Flat Zipper-Unfolding Pairs for Platonic Solids
|
|
- Marsha Pitts
- 5 years ago
- Views:
Transcription
1 Flat Zipper-Unfolding Pairs for Platonic Solids Joseph O Rourke October, 00 Abstract We show that four of the five Platonic solids surfaces ma be cut open with a Hamiltonian path along edges and unfolded to a polgonal net each of which can zipper-refold to a flat doubl covered parallelogram, forming a rather compact representation of the surface. Thus these regular polhedra have particular flat zipper pairs. No such zipper pair eists for a dodecahedron, whose Hamiltonian unfoldings are zip-rigid. This report is primaril an inventor of the possibilities, and raises more questions than it answers. Introduction It has been known since the time of Aleandrov and it was certainl known to him that the surface of a polhedron could sometimes be cut open to a net and refolded to a doubl covered polgon, which we will henceforth call a flat polhedron. Such flat polhedra are eplicitl countenanced in Aleandrov s 9 gluing theorem. Perhaps the first specific eample of this possibilit occurred in [LO9], which included the eample illustrated in Figure : the familiar Latin-cross unfolding of the cube ma be refolded to a flat conve quadrilateral polhedron. This is one of the two flat conve polhedron that ma be folded from the Latin cross [DO07, Fig..]. Let us sa that two polhedra Q and Q form a net pair if the ma be unfolded to a common polgonal net. In Figure, the cube is cut along edges to unfold to the Latin cross polgon, but the flat quadrilateral must have face cuts through the interior of its two faces to unfold to the same Latin cross. In general there is little understanding of which polhedra form net pairs. See, for eample, Open Problem. in [DO07]. Here we eplore a narrow question on net pairs, narrow enough to obtain a complete answer. The cuts to unfold a conve polhedron to a single polgon form a spanning tree of the polhedron s vertices [DO07, Sec...]. Shephard eplored the special case where the spanning tree is a Hamiltonian path of the -skeleton Department of Computer Science, Smith College, Northampton, MA 00, USA. orourke@cs.smith.edu. See [DO07, Sec..] and [Pak0, Sec. 7] for descriptions of this theorem.
2 Figure : Folding the Latin cross cube net to a flat quadrilateral polhedron. Points with the same label in are identified in the refolding. of the polhedron, i.e., all cuts are along polhedron edges [She7]. The result is a Hamiltonian unfolding of the polhedron. (Note the cube unfolding that produces Figure is not a Hamiltonian unfolding: the cut tree has four leaves.) Some combinatorial questions on Hamiltonian unfoldings were eplored in [DDLO0]; see [DO07, Fig..9 ]. In particular, there are polhedra that have an eponential number of combinatoriall distinct Hamiltonian unfoldings: Ω(n) for a polhedron with n vertices. Another variant is provided b the class of perimeter-halving foldings [DO07, Sec... ], which correspond to spanning cut paths that ma emplo face cuts rather than solel following polhedron edges. In [LDD + 0] these paths were memorabl rechristened as zipper paths, producing zipper unfoldings. We will adopt that nomenclature, including the verbs zip and unzip to mean folding and unfolding (respectivel) along zipper paths. We reserve Hamiltonian path to be a zipper path along polhedron edges. Finall, if two polhedra each unzip to a common polgonal net, we sa the form a zipper pair. The narrow question we eplore is this: Question: Does each of the Platonic solids form a zipper pair with a flat conve polhedron, with the zipper path on the regular polhedron forming a Hamiltonian path of its edges? We show that the tetrahedron, the cube, the octahedron, and the icosahedron all form such zipper pairs with flat parallelogram polhedra. The dodecahedron has no such zipper mate. Note that it would be too restrictive to insist We drop the modifier regular to shorten the names of the five regular polhedra.
