NONCONGRUENT EQUIDISSECTIONS OF THE PLANE
|
|
- Baldric Parrish
- 5 years ago
- Views:
Transcription
1 NONCONGRUENT EQUIDISSECTIONS OF THE PLANE D. FRETTLÖH Abstract. Nandakumar asked whether there is a tiling of the plane b pairwise non-congruent triangles of equal area and equal perimeter. Here a weaker result is obtained: there is a tiling of the plane b pairwise non-congruent triangles of equal area such that their perimeter is bounded b some common constant. Several variants of the problem are stated, some of them are answered.. Introduction There are several problems in Discrete Geometr, old and new, that can be stated easil but are hard to solve. Tilings and dissections provide a large number of such problems, see for instance [CFG, Chapter C]. On his blog [N], R. Nandakumar asked in 04: Question. Can the plane be filled b triangles of same area and perimeter with no two triangles congruent to each other? His webpage [N] contains several further interesting problems of this flavour. The main result of this paper, Theorem, answers a weaker form of the question above affirmativel. Section 4 is dedicated to the statement and the proof of this result, together with its analogues for quadrangles, pentagons and heagons. Section formulates several variants of this problem and gives a sstematic overview. Section contains some basic observations and a first result on a similar result for quadrangles... Notation. A tiling of R is a collection {T, T,...} of compact sets T i (the tiles) that is a packing of R (i.e., the interiors of distinct tiles are disjoint) as well as a covering of R (i.e. the union of the tiles equals R ). In general, tile shapes ma be prett complicated, but for the purpose of this paper tiles are alwas simple conve polgons. A tiling is called verte-to-verte, if the intersection of an two tiles is either an entire edge of both tiles, or a point, or empt. A tiling T is locall finite if an compact set in R intersects onl finitel man tiles of T. A tiling T is normal if there are R > r > 0 such that () each tile in T contains some ball of radius r, and () each tile is contained in some ball of radius R. B [GS,..] we have that each normal tiling is locall finite.. Basic observations In Question above, filled is to be understood in the sense that the plane is covered without overlaps. In other words: is there a tiling of the plane b pairwise noncongruent triangles all having the same area and the same perimeter? If one tries to find a solution one realises that the problem seems to be highl overdetermined. One possibilit to rela the problem is to drop the requirement on the perimeter. So one ma ask Is there a tiling of the plane b pairwise noncongruent triangles all having the same area? It is not too hard to find eamples. One possibilit is to partition the plane into half-strips and divide these half-strips into triangles, as shown in Figure. The image indicates how to fill the right half-plane b half-strips made of triangles. If all halfstrips have different widths then all triangles are distinct. The left half-plane can be filled in an analogous manner. Anwa, this tiling is not locall finite: the upper verte of an half-strip is contained in infinitel man triangles. So one ma ask Is there a locall finite tiling of the plane b pairwise noncongruent triangles of unit area? Even in this stronger form the question was alread Date: March 9, 06.
2 D. FRETTLÖH Figure. Tiling of the plane b pairwise non-congruent triangles of unit area. The perimeters of the triangles are unbounded. Moreover, the tiling is not locall finite. answered b R. Nandakumar. The image in Figure shows a solution, see also [W]. The idea is to dissect the upper right quadrant into triangles of area b zigzagging between the horizontal ais and the vertical ais. The triangles become ver long and thin soon. Nevertheless the are filling the quadrant. For the remaining three quadrants one uses an analogous construction, perturbing the coordinates slightl. (For instance, stretch a cop of the first quadrant b some irrational factor q > in the horizontal direction and shrink it b q in the vertical direction. See [W] for an alternative, more detailed eplanation.) This tiling is locall finite. Nevertheless, this eample is not reall satisfing. More precisel, this tiling is not normal, since in this solution the perimeters of the triangles become arbitrar large. (Hence the inradii become arbitrar small). So it seems natural to ask: Question. Is there a normal tiling of the plane b pairwise noncongruent triangles of unit area? This question was alread asked b Nandakumar, in the form whether there is a tiling of the plane b pairwise noncongruent triangles all having unit area such that the perimeter of the triangles is bounded b some common constant. Theorem below answers this question affirmativel. One possible approach to find a solution is the following. If one can partition a set S R into triangles of unit area such that () S tiles the plane, and () the triangles can be distorted Figure. Tiling of the plane b pairwise non-congruent triangles of area.
