DOCSLIB.ORG

6-1 Properties and Attributes of Polygons Warm Up

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
6-1 Properties and Attributes of Polygons Warm Up 66-1-1 PropertiesProperties andand Attributes of of Polygons Polygons Warm Up Lesson Presentation Lesson Quiz HoltHolt McDougal Geometry Geometry 6-1 Properties and Attributes of Polygons Warm Up 1. A ? is a three-sided polygon. triangle 2. A ? is a four-sided polygon. quadrilateral Evaluate each expression for n = 6. 3. (n – 4) 12 24 4. (n – 3) 90 270 Solve for a. 5. 12a + 4a + 9a = 100 4 Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Objectives Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex Holt McDougal Geometry 6-1 Properties and Attributes of Polygons In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons You can name a polygon by the number of its sides. The table shows the names of some common polygons. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Remember! A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 1A: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 1B: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, heptagon Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 1C: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. not a polygon Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 1a Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 1b Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. polygon, nonagon Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 1c Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon Holt McDougal Geometry 6-1 Properties and Attributes of Polygons All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 2A: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 2B: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 2C: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 2a Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 2b Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave Holt McDougal Geometry 6-1 Properties and Attributes of Polygons To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Remember! By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Holt McDougal Geometry 6-1 Properties and Attributes of Polygons In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n — 2)180°. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 3A: Finding Interior Angle Measures and Sums in Polygons Find the sum of the interior angle measures of a convex heptagon. (n – 2)180° Polygon Sum Thm. (7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 3B: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon Sum Thm. (16 – 2)180° = 2520° Substitute 16 for n and simplify. Step 2 Find the measure of one interior angle. The int. s are , so divide by 16. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. (5 – 2)180° = 540° Polygon Sum Thm. Polygon mA + mB + mC + mD + mE = 540° Sum Thm. 35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540 Combine like terms. c = 4 Divide both sides by 135. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 3C Continued mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128° Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 3a Find the sum of the interior angle measures of a convex 15-gon. (n – 2)180° Polygon Sum Thm. (15 – 2)180° A 15-gon has 15 sides, so substitute 15 for n. 2340° Simplify. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 3b Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon Sum Thm. (10 – 2)180° = 1440° Substitute 10 for n and simplify. Step 2 Find the measure of one interior angle. The int. s are , so divide by 10. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Remember! An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20-gon. A 20-gon has 20 sides and 20 vertices. sum of ext. s = 360°. Polygon Sum Thm. A regular 20-gon measure of one ext. = has 20 ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18°. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext. Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° 120b = 360 Combine like terms. b = 3 Divide both sides by 120. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 4a Find the measure of each exterior angle of a regular dodecagon. A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360°. Polygon Sum Thm. A regular dodecagon measure of one ext. has 12 ext. s, so divide the sum by 12. The measure of each exterior angle of a regular dodecagon is 30°. Holt McDougal Geometry 6-1 Properties and Attributes of Polygons Check It Out! Example 4b Find the value of r in polygon JKLM. 4r° + 7r° + 5r° + 8r° = 360° Polygon Ext. Sum Thm. 24r = 360 Combine like terms. r = 15 Divide both sides by 24. Holt McDougal Geometry .
Load more

Recommended publications
  • 1 Point for the Player Who Has the Shape with the Longest Side. Put
    Put three pieces together to create a new 1 point for the player who has the shape. 1 point for the player who has shape with the longest side. the shape with the smallest number of sides. 1 point for the player who has the 1 point for the player with the shape that shape with the shortest side. has the longest perimeter. 1 point for the player with the shape that 1 point for each triangle. has the largest area. 1 point for each shape with 1 point for each quadrilateral. a perimeter of 24 cm. Put two pieces together to create a new shape. 1 point for the player with 1 point for each shape with the shape that has the longest a perimeter of 12 cm. perimeter. Put two pieces together to create a new shape. 1 point for the player with 1 point for each shape with the shape that has the shortest a perimeter of 20 cm. perimeter. 1 point for each 1 point for each shape with non-symmetrical shape. a perimeter of 19 cm. 1 point for the shape that has 1 point for each the largest number of sides. symmetrical shape. Count the sides on all 5 pieces. 1 point for each pair of shapes where the 1 point for the player who has the area of one is twice the area of the other. largest total number of sides. Put two pieces together to create a new shape. 1 point for the player who has 1 point for each shape with at least two the shape with the fewest number of opposite, parallel sides.
