Angles of Polygons
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PERFORMED IDENTITIES: HEAVY METAL MUSICIANS BETWEEN 1984 and 1991 Bradley C. Klypchak a Dissertation Submitted to the Graduate
PERFORMED IDENTITIES: HEAVY METAL MUSICIANS BETWEEN 1984 AND 1991 Bradley C. Klypchak A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2007 Committee: Dr. Jeffrey A. Brown, Advisor Dr. John Makay Graduate Faculty Representative Dr. Ron E. Shields Dr. Don McQuarie © 2007 Bradley C. Klypchak All Rights Reserved iii ABSTRACT Dr. Jeffrey A. Brown, Advisor Between 1984 and 1991, heavy metal became one of the most publicly popular and commercially successful rock music subgenres. The focus of this dissertation is to explore the following research questions: How did the subculture of heavy metal music between 1984 and 1991 evolve and what meanings can be derived from this ongoing process? How did the contextual circumstances surrounding heavy metal music during this period impact the performative choices exhibited by artists, and from a position of retrospection, what lasting significance does this particular era of heavy metal merit today? A textual analysis of metal- related materials fostered the development of themes relating to the selective choices made and performances enacted by metal artists. These themes were then considered in terms of gender, sexuality, race, and age constructions as well as the ongoing negotiations of the metal artist within multiple performative realms. Occurring at the juncture of art and commerce, heavy metal music is a purposeful construction. Metal musicians made performative choices for serving particular aims, be it fame, wealth, or art. These same individuals worked within a greater system of influence. Metal bands were the contracted employees of record labels whose own corporate aims needed to be recognized. -
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. -
Square Rectangle Triangle Diamond (Rhombus) Oval Cylinder Octagon Pentagon Cone Cube Hexagon Pyramid Sphere Star Circle
SQUARE RECTANGLE TRIANGLE DIAMOND (RHOMBUS) OVAL CYLINDER OCTAGON PENTAGON CONE CUBE HEXAGON PYRAMID SPHERE STAR CIRCLE Powered by: www.mymathtables.com Page 1 what is Rectangle? • A rectangle is a four-sided flat shape where every angle is a right angle (90°). means "right angle" and show equal sides. what is Triangle? • A triangle is a polygon with three edges and three vertices. what is Octagon? • An octagon (eight angles) is an eight-sided polygon or eight-gon. what is Hexagon? • a hexagon is a six-sided polygon or six-gon. The total of the internal angles of any hexagon is 720°. what is Pentagon? • a plane figure with five straight sides and five angles. what is Square? • a plane figure with four equal straight sides and four right angles. • every angle is a right angle (90°) means "right ang le" show equal sides. what is Rhombus? • is a flat shape with four equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal what is Oval? • Many distinct curves are commonly called ovals or are said to have an "oval shape". • Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. Powered by: www.mymathtables.com Page 2 What is Cube? • Six equal square faces.tweleve edges and eight vertices • the angle between two adjacent faces is ninety. what is Sphere? • no faces,sides,vertices • All points are located at the same distance from the center. what is Cylinder? • two circular faces that are congruent and parallel • faces connected by a curved surface. -
Geometry Honors Mid-Year Exam Terms and Definitions Blue Class 1
Geometry Honors Mid-Year Exam Terms and Definitions Blue Class 1. Acute angle: Angle whose measure is greater than 0° and less than 90°. 2. Adjacent angles: Two angles that have a common side and a common vertex. 3. Alternate interior angles: A pair of angles in the interior of a figure formed by two lines and a transversal, lying on alternate sides of the transversal and having different vertices. 4. Altitude: Perpendicular segment from a vertex of a triangle to the opposite side or the line containing the opposite side. 5. Angle: A figure formed by two rays with a common endpoint. 6. Angle bisector: Ray that divides an angle into two congruent angles and bisects the angle. 7. Base Angles: Two angles not included in the legs of an isosceles triangle. 8. Bisect: To divide a segment or an angle into two congruent parts. 9. Coincide: To lie on top of the other. A line can coincide another line. 10. Collinear: Lying on the same line. 11. Complimentary: Two angle’s whose sum is 90°. 12. Concave Polygon: Polygon in which at least one interior angle measures more than 180° (at least one segment connecting two vertices is outside the polygon). 13. Conclusion: A result of summary of all the work that has been completed. The part of a conditional statement that occurs after the word “then”. 14. Congruent parts: Two or more parts that only have the same measure. In CPCTC, the parts of the congruent triangles are congruent. 15. Congruent triangles: Two triangles are congruent if and only if all of their corresponding parts are congruent. -