3 NONCONGRUENT EQUIDISSECTIONS 4 Figure. Partitioning a square into 4 distinct quadrangles of equal area. There are uncountabl man possibilities for such a dissection. continuousl, in a wa such that all triangles in S are distinct (but still having unit area), this solves the problem. We will illustrate this concept (where triangle is replaced b quadrangle ) in the proof of the following result. Theorem. There is a normal tiling of the plane b pairwise non-congruent conve quadrangles of unit area. Proof. Consider a square Q of edge length. Let Q be a point such that () for the distance d of to the centre of Q holds 0 < d < 0, and () is neither contained in the diagonals of Q nor in the line segments connecting mid-points of opposite edges of Q. Let be a point on the boundar of Q such that for the distance d of to the mid-point of the edge containing holds 0 < d < 0. The choice of and determines three further unique points,, 4 on the boundar of Q such that the line segments i ( i 4) partition Q into four quadrangles of unit area. B the choice of, avoiding the mirror aes of Q, it can alwas be achieved that all quadrangles in a single partition of Q are distinct. (In fact the author believes that no two congruent quadrangles can occur in a partition where is not contained in the mirror aes of Q, but this might be tedious to prove. Here we prefer rather to use the free parameters to achieve that all quadrangles are distinct.) Figure indicates such a partition. The two coordinates determining can be changed continuousl within a small range independentl, ielding two free parameters. One coordinate of can be changed continuousl, too, within some small range. Hence we obtain the desired tiling as follows: Tile the plane R with copies of the square Q. Dissect each cop of Q into four quadrangles of area, in some order. In each dissection, choose and such that the resulting quadrangles have not shown up earlier in the construction. This is alwas possible since, in each step, there are onl finitel man quadrangles constructed alread, whereas there are uncountabl man choices for and. At this point it becomes obvious that Questions and lead to several variants. The eample in the proof above ields a normal tiling, but in general not one that is verte-to-verte. One ma ask the questions for triangles, for quadrangles, for pentagons, and in each case with or without requiring equal perimeter, or normal, or verte-to-verte. The net section aims to give a sstematic overview of the questions.. Variants of the problem The general propert we will require throughout the paper is that a tiling consists of conve tiles of unit area such that all tiles are pairwise distinct. The tiles can be triangles (as in the original question), but also quadrangles, rectangles, pentagons or heagons. We ma or ma not require additionall that all tiles have equal perimeter, or that the perimeter is bounded b some common constant, or that the tilings are normal, or just locall finite. Furthermore, it ma be possible to construct a tiling analogous to the proof of Theorem, that is, b tiling a tile S in infinitel man was, where S in turn can tile the plane. The connections between these properties is shown in the following diagram. } equal perimeter () perimeter is bounded normal locall finite tiling a tile
4 4 D. FRETTLÖH For instance, Equation () tells that if there is tiling obtained b tiling a tile S in infinitel man was, then the perimeters of the tiles in this tiling are bounded b some common constant. In turn, the latter is equivalent to the tiling being normal (since all tiles are conve and have unit area), which in turn implies (b [GS,..]) that the tiling is locall finite. These implications help to give an overview of the several variants of the questions. The following tables list, for each of the cases of triangles, conve quadrangles, conve pentagons, and conve heagons, whether there is some tiling known fulfilling the properties in Equation (), and whether there is such a tiling that is even verte-to-verte. Because of the implications in Equation (), if there is es in some column, then the entries above in the same column contain also a es. Note that not vtv is usuall a weaker condition than vtv, but a tiling b conve heagons that is not vtv is much harder to find than one that is vtv. Triangles vtv not vtv locall finite? es bounded perimeter? es tiling a tile?? Pentagons vtv not vtv locall finite? es bounded perimeter? es tiling a tile? es Quadrangles vtv not vtv locall finite es es bounded perimeter es es tiling a tile es es Heagons vtv not vtv locall finite es? bounded perimeter es? tiling a tile es? Theorem above proves the case quadrangles: tiling a tile, not vtv (thus also the two cases above it in the same column in the corresponding table). Theorem below proves triangles: bounded perimeter, not vtv, Theorem proves quadrangles: tiling a tile, vtv, Theorem 4 proves heagons: tiling a tile, vtv, and Theorem 5 proves heagons: tiling a tile, vtv. 4. Main results Theorem. There is a normal tiling of the plane b pairwise non-congruent triangles of unit area. Proof. The idea of the proof is a refinement of the construction in Figure. Basicall we add additional fault lines in each quadrant. Moreover, we make use of some free parameter in some range, allowing for uncountabl man choices, where in each step of the construction onl finitel man triangle shapes must be avoided. Choose some constant c big enough. This serves as the upper bound on the perimeter of the triangles. For our purposes c = 00 will do. Consider the upper right quadrant Q. Pick a point 0 on the positive horizontal ais with 0 < c. Let T be the unique triangle in Q with vertices 0, 0 and area. (For this and what follows compare Figure 4.) Denote the third verte of T b. Choose on the horizontal ais such that the triangle T with vertices 0, and has area. Continue zigzagging in this wa between horizontal and vertical ais. I.e., choose i+ on the ais not containing i such that the triangle T i+ with vertices i, i, i+ has area. Repeat this until the net triangle T i+ would have perimeter larger than c. Omit T i+. Pick such that the triangle i, i+, has area. There are uncountabl man choices for. For the sake of smmetr let be close to the bisector {(, ) R} of Q. Choose a half-line l proceeding from. There are uncountabl man choices for l. Again, for the sake of smmetr, let l be close to the bisector of Q. Continue b zigzagging in two regions, between the horizontal ais and l, and between the vertical ais and l. I.e., if k is the last point on the horizontal ais, pick t on the horizontal ais such that the triangle k,, t has area. Continue b choosing t on l such that the triangle, t, t has area and so on, until the perimeter of the net triangle t i, t i+, t i+ would be larger than c. Omit this triangle. Choose such that the new triangle t i, t i+, has area. Choose a half-line l proceeding from. Again there are uncountabl man choices for and l.