    [Show full text]
  • Extremal Problems for Convex Polygons ∗
    Extremal Problems for Convex Polygons ∗ Charles Audet Ecole´ Polytechnique de Montreal´ Pierre Hansen HEC Montreal´ Fred´ eric´ Messine ENSEEIHT-IRIT Abstract. Consider a convex polygon Vn with n sides, perimeter Pn, diameter Dn, area An, sum of distances between vertices Sn and width Wn. Minimizing or maximizing any of these quantities while fixing another defines ten pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization meth- ods. Numerous open problems are mentioned, as well as series of test problems for global optimization and nonlinear programming codes. Keywords: polygon, perimeter, diameter, area, sum of distances, width, isoperimeter problem, isodiametric problem. 1. Introduction Plane geometry is replete with extremal problems, many of which are de- scribed in the book of Croft, Falconer and Guy [12] on Unsolved problems in geometry. Traditionally, such problems have been solved, some since the Greeks, by geometrical reasoning. In the last four decades, this approach has been increasingly complemented by global optimization methods. This allowed solution of larger instances than could be solved by any one of these two approaches alone. Probably the best known type of such problems are circle packing ones: given a geometrical form such as a unit square, a unit-side triangle or a unit- diameter circle, find the maximum radius and configuration of n circles which can be packed in its interior (see [46] for a recent survey and the site [44] for a census of exact and approximate results with up to 300 circles).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Simple Polygons Scribe: Michael Goldwasser
    CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons Scribe: Michael Goldwasser Algorithms for triangulating polygons are important tools throughout computa- tional geometry. Many problems involving polygons are simplified by partitioning the complex polygon into triangles, and then working with the individual triangles. The applications of such algorithms are well documented in papers involving visibility, motion planning, and computer graphics. The following notes give an introduction to triangulations and many related definitions and basic lemmas. Most of the definitions are based on a simple polygon, P, containing n edges, and hence n vertices. However, many of the definitions and results can be extended to a general arrangement of n line segments. 1 Diagonals Definition 1. Given a simple polygon, P, a diagonal is a line segment between two non-adjacent vertices that lies entirely within the interior of the polygon. Lemma 2. Every simple polygon with jPj > 3 contains a diagonal. Proof: Consider some vertex v. If v has a diagonal, it’s party time. If not then the only vertices visible from v are its neighbors. Therefore v must see some single edge beyond its neighbors that entirely spans the sector of visibility, and therefore v must be a convex vertex. Now consider the two neighbors of v. Since jPj > 3, these cannot be neighbors of each other, however they must be visible from each other because of the above situation, and thus the segment connecting them is indeed a diagonal.
