Circles, Polygons, and Star Polygons
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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. -
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. -
Interior and Exterior Angles of Polygons 2A
Regents Exam Questions Name: ________________________ G.CO.C.11: Interior and Exterior Angles of Polygons 2a www.jmap.org G.CO.C.11: Interior and Exterior Angles of Polygons 2a 1 Which type of figure is shown in the accompanying 5 In the diagram below of regular pentagon ABCDE, diagram? EB is drawn. 1) hexagon 2) octagon 3) pentagon 4) quadrilateral What is the measure of ∠AEB? 2 What is the measure of each interior angle of a 1) 36º regular hexagon? 2) 54º 1) 60° 3) 72º 2) 120° 4) 108º 3) 135° 4) 270° 6 What is the measure, in degrees, of each exterior angle of a regular hexagon? 3 Determine, in degrees, the measure of each interior 1) 45 angle of a regular octagon. 2) 60 3) 120 4) 135 4 Determine and state the measure, in degrees, of an interior angle of a regular decagon. 7 A stop sign in the shape of a regular octagon is resting on a brick wall, as shown in the accompanying diagram. What is the measure of angle x? 1) 45° 2) 60° 3) 120° 4) 135° 1 Regents Exam Questions Name: ________________________ G.CO.C.11: Interior and Exterior Angles of Polygons 2a www.jmap.org 8 One piece of the birdhouse that Natalie is building 12 The measure of an interior angle of a regular is shaped like a regular pentagon, as shown in the polygon is 120°. How many sides does the polygon accompanying diagram. have? 1) 5 2) 6 3) 3 4) 4 13 Melissa is walking around the outside of a building that is in the shape of a regular polygon. -
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. -
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°. -
Geometric Constructions of Regular Polygons and Applications to Trigonometry Introduction
Geometric Constructions of Regular Polygons and Applications to Trigonometry José Gilvan de Oliveira, Moacir Rosado Filho, Domingos Sávio Valério Silva Abstract: In this paper, constructions of regular pentagon and decagon, and the calculation of the main trigonometric ratios of the corresponding central angles are approached. In this way, for didactic purposes, it is intended to show the reader that it is possible to broaden the study of Trigonometry by addressing new applications and exercises, for example, with angles 18°, 36° and 72°. It is also considered constructions of other regular polygons and a relation to a construction of the regular icosahedron. Introduction The main objective of this paper is to approach regular pentagon and decagon constructions, together with the calculation of the main trigonometric ratios of the corresponding central angles. In textbooks, examples and exercises involving trigonometric ratios are usually restricted to so-called notable arcs of angles 30°, 45° and 60°. Although these angles are not the main purpose of this article, they will be addressed because the ingredients used in these cases are exactly the same as those we will use to build the regular pentagon and decagon. The difference to the construction of these last two is restricted only to the number of steps required throughout the construction process. By doing so, we intend to show the reader that it is possible to broaden the study of Trigonometry by addressing new applications and exercises, for example, with angles 18°, 36° and 72°. The ingredients used in the text are those found in Plane Geometry and are very few, restricted to intersections of lines and circumferences, properties of triangles, and Pythagorean Theorem. -
A Note on Soenergy of Stars, Bi-Stars and Double Star Graphs
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 6(2016), 105-113 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) A NOTE ON soENERGY OF STARS, BISTAR AND DOUBLE STAR GRAPHS S. P. Jeyakokila and P. Sumathi Abstract. Let G be a finite non trivial connected graph.In the earlier paper idegree, odegree, oidegree of a minimal dominating set of G were introduced, oEnergy of a graph with respect to the minimal dominating set was calculated in terms of idegree and odegree. Algorithm to get the soEnergy was also introduced and soEnergy was calculated for some standard graphs in the earlier papers. In this paper soEnergy of stars, bistars and double stars with respect to the given dominating set are found out. 1. INTRODUCTION Let G = (V; E) be a finite non trivial connected graph. A set D ⊂ V is a dominating set of G if every vertex in V − D is adjacent to some vertex in D.A dominating set D of G is called a minimal dominating set if no proper subset of D is a dominating set. Star graph is a tree consisting of one vertex adjacent to all the others. Bistar is a graph obtained from K2 by joining n pendent edges to both the ends of K2. Double star is the graph obtained from K2 by joining m pendent edges to one end and n pendent edges to the other end of K2. -
Total Chromatic Number of Star and Bistargraphs 1D