5 NONCONGRUENT EQUIDISSECTIONS 5 5 t 4 t l l l 0 0 t 4 t Figure 4. Tiling the upper right quadrant Q b pairwise non-congruent triangles of unit area and bounded perimeter. Do the analogous construction in the upper region between l and the vertical ais. Continue in this manner. Whenever a triangle occurs with perimeter larger than c choose a new point k and a new line l k dividing the old region into two. The uncountabilit of choices for k and l k ensures that we can alwas avoid to add a triangle that is congruent to some triangle added earlier. Indeed, whenever we are in the situation to choose k and l k there are at most countabl man triangles constructed alread. Hence k can be chosen such that no triangle with verte k is congruent to an alread constructed one, and l k can be chosen such that no triangle occurring in the two new regions defined b l k is congruent to an alread constructed one. Hence the quadrant Q can be tiled b pairwise non-congruent triangles with area and perimeter less than c. The other quadrants can be tiled accordingl. Whenever a choice of a new point and a new half-line happens there are uncountabl man possibilities, hence all (at most countabl man) alread constructed triangles can be avoided. Theorem. There is a normal vtv tiling of the plane b pairwise non-congruent quadrangles of unit area. The tiling consists of squares that are dissected into four distinct quadrangles of equal area. Proof. The idea is to use the construction in the proof of Theorem, adding (dissected) squares consecutivel, using the degrees of freedom to achieve verte-to-verte in neighbouring squares. Figure 5 indicates the order in which squares are added, and the degrees of freedom in the dissection of each square. Start with some square S, dissected as in the proof of Theorem. There are three degrees of freedom how to dissect S into four quadrangles, two for placing the centre of dissection, one for a point on the boundar. This square is indicated b a circled in Figure 5. Add four more dissected squares adjacent to S, such that the quadrangles are verte-to-verte. These squares are numbers to 5 in the figure. In each of these squares there are still two degrees of freedom for placing the centre. The third parameter is determined uniquel b the verte-to-verte condition. Still one ma use the two degrees of freedom to avoid adding a quadrangles that is congruent to one added alread. Now add four more squares (6 to 9), each one adjacent to two edges of squares to 5, respectivel. Now the position of two points of the dissection are determined for each of the squares 6 to 9. Hence, b the area condition, the centre of the square is restricted to some line. Anwa, it can
6 6 D. FRETTLÖH Figure 5. Tiling a square with distinct quadrangles of unit area (compare Figure ) can be done in a wa such that partitions of adjacent squares are verte-toverte. Circled numbers indicate the order in which squares are added consecutivel, ordinar numbers indicate the degrees of freedom in each square. A means that the centre can be wiggled within a small ball. A means that the centre can be shifted along some line b a small amount. The (approimate) directions of these lines are indicated b dashed line segments. be shifted along a small segment of this line continuousl. Hence there is still one free parameter that we can use to avoid adding a quadrangle that is congruent to some quadrangle added earlier. In this wa we continue filling the plane: add four squares along the horizontal and vertical aes (the net step would be adding squares 0 to in the figure), add more squares to the pattern to complete a square pattern. Proceeding in this wa ensures that in each step there is at least one free parameter that can be used to avoid adding a square congruent to one added earlier. Theorem 4. There is a normal tiling of the plane b pairwise non-congruent pentagons of unit area. The tiling consists of heagons that are dissected into three distinct pentagons of equal area. Proof. A regular heagon of area three can be divided into three heagons of unit area in uncountabl man was, compare the left part of Figure 6. a a Figure 6. A regular heagon can be divided into three distinct pentagons of equal area in uncountabl man was (left). A non-conve 4-gon can be divided into four distinct heagons of equal area in uncountabl man was.
7 NONCONGRUENT EQUIDISSECTIONS 7 Theorem 5. There is a normal vtv tiling of the plane b pairwise non-congruent heagons of unit area. The tiling consists of non-conve 4-gons that are dissected into four distinct heagons of equal area. Proof. Consider a non-conve 4-gon assembled from three regular heagons and a fourth heagon that is obtained from a regular heagon b stretching it slightl in the direction of one of the edges, see Figure 6 right. The longer edges are labelled with a in the figure. This 4-gon can be dissected into four heagons of equal area. There is still one parameter of freedom: one verte of the dissection can be shifted continuousl along a line segment, the other interior verte of the dissection is then determined uniquel b the area condition. The 4-gons ield a tiling of the plane: gluing 4-gons together at their edges of length a ields biinfinite strips. These strips in turn can be assembled into a tiling. During working on the problem the author tried several approaches. Based on this eperience we want to highlight the following problems for further stud. () Is there a compact conve region in the plane that can be tiled b non-congruent triangles of unit area in infinitel man (uncountabl man) was? () Is there a compact region in the plane that (a) can be tiled b non-congruent triangles of unit area in infinitel man (uncountabl man) was, and (b) tiles the plane? () Is there a verte-to-verte tiling of the plane b pairwise non-congruent triangles of unit area? (4) Is there a verte-to-verte tiling of the plane b pairwise non-congruent triangles of unit area such that the perimeter of the triangles is bounded b some common constant? (5) Is there a tiling of the plane b pairwise non-congruent rectangles of unit area such that the perimeter of the rectangles is bounded b some common constant? Acknowledgments The author epresses his gratitude to R. Nandakumar for providing several interesting problems. Special thanks to Jens Schubert for helpful discussions on this topic during a pleasant summer weekend in Bochum. References [CFG] H.T. Croft, K.J. Falconer, R.K. Gu: Unsolved Problems in Geometr, Springer, New York (99). [GS] B. Grünbaum, G.C. Shephard: Tilings and Patterns, W.H. Freeman, New York (987). [N] R. Nandakumar: post from Dec 04 and Jan 05. [W] S. Wagon: Bielefeld Universit, Postfach 00, 50 Bielefeld, German
Translations, Reflections, and Rotations
Translations, Reflections, and Rotations The Marching Cougars Lesson 9-1 Transformations Learning Targets: Perform transformations on and off the coordinate plane. Identif characteristics of transformations
More information2.2 Absolute Value Functions
. Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number
More informationTransformations of Functions. 1. Shifting, reflecting, and stretching graphs Symmetry of functions and equations
Chapter Transformations of Functions TOPICS.5.. Shifting, reflecting, and stretching graphs Smmetr of functions and equations TOPIC Horizontal Shifting/ Translation Horizontal Shifting/ Translation Shifting,
More informationSection 2.2: Absolute Value Functions, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a
Section.: Absolute Value Functions, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license.