    [Show full text]
  • Polygons and Convexity
    Geometry Week 4 Sec 2.5 to ch. 2 test section 2.5 Polygons and Convexity Definitions: convex set – has the property that any two of its points determine a segment contained in the set concave set – a set that is not convex concave concave convex convex concave Definitions: polygon – a simple closed curve that consists only of segments side of a polygon – one of the segments that defines the polygon vertex – the endpoint of the side of a polygon 1 angle of a polygon – an angle with two properties: 1) its vertex is a vertex of the polygon 2) each side of the angle contains a side of the polygon polygon not a not a polygon (called a polygonal curve) polygon Definitions: polygonal region – a polygon together with its interior equilateral polygon – all sides have the same length equiangular polygon – all angels have the same measure regular polygon – both equilateral and equiangular Example: A square is equilateral, equiangular, and regular. 2 diagonal – a segment that connects 2 vertices but is not a side of the polygon C B C B D A D A E AC is a diagonal AC is a diagonal AB is not a diagonal AD is a diagonal AB is not a diagonal Notation: It does not matter which vertex you start with, but the vertices must be listed in order. Above, we have square ABCD and pentagon ABCDE. interior of a convex polygon – the intersection of the interiors of is angles exterior of a convex polygon – union of the exteriors of its angles 3 Polygon Classification Number of sides Name of polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon
    [Show full text]
  • Englewood Public School District Geometry Second Marking Period
    Englewood Public School District Geometry Second Marking Period Unit 2: Polygons, Triangles, and Quadrilaterals Overview: During this unit, students will learn how to prove two triangles are congruent, relationships between angle measures and side lengths within a triangle, and about different types of quadrilaterals. Time Frame: 35 to 45 days (One Marking Period) Enduring Understandings: • Congruent triangles can be visualized by placing one on top of the other. • Corresponding sides and angles can be marked using tic marks and angle marks. • Theorems can be used to prove triangles congruent. • The definitions of isosceles and equilateral triangles can be used to classify a triangle. • The Midpoint Formula can be used to find the midsegment of a triangle. • The Distance Formula can be used to examine relationships in triangles. • Side lengths of triangles have a relationship. • The negation statement can be proved and used to show a counterexample. • The diagonals of a polygon can be used to derive the formula for the angle measures of the polygon. • The properties of parallel and perpendicular lines can be used to classify quadrilaterals. • Coordinate geometry can be used to classify special parallelograms. • Slope and the distance formula can be used to prove relationships in the coordinate plane. Essential Questions: • How do you identify corresponding parts of congruent triangles? • How do you show that two triangles are congruent? • How can you tell if a triangle is isosceles or equilateral? • How do you use coordinate geometry
    [Show full text]
  • Midsegment of a Trapezoid Parallelogram
    Vocabulary Flash Cards base angles of a trapezoid bases of a trapezoid Chapter 7 (p. 402) Chapter 7 (p. 402) diagonal equiangular polygon Chapter 7 (p. 364) Chapter 7 (p. 365) equilateral polygon isosceles trapezoid Chapter 7 (p. 365) Chapter 7 (p. 402) kite legs of a trapezoid Chapter 7 (p. 405) Chapter 7 (p. 402) Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards The parallel sides of a trapezoid Either pair of consecutive angles whose common side is a base of a trapezoid A polygon in which all angles are congruent A segment that joins two nonconsecutive vertices of a polygon A trapezoid with congruent legs A polygon in which all sides are congruent The nonparallel sides of a trapezoid A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards midsegment of a trapezoid parallelogram Chapter 7 (p. 404) Chapter 7 (p. 372) rectangle regular polygon Chapter 7 (p. 392) Chapter 7 (p. 365) rhombus square Chapter 7 (p. 392) Chapter 7 (p. 392) trapezoid Chapter 7 (p. 402) Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards A quadrilateral with both pairs of opposite sides The segment that connects the midpoints of the parallel legs of a trapezoid PQRS A convex polygon that is both equilateral and A parallelogram with four right angles equiangular A parallelogram with four congruent sides and four A parallelogram with four congruent sides right angles A quadrilateral with exactly one pair of parallel sides Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved.