International Journal of Pure and Applied Mathematics Volume 117 No. 21 2017, 699-708 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu Total Chromatic Number of Star and BistarGraphs 1D. Muthuramakrishnanand 2G. Jayaraman 1Department of Mathematics, National College, Trichy, Tamil Nadu, India. [email protected] 2Research Scholar, National College, Trichy, Tamil Nadu, India. [email protected] Abstract A total coloring of a graph G is an assignment of colors to both the vertices and edges of G, such that no two adjacent or incident vertices and edges of G are assigned the same colors. In this paper, we have discussed 2 the total coloring of 푀 퐾1,푛 , 푇 퐾1,푛 , 퐿 퐾1,푛 and 퐵푛,푛and we obtained the 2 total chromatic number of 푀 퐾1,푛 , 푇 퐾1,푛 , 퐿 퐾1,푛 and 퐵푛,푛. AMS Subject Classification: 05C15 Keywords and Phrases:Middle graph, line graph, total graph, star graph, square graph, total coloring and total chromatic number. 699 International Journal of Pure and Applied Mathematics Special Issue 1. Introduction In this paper, we have chosen finite, simple and undirected graphs. Let퐺 = (푉 퐺 , 퐸 퐺 ) be a graph with the vertex set 푉(퐺) and the edge set E(퐺), respectively. In 1965, the concept of total coloring was introduced by Behzad [1] and in 1967 he [2] came out new ideology that, the total chromatic number of complete graph and complete bi-partite graph. A total coloring of퐺, is a function 푓: 푆 → 퐶, where 푆 = 푉 퐺 ∪ 퐸 퐺 and C is a set of colors to satisfies the given conditions. -
Achromatic Coloring on Double Star Graph Families
International J.Math. Combin. Vol.3 (2009), 71-81 Achromatic Coloring on Double Star Graph Families Vernold Vivin J. Department of Mathematics, Sri Shakthi Institute of Engineering and Technology, Coimbatore - 641 062, Tamil Nadu, India. E-mail: vernold [email protected], [email protected] Venkatachalam M. Department of Mathematics, SSK College of Engineering and Technology, Coimbatore - 641 105, Tamil Nadu, India. E-mail: [email protected] Akbar Ali M.M. Department of Mathematics, Sri Shakthi Institute of Engineering and Technology, Coimbatore - 641 062, Tamil Nadu, India. E-mail: um [email protected] Abstract: The purpose of this article is to find the achromatic number, i.e., Smarandachely achromatic 1-coloring for the central graph, middle graph, total graph and line graph of double star graph K1,n,n denoted by C(K1,n,n), M(K1,n,n), T (K1,n,n) and L(K1,n,n) re- spectively. Keywords: Smarandachely achromatic k-coloring, Smarandachely achromatic number, central graph, middle graph, total graph, line graph and achromatic coloring. AMS(2000): 05C15 §1. Preliminaries For a given graph G = (V, E) we do an operation on G, by subdividing each edge exactly once and joining all the non adjacent vertices of G. The graph obtained by this process is called central graph [10] of G denoted by C(G). Let G be a graph with vertex set V (G) and edge set E(G). The middle graph [4] of G, denoted by M(G) is defined as follows. The vertex set of M(G) is V (G) E(G). Two vertices ∪ x, y in the vertex set of M(G) are adjacent in M(G) in case one of the following holds: (i) x, y are in E(G) and x, y are adjacent in G. -
Symmetry Is a Manifestation of Structural Harmony and Transformations of Geometric Structures, and Lies at the Very Foundation
Bridges Finland Conference Proceedings Some Girihs and Puzzles from the Interlocks of Similar or Complementary Figures Treatise Reza Sarhangi Department of Mathematics Towson University Towson, MD 21252 E-mail: [email protected] Abstract This paper is the second one to appear in the Bridges Proceedings that addresses some problems recorded in the Interlocks of Similar or Complementary Figures treatise. Most problems in the treatise are sketchy and some of them are incomprehensible. Nevertheless, this is the only document remaining from the medieval Persian that demonstrates how a girih can be constructed using compass and straightedge. Moreover, the treatise includes some puzzles in the transformation of a polygon into another one using mathematical formulas or dissection methods. It is believed that the document was written sometime between the 13th and 15th centuries by an anonymous mathematician/craftsman. The main intent of the present paper is to analyze a group of problems in this treatise to respond to questions such as what was in the mind of the treatise’s author, how the diagrams were constructed, is the conclusion offered by the author mathematically provable or is incorrect. All images, except for photographs, have been created by author. 1. Introduction There are a few documents such as treatises and scrolls in Persian mosaic design, that have survived for centuries. The Interlocks of Similar or Complementary Figures treatise [1], is one that is the source for this article. In the Interlocks document, one may find many interesting girihs and also some puzzles that are solved using mathematical formulas or dissection methods. Dissection, in the present literature, refers to cutting a geometric, two-dimensional shape, into pieces that can be rearranged to compose a different shape.