More informationMath 1050 Lab Activity: Graphing Transformations
Math 00 Lab Activit: Graphing Transformations Name: We'll focus on quadratic functions to eplore graphing transformations. A quadratic function is a second degree polnomial function. There are two common
More information12.1. Angle Relationships. Identifying Complementary, Supplementary Angles. Goal: Classify special pairs of angles. Vocabulary. Complementary. angles.
. Angle Relationships Goal: Classif special pairs of angles. Vocabular Complementar angles: Supplementar angles: Vertical angles: Eample Identifing Complementar, Supplementar Angles In quadrilateral PQRS,
More informationBEHIND THE INTUITION OF TILINGS
BEHIND THE INTUITION OF TILINGS EUGENIA FUCHS Abstract. It may seem visually intuitive that certain sets of tiles can be used to cover the entire plane without gaps or overlaps. However, it is often much
More informationFair Game Review. Chapter 11. Name Date. Reflect the point in (a) the x-axis and (b) the y-axis. 2. ( 2, 4) 1. ( 1, 1 ) 3. ( 3, 3) 4.
Name Date Chapter Fair Game Review Reflect the point in (a) the -ais and (b) the -ais.. (, ). (, ). (, ). (, ) 5. (, ) 6. (, ) Copright Big Ideas Learning, LLC Name Date Chapter Fair Game Review (continued)
More informationIsometry: When the preimage and image are congruent. It is a motion that preserves the size and shape of the image as it is transformed.
Chapter Notes Notes #36: Translations and Smmetr (Sections.1,.) Transformation: A transformation of a geometric figure is a change in its position, shape or size. Preimage: The original figure. Image:
More information20 Calculus and Structures
0 Calculus and Structures CHAPTER FUNCTIONS Calculus and Structures Copright LESSON FUNCTIONS. FUNCTIONS A function f is a relationship between an input and an output and a set of instructions as to how
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem
More information3.6 Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions
76 CHAPTER Graphs and Functions Find the equation of each line. Write the equation in the form = a, = b, or = m + b. For Eercises through 7, write the equation in the form f = m + b.. Through (, 6) and
More informationy = f(x) x (x, f(x)) f(x) g(x) = f(x) + 2 (x, g(x)) 0 (0, 1) 1 3 (0, 3) 2 (2, 3) 3 5 (2, 5) 4 (4, 3) 3 5 (4, 5) 5 (5, 5) 5 7 (5, 7)
0 Relations and Functions.7 Transformations In this section, we stud how the graphs of functions change, or transform, when certain specialized modifications are made to their formulas. The transformations
More informationtriangles leaves a smaller spiral polgon. Repeating this process covers S with at most dt=e lights. Generaliing the polgon shown in Fig. 1 establishes
Verte -Lights for Monotone Mountains Joseph O'Rourke Abstract It is established that dt=e = dn=e?1 verte -lights suce to cover a monotone mountain polgon of t = n? triangles. A monotone mountain is a monotone
More informationReteaching Golden Ratio
Name Date Class Golden Ratio INV 11 You have investigated fractals. Now ou will investigate the golden ratio. The Golden Ratio in Line Segments The golden ratio is the irrational number 1 5. c On the line
More informationTransformations. which the book introduces in this chapter. If you shift the graph of y 1 x to the left 2 units and up 3 units, the
CHAPTER 8 Transformations Content Summar In Chapter 8, students continue their work with functions, especiall nonlinear functions, through further stud of function graphs. In particular, the consider three
More informationM O T I O N A N D D R A W I N G
2 M O T I O N A N D D R A W I N G Now that ou know our wa around the interface, ou re read to use more of Scratch s programming tools. In this chapter, ou ll do the following: Eplore Scratch s motion and
More informationAppendix C: Review of Graphs, Equations, and Inequalities
Appendi C: Review of Graphs, Equations, and Inequalities C. What ou should learn Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real numbers b points
More informationChapter 9 Transformations
Section 9-1: Reflections SOL: G.2 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving smmetr and transformation.
More informationACTIVITY 9. Learning Targets: 112 SpringBoard Mathematics Geometry, Unit 2 Transformations, Triangles, and Quadrilaterals. Reflection.