    [Show full text]
  • Angles of Polygons
    5.3 Angles of Polygons How can you fi nd a formula for the sum of STATES the angle measures of any polygon? STANDARDS MA.8.G.2.3 1 ACTIVITY: The Sum of the Angle Measures of a Polygon Work with a partner. Find the sum of the angle measures of each polygon with n sides. a. Sample: Quadrilateral: n = 4 A Draw a line that divides the quadrilateral into two triangles. B Because the sum of the angle F measures of each triangle is 180°, the sum of the angle measures of the quadrilateral is 360°. C D E (A + B + C ) + (D + E + F ) = 180° + 180° = 360° b. Pentagon: n = 5 c. Hexagon: n = 6 d. Heptagon: n = 7 e. Octagon: n = 8 196 Chapter 5 Angles and Similarity 2 ACTIVITY: The Sum of the Angle Measures of a Polygon Work with a partner. a. Use the table to organize your results from Activity 1. Sides, n 345678 Angle Sum, S b. Plot the points in the table in a S coordinate plane. 1080 900 c. Write a linear equation that relates S to n. 720 d. What is the domain of the function? 540 Explain your reasoning. 360 180 e. Use the function to fi nd the sum of 1 2 3 4 5 6 7 8 n the angle measures of a polygon −180 with 10 sides. −360 3 ACTIVITY: The Sum of the Angle Measures of a Polygon Work with a partner. A polygon is convex if the line segment connecting any two vertices lies entirely inside Convex the polygon.
    [Show full text]
  • Some Polygon Facts Student Understanding the Following Facts Have Been Taken from Websites of Polygons
    eachers assume that by the end of primary school, students should know the essentials Tregarding shape. For example, the NSW Mathematics K–6 syllabus states by year six students should be able manipulate, classify and draw two- dimensional shapes and describe side and angle properties. The reality is, that due to the pressure for students to achieve mastery in number, teachers often spend less time teaching about the other aspects of mathematics, especially shape (Becker, 2003; Horne, 2003). Hence, there is a need to modify the focus of mathematics education to incorporate other aspects of JILLIAN SCAHILL mathematics including shape and especially polygons. The purpose of this article is to look at the teaching provides some and learning of polygons in primary classrooms by providing some essential information about polygons teaching ideas and some useful teaching strategies and resources. to increase Some polygon facts student understanding The following facts have been taken from websites of polygons. and so are readily accessible to both teachers and students. “The word ‘polygon’ derives from the Greek word ‘poly’, meaning ‘many’ and ‘gonia’, meaning ‘angle’” (Nation Master, 2004). “A polygon is a closed plane figure with many sides. If all sides and angles of a polygon are equal measures then the polygon is called regular” 30 APMC 11 (1) 2006 Teaching polygons (Weisstein, 1999); “a polygon whose sides and angles that are not of equal measures are called irregular” (Cahir, 1999). “Polygons can be convex, concave or star” (Weisstein, 1999). A star polygon is a figure formed by connecting straight lines at every second point out of regularly spaced points lying on a circumference.
    [Show full text]
  • Convex Polytopes and Tilings with Few Flag Orbits
    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.
    [Show full text]
  • Approximation of Convex Figures by Pairs of Rectangles
    Approximation of Convex Figures byPairs of Rectangles y z x Otfried Schwarzkopf UlrichFuchs Gunter Rote { Emo Welzl Abstract We consider the problem of approximating a convex gure in the plane by a pair (r;R) of homothetic (that is, similar and parallel) rectangles with r C R.We show the existence of such a pair where the sides of the outer rectangle are at most twice as long as the sides of the inner rectangle, thereby solving a problem p osed byPolya and Szeg}o. If the n vertices of a convex p olygon C are given as a sorted array, such 2 an approximating pair of rectangles can b e computed in time O (log n). 1 Intro duction Let C b e a convex gure in the plane. A pair of rectangles (r;R) is called an approximating pair for C ,ifrC Rand if r and R are homothetic, that is, they are parallel and have the same asp ect ratio. Note that this is equivalentto the existence of an expansion x 7! (x x )+x (with center x and expansion 0 0 0 factor ) which maps r into R. We measure the quality (r;R) of our approximating pair (r;R) as the quotient of the length of a side of R divided by the length of the corresp onding side of r . This is just the expansion factor used in the ab ove expansion mapping. The motivation for our investigation is the use of r and R as simple certi cates for the imp ossibility or p ossibility of obstacle-avoiding motions of C .IfRcan b e moved along a path without hitting a given set of obstacles, then this is also p ossible for C .
    [Show full text]