3 that both zippers are Hamiltonian paths of the -skeletons, because for a flat polhedron, the -skeleton is the single ccle bounding the polgon, and so a Hamiltonian unfolding is just two copies of the conve polgon joined along one edge. Flat Zipper Pairs. Tetrahedron The regular tetrahedron has onl one Hamiltonian path (up to smmetries), which unfolds to the parallelogram shown in Figure (in Fig. in [LDD + 0]). Because this net is a conve polgon, Thm... in [DO07] establishes that it has an infinite number of zippings to various conve polhedra. The zipping shown in Figure (c) folds it to a doubl covered rhombus. (c) Figure : The Hamiltonian cut path on a tetrahedron leads to the Hamiltonian unfolding, which zips from to (identifing the labeled points) to a flat rhombus polhedron of side length (c).. Cube The cube has three distinct Hamiltonian unfoldings (Fig. in [LDD + 0]): one with the path endpoints at opposite cube corners, and two with the path endpoints at either end of a cube edge. One of the latter (shown in Figure ) produces a T -shape that has no zippings ecept back to the cube. We call such a zipper unfolding zip-rigid. We defer an eplanation of how it is known that this unfolding is zip-rigid to Section. below.
4 Figure : The first Hamiltonian cut path leads to a zip-rigid Hamiltonian T -unfolding of the cube. The other two Hamiltonian unfoldings of the cube, which we call the S - and the Z -unfoldings, both zip to the same doubl covered parallelogram, as shown in Figures and. An animation of the S -folding is shown in Figure.. A Zipping Algorithm Let P be a polgon, the polgonal net corresponding to a zip-pair of polhedra Q and Q. Each of the two zippings of P are perimeter-halving foldings, with the endpoints of the zip path bisecting the perimeter. If we normalize the perimeter of P to and parametrize it from 0 to, we can view the two zippings abstractl as in Figure 7. One of the zip-path endpoints are at 0 and, and the other zip-path endpoints are at and = +. We seek to find all the locations that determine a zipping to some conve polhedron. As previousl mentioned, if P is conve, then ever determines a conve polhedron (Thm... in [DO07]), so we henceforth eclude that case. If P is not conve, it has at least one refle verte v with internal angle β > π. Now there are onl two options at v: () v can serve as, so the zipping starts at v = ; or () Some strictl conve verte u i whose internal angle α i satisfies α i + β π is glued to v. If more than one verte is glued to v, then the folding would not be a zipping, as v would then constitute a junction of degree > in the gluing tree ([DO07, Sec..]). Note that if u i glues to v, then is determined: halfwa between u i and v along the perimeter of P. Thus we onl need tr each u i in turn, and check that Aleandrov s conditions hold for the uniquel determined zipping [DO07, Thm...]). This incidentall shows that an P with a refle verte admits onl O(n) zippings. For eample, appling this algorithm to the cube Z -unfolding in Figure results in si zippings: two copies of the one shown in that figure, two copies of a tetrahedron, one -verte and one -verte polhedron. Although this provides a linear-time algorithm for determining all zippings of P, it does not tell us which of these zippings lead to flat polhedra. Although
5 (c) Figure : The second Hamiltonian cut path on a cube. The resulting Hamiltonian S unfolding. (c) Zipped according to the indicated point identifications to a parallelogram polhedron of side lengths and.