Learning Targets: Perform reflections on and off the coordinate plane. Identif reflectional smmetr in plane figures. SUGGESTED LERNING STRTEGIES: Visualization, Create Representations, Predict and Confirm,
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationGlossary alternate interior angles absolute value function Example alternate exterior angles Example angle of rotation Example
Glossar A absolute value function An absolute value function is a function that can be written in the form, where is an number or epression. alternate eterior angles alternate interior angles Alternate
More informationPre-Algebra Notes Unit 13: Angle Relationships and Transformations
Pre-Algebra Notes Unit 13: Angle Relationships and Transformations Angle Relationships Sllabus Objectives: (7.1) The student will identif measures of complementar, supplementar, and vertical angles. (7.2)
More informationGrade 6 Math Circles February 29, D Geometry
1 Universit of Waterloo Facult of Mathematics Centre for Education in Mathematics and Computing Grade 6 Math Circles Feruar 29, 2012 2D Geometr What is Geometr? Geometr is one of the oldest ranches in
More informationIntermediate Algebra. Gregg Waterman Oregon Institute of Technology
Intermediate Algebra Gregg Waterman Oregon Institute of Technolog c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license
More informationAppendix F: Systems of Inequalities
A0 Appendi F Sstems of Inequalities Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit The statements < and are inequalities in two variables. An ordered pair
More informationName Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n =
Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equalit properties of real numbers and inverse operations
More informationConnecting Algebra and Geometry with Polygons
Connecting Algebra and Geometr with Polgons 15 Circles are reall important! Once ou know our wa around a circle, ou can use this knowledge to figure out a lot of other things! 15.1 Name That Triangle!
More informationPolar Functions Polar coordinates
548 Chapter 1 Parametric, Vector, and Polar Functions 1. What ou ll learn about Polar Coordinates Polar Curves Slopes of Polar Curves Areas Enclosed b Polar Curves A Small Polar Galler... and wh Polar
More information2.8 Distance and Midpoint Formulas; Circles
Section.8 Distance and Midpoint Formulas; Circles 9 Eercises 89 90 are based on the following cartoon. B.C. b permission of Johnn Hart and Creators Sndicate, Inc. 89. Assuming that there is no such thing
More information2.4 Coordinate Proof Using Distance with Quadrilaterals
Name Class Date.4 Coordinate Proof Using Distance with Quadrilaterals Essential Question: How can ou use slope and the distance formula in coordinate proofs? Resource Locker Eplore Positioning a Quadrilateral
More informationCS 157: Assignment 6
CS 7: Assignment Douglas R. Lanman 8 Ma Problem : Evaluating Conve Polgons This write-up presents several simple algorithms for determining whether a given set of twodimensional points defines a conve
More informationPoint A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled.
Test Date: November 3, 2016 Format: Scored out of 100 points. 8 Multiple Choice (40) / 8 Short Response (60) Topics: Points, Angles, Linear Objects, and Planes Recognizing the steps and procedures for
More informationThe Graph Scale-Change Theorem
Lesson 3-5 Lesson 3-5 The Graph Scale-Change Theorem Vocabular horizontal and vertical scale change, scale factor size change BIG IDEA The graph of a function can be scaled horizontall, verticall, or in
More informationThree-Dimensional Coordinates
CHAPTER Three-Dimensional Coordinates Three-dimensional movies superimpose two slightl different images, letting viewers with polaried eeglasses perceive depth (the third dimension) on a two-dimensional
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;
More informationImage Metamorphosis By Affine Transformations
Image Metamorphosis B Affine Transformations Tim Mers and Peter Spiegel December 16, 2005 Abstract Among the man was to manipulate an image is a technique known as morphing. Image morphing is a special
More informationLines and Their Slopes
8.2 Lines and Their Slopes Linear Equations in Two Variables In the previous chapter we studied linear equations in a single variable. The solution of such an equation is a real number. A linear equation
More informationGeometry. Origin of Analytic Geometry. Slide 1 / 202 Slide 2 / 202. Slide 4 / 202. Slide 3 / 202. Slide 5 / 202. Slide 6 / 202.
Slide 1 / Slide / Geometr Analtic Geometr 1-- www.njctl.org Slide 3 / Table of Contents Origin of Analtic Geometr The Distance Formula The Midpoint Formula Partitions of a Line Segment Slopes of Parallel
More information8.6 Three-Dimensional Cartesian Coordinate System
SECTION 8.6 Three-Dimensional Cartesian Coordinate Sstem 69 What ou ll learn about Three-Dimensional Cartesian Coordinates Distance and Midpoint Formulas Equation of a Sphere Planes and Other Surfaces
More informationUnit 2: Function Transformation Chapter 1
Basic Transformations Reflections Inverses Unit 2: Function Transformation Chapter 1 Section 1.1: Horizontal and Vertical Transformations A of a function alters the and an combination of the of the graph.
