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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. -
Vertex-Transplants on a Convex Polyhedron
CCCG 2020, Saskatoon, Canada, August 5{7, 2020 Vertex-Transplants on a Convex Polyhedron Joseph O'Rourke Abstract Regular Tetrahedron. Let the four vertices of a regu- lar tetrahedron of unit edge length be v1; v2; v3 forming Given any convex polyhedron P of sufficiently many ver- the base, and apex v0. Place a point x on the v3v0 edge, tices n, and with no vertex's curvature greater than π, close to v3. Then one can form a digon starting from x it is possible to cut out a vertex, and paste the excised and surrounding v0 with geodesics γ1 and γ2 to a point y portion elsewhere along a vertex-to-vertex geodesic, cre- on 4v1v2v0, with jγ1j = jγ2j = 1. See Fig. 1(a,b). This ating a new convex polyhedron P0 of n + 2 vertices. digon can then be cut out and its hole sutured closed. The removed digon surface can be folded to a doubly covered triangle, and pasted into edge v1v2. The re- 1 Introduction sulting convex polyhedron guaranteed by Alexandrov's Theorem is a 6-vertex irregular octahedron P0. The goal of this paper is to prove the following theorem: Theorem 1 For any convex polyhedron P of n > N vertices, none of which have curvature greater than π, there is a vertex v0 that can be cut out along a digon of geodesics, and the excised surface glued to a geodesic on P connecting two vertices v1; v2. The result is a new convex polyhedron P0 with n + 2 vertices. N = 16 suffices. -
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. -
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. -
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. -
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°. -
Winding Numbers and Solid Angles
Palestine Polytechnic University Deanship of Graduate Studies and Scientific Research Master Program of Mathematics Winding Numbers and Solid Angles Prepared by Warda Farajallah M.Sc. Thesis Hebron - Palestine 2017 Winding Numbers and Solid Angles Prepared by Warda Farajallah Supervisor Dr. Ahmed Khamayseh M.Sc. Thesis Hebron - Palestine Submitted to the Department of Mathematics at Palestine Polytechnic University as a partial fulfilment of the requirement for the degree of Master of Science. Winding Numbers and Solid Angles Prepared by Warda Farajallah Supervisor Dr. Ahmed Khamayseh Master thesis submitted an accepted, Date September, 2017. The name and signature of the examining committee members Dr. Ahmed Khamayseh Head of committee signature Dr. Yousef Zahaykah External Examiner signature Dr. Nureddin Rabie Internal Examiner signature Palestine Polytechnic University Declaration I declare that the master thesis entitled "Winding Numbers and Solid Angles " is my own work, and hereby certify that unless stated, all work contained within this thesis is my own independent research and has not been submitted for the award of any other degree at any institution, except where due acknowledgment is made in the text. Warda Farajallah Signature: Date: iv Statement of Permission to Use In presenting this thesis in partial fulfillment of the requirements for the master degree in mathematics at Palestine Polytechnic University, I agree that the library shall make it available to borrowers under rules of library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledg- ment of the source is made. Permission for the extensive quotation from, reproduction, or publication of this thesis may be granted by my main supervisor, or in his absence, by the Dean of Graduate Studies and Scientific Research when, in the opinion either, the proposed use of the material is scholarly purpose. -
Petrie Schemes
Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes. -
Uniform Panoploid Tetracombs
Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs. -
Habilitation Thesis
HABILITATION THESIS FLEXIBLE OCTAHEDRA IN THE PROJECTIVE EXTENSION OF THE EUCLIDEAN 3-SPACE AND THEIR APPLICATION Die vorliegende Habilitationsschrift wurde ausgeführt zum Zweck der Erlangung der Lehrbefugnis für das Fach GEOMETRIE eingereicht an der Technischen Universität Wien Fakultät für Mathematik und Geoinformation von Mag.rer.nat. Dr.techn. Georg NAWRATIL 7. Jänner 2011 Danksagung Mein Dank geb¨uhrt O.Univ.Prof. Dr. Hellmuth Stachel, der w¨ahrend meiner bisherigen wis- senschaftlichen T¨atigkeit stets Zeit f¨ur die Beantwortung von Fragen und die Diskussion von Problemen fand. Seine dabei vermittelte geometrische und kinematische Sicht der Dinge, haben meine Denkweise nachhaltig gepr¨agt. Weiters gesch¨atzt habe ich seinen mir stets anvertrauten Gestaltungsfreiraum in Bezug auf Forschung sowie Lehre. Bei dieser Gelegenheit m¨ochte ich mich auch bei all jenen Kollegen der Abteilung f¨ur Differential- geometrie und Geometrische Strukturen sowie jenen der Abteilung f¨ur Geometrische Model- lierung und Industrielle Geometrie des Instituts f¨ur Diskrete Mathematik und Geometrie der TU Wien bedanken, die maßgeblich zur Schaffung einer angenehmen Arbeitsatmosph¨are beigetragen haben. Der gr¨oßte Dank richtet sich jedoch an meine Eltern sowie an meine Freundin Petra, die mich w¨ahrend der Entstehungszeit dieser Habilitationsschrift in allen Belangen tatkr¨aftig unterst¨utzt haben. I Contents 1 Introduction 1 1.1 Application and future research . ....... 3 1.2 References...................................... 4 2 Composition of spherical four-bar-mechanisms (with H. Stachel) 5 2.1 Introduction.................................... 5 2.2 Transmission by a spherical four-bar linkage . ........... 6 2.3 Composition of two spherical four-bar linkages . ........... 9 2.4 Conclusions..................................... 12 2.5 References.....................................