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. -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
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. -
Chapter 6: Things to Know
MAT 222 Chapter 6: Things To Know Section 6.1 Polygons Objectives Vocabulary 1. Define and Name Polygons. • polygon 2. Find the Sum of the Measures of the Interior • vertex Angles of a Quadrilateral. • n-gon • concave polygon • convex polygon • quadrilateral • regular polygon • diagonal • equilateral polygon • equiangular polygon Polygon Definition A figure is a polygon if it meets the following three conditions: 1. 2. 3. The endpoints of the sides of a polygon are called the ________________________ (Singular form: _______________ ). Polygons must be named by listing all of the vertices in order. Write two different ways of naming the polygon to the right below: _______________________ , _______________________ Example Identifying Polygons Identify the polygons. If not a polygon, state why. a. b. c. d. e. MAT 222 Chapter 6 Things To Know Number of Sides Name of Polygon 3 4 5 6 7 8 9 10 12 n Definitions In general, a polygon with n sides is called an __________________________. A polygon is __________________________ if no line containing a side contains a point within the interior of the polygon. Otherwise, a polygon is _________________________________. Example Identifying Convex and Concave Polygons. Identify the polygons. If not a polygon, state why. a. b. c. Definition An ________________________________________ is a polygon with all sides congruent. An ________________________________________ is a polygon with all angles congruent. A _________________________________________ is a polygon that is both equilateral and equiangular. MAT 222 Chapter 6 Things To Know Example Identifying Regular Polygons Determine if each polygon is regular or not. Explain your reasoning. a. b. c. Definition A segment joining to nonconsecutive vertices of a convex polygon is called a _______________________________ of the polygon. -
The Construction, by Euclid, of the Regular Pentagon
THE CONSTRUCTION, BY EUCLID, OF THE REGULAR PENTAGON Jo˜ao Bosco Pitombeira de CARVALHO Instituto de Matem´atica, Universidade Federal do Rio de Janeiro, Cidade Universit´aria, Ilha do Fund˜ao, Rio de Janeiro, Brazil. e-mail: [email protected] ABSTRACT We present a modern account of Ptolemy’s construction of the regular pentagon, as found in a well-known book on the history of ancient mathematics (Aaboe [1]), and discuss how anachronistic it is from a historical point of view. We then carefully present Euclid’s original construction of the regular pentagon, which shows the power of the method of equivalence of areas. We also propose how to use the ideas of this paper in several contexts. Key-words: Regular pentagon, regular constructible polygons, history of Greek mathe- matics, equivalence of areas in Greek mathematics. 1 Introduction This paper presents Euclid’s construction of the regular pentagon, a highlight of the Elements, comparing it with the widely known construction of Ptolemy, as presented by Aaboe [1]. This gives rise to a discussion on how to view Greek mathematics and shows the care on must have when adopting adapting ancient mathematics to modern styles of presentation, in order to preserve not only content but the very way ancient mathematicians thought and viewed mathematics. 1 The material here presented can be used for several purposes. First of all, in courses for prospective teachers interested in using historical sources in their classrooms. In several places, for example Brazil, the history of mathematics is becoming commonplace in the curricula of courses for prospective teachers, and so one needs materials that will awaken awareness of the need to approach ancient mathematics as much as possible in its own terms, and not in some pasteurized downgraded versions.