More informationTilings of the plane with unit area triangles of bounded diameter
Tilings of the plane with unit area triangles of bounded diameter Andrey Kupavskii János Pach Gábor Tardos Dedicated to the 70th birthdays of Ted Bisztriczky, Gábor Fejes Tóth, and Endre Makai Abstract
More informationTILING RECTANGLES SIMON RUBINSTEIN-SALZEDO
TILING RECTANGLES SIMON RUBINSTEIN-SALZEDO. A classic tiling problem Question.. Suppose we tile a (large) rectangle with small rectangles, so that each small rectangle has at least one pair of sides with
More informationChain Pattern Scheduling for nested loops
Chain Pattern Scheduling for nested loops Florina Ciorba, Theodore Andronikos and George Papakonstantinou Computing Sstems Laborator, Computer Science Division, Department of Electrical and Computer Engineering,
More informationComputer Graphics. Modelling in 2D. 2D primitives. Lines and Polylines. OpenGL polygon primitives. Special polygons
Computer Graphics Modelling in D Lecture School of EECS Queen Mar, Universit of London D primitives Digital line algorithms Digital circle algorithms Polgon filling CG - p.hao@qmul.ac.uk D primitives Line
More informationMATH STUDENT BOOK. 10th Grade Unit 9
MATH STUDENT BOOK 10th Grade Unit 9 Unit 9 Coordinate Geometr MATH 1009 Coordinate Geometr INTRODUCTION 3 1. ORDERED PAIRS 5 POINTS IN A PLANE 5 SYMMETRY 11 GRAPHS OF ALGEBRAIC CONDITIONS 19 SELF TEST
More informationangle The figure formed by two lines with a common endpoint called a vertex. angle bisector The line that divides an angle into two equal parts.
A angle The figure formed b two lines with a common endpoint called a verte. verte angle angle bisector The line that divides an angle into two equal parts. circle A set of points that are all the same
More informationThe Farey Tessellation
The Farey Tessellation Seminar: Geometric Structures on manifolds Mareike Pfeil supervised by Dr. Gye-Seon Lee 15.12.2015 Introduction In this paper, we are going to introduce the Farey tessellation. Since
More informationModule 2, Section 2 Graphs of Trigonometric Functions
Principles of Mathematics Section, Introduction 5 Module, Section Graphs of Trigonometric Functions Introduction You have studied trigonometric ratios since Grade 9 Mathematics. In this module ou will
More informationMF9SB_CH08_p pp7.qxd 4/10/09 11:44 AM Page NEL
362 NEL Chapter 8 Smmetr GOLS You will be able to identif and appl line smmetr identif and appl rotation smmetr relate smmetr to transformations solve problems b using diagrams Smmetr is often seen in
More information4.2 Properties of Rational Functions. 188 CHAPTER 4 Polynomial and Rational Functions. Are You Prepared? Answers
88 CHAPTER 4 Polnomial and Rational Functions 5. Obtain a graph of the function for the values of a, b, and c in the following table. Conjecture a relation between the degree of a polnomial and the number
More informationShifting, Reflecting, and Stretching Graphs
Shifting, Reflecting, and Stretching s Shifting s 1 ( ) ( ) This is f ( ) This is f ( ) This is f ( ) What happens to the graph? f ( ) is f () shifted units to the right. f ( ) is f () shifted units to
More informationCourse Number: Course Title: Geometry
Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line
More informationGEOMETRY CURRICULUM MAP
2017-2018 MATHEMATICS GEOMETRY CURRICULUM MAP Department of Curriculum and Instruction RCCSD Congruence Understand congruence in terms of rigid motions Prove geometric theorems Common Core Major Emphasis
More informationANGLES See the Math Notes boxes in Lessons and for more information about angle relationships.
CC1 Basic Definitions Defense Practice ANGLES 2.1.1 2.1.5 Applications of geometr in everda settings often involve the measures of angles. In this chapter we begin our stud of angle measurement. After
More information5 and Parallel and Perpendicular Lines
Ch 3: Parallel and Perpendicular Lines 3 1 Properties of Parallel Lines 3 Proving Lines Parallel 3 3 Parallel and Perpendicular Lines 3 Parallel Lines and the Triangle Angles Sum Theorem 3 5 The Polgon
More information12.4 The Ellipse. Standard Form of an Ellipse Centered at (0, 0) (0, b) (0, -b) center
. The Ellipse The net one of our conic sections we would like to discuss is the ellipse. We will start b looking at the ellipse centered at the origin and then move it awa from the origin. Standard Form
More information(ii) Explain how the trapezium rule could be used to obtain a more accurate estimate of the area. [1]
C Integration. June 00 qu. Use the trapezium rule, with strips each of width, to estimate the area of the region bounded by the curve y = 7 +, the -ais, and the lines = and = 0. Give your answer correct
More informationImplicit differentiation
Roberto s Notes on Differential Calculus Chapter 4: Basic differentiation rules Section 5 Implicit differentiation What ou need to know alread: Basic rules of differentiation, including the chain rule.
