidth. A. Initial Pentagram Star-shaped Antenna Index Terms—Golden angle, pentagram, pentagon, star- The antenna presented in this paper was designed on a shaped antenna FR-4 substrate, which was a partial ground on one side and a conducting patch on the opposite side. The antenna design is initially started with a pentagram star shape and I. INTRODUCTION then the shape was modified into a pentagon shape. The Nowadays, wireless communications and their high- pentagram is also called a pentangle, i.e., 5-pointed star speed data rate are becoming increasingly popular. The [13]. The geometry of the microstrip line fed star-shaped research in the field of microstrip antennas has also been monopole antenna is shown in Fig. -
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]. -
Emoji Symbols
Background data for Proposal for Encoding Emoji Symbols N3681 Date: 2009-Sep-17 Author: Markus Scherer (Google Inc.) This document reflects proposed Emoji symbols data as shown in PDAM8 (N3658), plus changes made in the UTC #120/L2 #217 meeting on 2009-Aug-11. Note: The glyphs shown in this document in the second column ("Symbol") are out of date. They are the glyphs from the original US proposal (N3583) and do not reflect modified glyphs in PDAM8, agreed during the UTC #120/L2 #217 meeting, or agreed or suggested since then. The carrier symbol images in this file point to images on other sites. The images are only for comparison and may change. See the chart legend for an explanation of the data presentation in this chart. In the HTML version of this document, each symbol row has an anchor to allow direct linking by appending #e-4B0 (for example) to this page's URL in the address bar. Internal Symbol Name & Annotations DoCoMo KDDI SoftBank Google ID 1. Nature Weather and landscape symbols (1. Nature) BLACK SUN WITH RAYS #1 #44 #81 #old74 = ARIB-9364 'Fine' re e-000 Temporary Notes: clear weather for 晴れ 「晴 」 太陽 晴れ 「晴re」 U+FE000 Japanese mobile carriers, usually in red U+E63E U+E488 U+E04A U+2600 color SJIS-F89F JIS-7541 SJIS-F660 JIS-7541 SJIS-F98B unified #2 #107 #83 #old73 e-001 CLOUD 'Cloudy' 曇り 「曇ri」 くもり 「kumori」 くもり 「kumori」 U+FE001 = ARIB-9365 U+E63F U+E48D U+E049 U+2601 SJIS-F8A0 JIS-7546 SJIS-F665 JIS-7546 SJIS-F98A unified #3 #95 #82 #old75 e-002 UMBRELLA WITH RAIN DROPS 'Rain' 雨 雨 雨 U+FE002 = ARIB-9381 U+E640 U+E48C U+E04B U+2614 unified SJIS-F8A1 JIS-7545 SJIS-F664 JIS-7545 SJIS-F98C SNOWMAN WITHOUT SNOW #4 #191 #84 #old72 = ARIB-9367 'Snow' yukidaruma 雪(雪だるま) 「雪(雪 e-003 Temporary Note: Unified with an 雪 ゆきだるま 「 」 U+FE003 U+E641 U+E485 daruma)」 U+26C4 upcoming Unicode 5.2/AMD6 character; code point and name are preliminary.