More informationLINEAR PROGRAMMING. Straight line graphs LESSON
LINEAR PROGRAMMING Traditionall we appl our knowledge of Linear Programming to help us solve real world problems (which is referred to as modelling). Linear Programming is often linked to the field of
More informationGraphing Quadratics: Vertex and Intercept Form
Algebra : UNIT Graphing Quadratics: Verte and Intercept Form Date: Welcome to our second function famil...the QUADRATIC FUNCTION! f() = (the parent function) What is different between this function and
More informationGraphing Equations Case 1: The graph of x = a, where a is a constant, is a vertical line. Examples a) Graph: x = x
06 CHAPTER Algebra. GRAPHING EQUATIONS AND INEQUALITIES Tetbook Reference Section 6. &6. CLAST OBJECTIVE Identif regions of the coordinate plane that correspond to specific conditions and vice-versa Graphing
More informationSECTION 3-4 Rational Functions
20 3 Polnomial and Rational Functions 0. Shipping. A shipping bo is reinforced with steel bands in all three directions (see the figure). A total of 20. feet of steel tape is to be used, with 6 inches
More informationSect Linear Inequalities in Two Variables
Sect 9. - Linear Inequalities in Two Variables Concept # Graphing a Linear Inequalit in Two Variables Definition Let a, b, and c be real numbers where a and b are not both zero. Then an inequalit that
More information14-1. Translations. Vocabulary. Lesson
Chapter 1 Lesson 1-1 Translations Vocabular slide, translation preimage translation image congruent figures Adding fied numbers to each of the coordinates of a figure has the effect of sliding or translating
More informationUnit 4 Trigonometry. Study Notes 1 Right Triangle Trigonometry (Section 8.1)
Unit 4 Trigonometr Stud Notes 1 Right Triangle Trigonometr (Section 8.1) Objective: Evaluate trigonometric functions of acute angles. Use a calculator to evaluate trigonometric functions. Use trigonometric
More informationFractal Tilings Based on Dissections of Polyominoes, Polyhexes, and Polyiamonds
Fractal Tilings Based on Dissections of Polyominoes, Polyhexes, and Polyiamonds Robert W. Fathauer Abstract Fractal tilings ("f-tilings") are described based on single prototiles derived from dissections
More information8.5 Quadratic Functions and Their Graphs
CHAPTER 8 Quadratic Equations and Functions 8. Quadratic Functions and Their Graphs S Graph Quadratic Functions of the Form f = + k. Graph Quadratic Functions of the Form f = - h. Graph Quadratic Functions
More informationPartial Fraction Decomposition
Section 7. Partial Fractions 53 Partial Fraction Decomposition Algebraic techniques for determining the constants in the numerators of partial fractions are demonstrated in the eamples that follow. Note
More information1. y = f(x) y = f(x + 3) 3. y = f(x) y = f(x 1) 5. y = 3f(x) 6. y = f(3x) 7. y = f(x) 8. y = f( x) 9. y = f(x 3) + 1
.7 Transformations.7. Eercises To see all of the help resources associated with this section, click OSttS Chapter b. Suppose (, ) is on the graph of = f(). In Eercises - 8, use Theorem.7 to find a point
More informationPROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS
Topic 21: Problem solving with eponential functions 323 PROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS Lesson 21.1 Finding function rules from graphs 21.1 OPENER 1. Plot the points from the table onto the
More informationSTRAND I: Geometry and Trigonometry. UNIT 37 Further Transformations: Student Text Contents. Section Reflections. 37.
MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet ontents STRN I: Geometr and Trigonometr Unit 7 Further Transformations Student Tet ontents Section 7. Reflections 7. Rotations 7. Translations
More informationGeometry Vocabulary. acute angle-an angle measuring less than 90 degrees
Geometry Vocabulary acute angle-an angle measuring less than 90 degrees angle-the turn or bend between two intersecting lines, line segments, rays, or planes angle bisector-an angle bisector is a ray that
More information9. f(x) = x f(x) = x g(x) = 2x g(x) = 5 2x. 13. h(x) = 1 3x. 14. h(x) = 2x f(x) = x x. 16.
Section 4.2 Absolute Value 367 4.2 Eercises For each of the functions in Eercises 1-8, as in Eamples 7 and 8 in the narrative, mark the critical value on a number line, then mark the sign of the epression
More informationPolygons in the Coordinate Plane
. Polgons in the Coordinate Plane How can ou find the lengths of line segments in a coordinate plane? ACTIVITY: Finding Distances on a Map Work with a partner. The coordinate grid shows a portion of a
More informationscience. In this course we investigate problems both algebraically and graphically.
Section. Graphs. Graphs Much of algebra is concerned with solving equations. Man algebraic techniques have been developed to provide insights into various sorts of equations and those techniques are essential
More information14 Loci and Transformations
MEP Pupil Tet 1 1 Loci and Transformations 1.1 rawing and Smmetr This section revises the ideas of smmetr first introduced in Unit and gives ou practice in drawing simple shapes. Worked Eample 1 escribe
More informationWhat and Why Transformations?
2D transformations What and Wh Transformations? What? : The geometrical changes of an object from a current state to modified state. Changing an object s position (translation), orientation (rotation)
More informationEXTREME POINTS AND AFFINE EQUIVALENCE
EXTREME POINTS AND AFFINE EQUIVALENCE The purpose of this note is to use the notions of extreme points and affine transformations which are studied in the file affine-convex.pdf to prove that certain standard
More informationThe Graph of an Equation
60_0P0.qd //0 :6 PM Page CHAPTER P Preparation for Calculus Archive Photos Section P. RENÉ DESCARTES (96 60) Descartes made man contributions to philosoph, science, and mathematics. The idea of representing
More informationAppendix F: Systems of Inequalities
Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit What ou should learn The statements < and ⱖ are inequalities in two variables. An ordered pair 共a, b兲 is a
More informationGeometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review
Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Polygon a closed plane figure with at least 3 sides that are segments -the sides do not intersect except at the vertices N-gon -
More informationAnswers. Investigation 4. ACE Assignment Choices. Applications
Answers Investigation ACE Assignment Choices Problem. Core Other Connections, ; Etensions ; unassigned choices from previous problems Problem. Core, 7 Other Applications, ; Connections ; Etensions ; unassigned
More informationSummary Of Topics covered in Year 7. Topic All pupils should Most pupils should Some pupils should Learn formal methods for
Summary Of Topics covered in Year 7 Topic All pupils should Most pupils should Some pupils should Learn formal methods for Have a understanding of computing multiplication Use the order of basic number
More informationAlgebra I. Linear Equations. Slide 1 / 267 Slide 2 / 267. Slide 3 / 267. Slide 3 (Answer) / 267. Slide 4 / 267. Slide 5 / 267
Slide / 67 Slide / 67 lgebra I Graphing Linear Equations -- www.njctl.org Slide / 67 Table of ontents Slide () / 67 Table of ontents Linear Equations lick on the topic to go to that section Linear Equations
More informationPolynomials. Math 4800/6080 Project Course
Polnomials. Math 4800/6080 Project Course 2. The Plane. Boss, boss, ze plane, ze plane! Tattoo, Fantas Island The points of the plane R 2 are ordered pairs (x, ) of real numbers. We ll also use vector
More informationMATHEMATICAL METHODS for Scientists and Engineers
Chapter 18 Figures From MATHEMATICAL METHODS for Scientists and Engineers Donald A. McQuarrie For the Novice Acrobat User or the Forgetful When ou opened this file ou should have seen a slightl modified
More informationThe Cartesian plane 15B
Weblink attleship Game e A bishop can move diagonall an number of (unoccupied) squares at a time. omplete the following sentence: With its net move, the bishop at b could capture the........ (name the
More informationTransforming Coordinates
# Transforming Coordinates The drawing window in man computer geometr programs is a coordinate grid. You make designs b specifing the endpoints of line segments. When ou transform a design, the coordinates
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 5. Graph sketching
Roberto s Notes on Differential Calculus Chapter 8: Graphical analsis Section 5 Graph sketching What ou need to know alread: How to compute and interpret limits How to perform first and second derivative
More informationL3 Rigid Motion Transformations 3.1 Sequences of Transformations Per Date
3.1 Sequences of Transformations Per Date Pre-Assessment Which of the following could represent a translation using the rule T (, ) = (, + 4), followed b a reflection over the given line? (The pre-image
More informationAnswers to Exercises 11.
CHAPTER 7 CHAPTER LESSON 7.1 CHAPTER 7 CHAPTER 1. Rigid; reflected, but the size and shape do not change. 2. Nonrigid; the shape changes. 3. Nonrigid; the size changes. 4.. 6. 7 7. possible answer: a boat
More informationLesson 5.3 Exercises, pages
Lesson 5.3 Eercises, pages 37 3 A. Determine whether each ordered pair is a solution of the quadratic inequalit: 3 - a) (-3, ) b) (, 5) Substitute each ordered pair in» 3. L.S. ; R.S.: 3( 3) 3 L.S. 5;
More informationRotate. A bicycle wheel can rotate clockwise or counterclockwise. ACTIVITY: Three Basic Ways to Move Things
. Rotations object in a plane? What are the three basic was to move an Rotate A biccle wheel can rotate clockwise or counterclockwise. 0 0 0 9 9 9 8 8 8 7 6 7 6 7 6 ACTIVITY: Three Basic Was to Move Things
More informationCSC Computer Graphics
7//7 CSC. Computer Graphics Lecture Kasun@dscs.sjp.ac.l Department of Computer Science Universit of Sri Jaewardanepura Line drawing algorithms DDA Midpoint (Bresenham s) Algorithm Circle drawing algorithms
More informationCHAPTER 5 LESSON 5.1 EXERCISES. 84 CHAPTER 5 Discovering Geometry Solutions Manual 2008 Kendall Hunt Publishing
LESSON.1 CHAPTER EXERCISES 1. (See table at bottom of page.) For each given value of n (the number of sides of the polgon), calculate 180 (n ) (the sum of the measures of the interior angles).. (See table
More informationThink About. Unit 5 Lesson 3. Investigation. This Situation. Name: a Where do you think the origin of a coordinate system was placed in creating this
Think About This Situation Unit 5 Lesson 3 Investigation 1 Name: Eamine how the sequence of images changes from frame to frame. a Where do ou think the origin of a coordinate sstem was placed in creating
More informationNotes #36: Solving Ratios and Proportions and Similar Triangles (Sections 7.1 and 7.2) , 3 to 4, 3:4
Name: Geometr Rules! Period: Chapter 7 Notes - 1 - Notes #3: Solving Ratios and Proportions and Similar Triangles (Sections 7.1 and 7.) Ratio: a comparison of two quantities. 3, 3 to, 3: Proportion: two
More informationToday s class. Geometric objects and transformations. Informationsteknologi. Wednesday, November 7, 2007 Computer Graphics - Class 5 1
Toda s class Geometric objects and transformations Wednesda, November 7, 27 Computer Graphics - Class 5 Vector operations Review of vector operations needed for working in computer graphics adding two
More informationMatrix Representations
CONDENSED LESSON 6. Matri Representations In this lesson, ou Represent closed sstems with transition diagrams and transition matrices Use matrices to organize information Sandra works at a da-care center.
More information