6 Snapshots from an animation folding the parallelogram in Fig- Figure : ure (b,c).
7 (c) Figure : The third Hamiltonian cut path on a cube. The resulting Hamiltonian Z unfolding. (c) Zipped according to the indicated point identifications to a parallelogram polhedron of side lengths and. ½ 0 Figure 7: A zip-pair, abstractl. The perimeter has been normalized to. 7
8 there is an O(n ) algorithm for deciding if an Aleandrov gluing is flat [O R0], this remains unimplemented. We resorted to manual folding of the zippings.. Octahedron The octahedron also has three distinct Hamiltonian paths, one between the top and bottom vertices (separated b distance in the -skeleton), and two paths between adjacent (distance-) vertices. The first Hamiltonian unfolding both zips to a rectangle as shown in Figure 8, and zips to a parallelogram, Figure 9. I find the rectangle zipping especiall surprising, as it derives from a shape all of whose angles are multiples of π/ = 0. (c) Figure 8: Hamiltonian cut path on an octahedron. Its corresponding Hamiltonian unfolding. (c) Zipping folds it to a flat doubl covered rectangle of dimensions. One of the other Hamiltonian unfoldings of the octahedron, shown in Figure 0, zips to a parallelogram. The other Hamiltonian unfolding does not have This is natural because the cube and octahedron are duals. However, it is shown in [LDD + 0, Fig. ] that the dual of a Hamiltonian unfolding is not necessaril a Hamiltonian path through the faces of that unfolding. 8
9 Figure 9: Another zipping of the same unfolding from Figure 8 leads to a parallelogram polhedron. a flat zipping, although it does have zippings, e.g., to a tetrahedron all four of whose vertices have curvature π. I cannot resist mentioning that this last net folds to a flat rectangular polhedron, whose cut tree, however, is not a zipping: Figure.. Dodecahedron Ever Hamiltonian unfolding of the dodecahedron is zip-rigid, and therefore it has no flat zip pair in the sense posed in our Question above. The reason is as follows. Let and be the endpoints of the Hamiltonian path that unfolds the dodecahedron. Then the refle angle of the net at and is π =, leaving an eternal angle of there. The smallest conve angle in an edge unfolding of the dodecahedron is π = 08, so no verte can glue into or. Therefore, a zipping must zip at and, leading directl back to the dodecahedron. We should mention that loosening the criteria posed in our Question leads to a flat refolding of a Hamiltonian net for the dodecahedron. Figure illustrates one such, using the unfolding in Fig. in [LDD + 0]. Here the refolding in Figure (c) is neither conve nor a zipper folding. 9
10 (c) Figure 0: Hamiltonian cut path on an octahedron. Its corresponding Hamiltonian unfolding. (c) Zipping folds it to a flat doubl covered parallelogram of dimensions. 0
11 Figure : The same Hamiltonian unfolding from Figure 0 folds to a doubl covered rectangle, but this folding is not a zipping.
12 (c) a b Figure : Hamiltonian cut path on a dodecahedron. Its corresponding Hamiltonian unfolding. (c) A non-zipper refolding to a doubl covered flat nonconve polgon. The cut tree has degree at vertices a and b.. Icosahedron For the tetrahedron, cube, and octahedron, it was eas to eplore all the Hamiltonian unfoldings, because there are so few (,, and respectivel). The icosahedron, however, has hundreds of Hamiltonian unfoldings. At this writing, I do not know precisel how man geometricall distinct Hamiltonian unfoldings it possesses. The diameter of the icosahedral graph is, so the end points of a Hamiltonian path are a distance,, or apart. Fiing two vertices separated b a distance d {,, }, I found that there are, respectivel,, 08, and 70 labeled Hamiltonian paths between them. Of course not all these labeled paths are distinct geometric paths because of smmetries. However, I have not carried out the more difficult enumeration of the number of geometricall distinct (incongruent as paths in R ) Hamiltonian paths on an icosahedron. But certainl this number is less than = 80. For each of these 80 Hamiltonian unfoldings, I ran the zipping algorithm in Section., which determined that 8 of the unfoldings had at least one zipping, while all the others are zip-rigid (8 = in the three classes, respectivel). B visual inspection, it appears that of these Hamiltonian It is a curious fact that the number of labeled Hamiltonian ccles through an fied edge is 9 =. The simplicit of this epression suggests there might be a combinatorial eplanation, a question I asked on MathOverflow,
13 unfoldings are distinct; the are displaed in Figure. Figure : The distinct Hamiltonian unfolding of the icosahedron that each have at least one zipping to another conve polhedron (Not all are displaed to the same scale.) At least one of these unfoldings (the leftmost in the first row) zips to a parallelogram, as shown in Figure. At this writing we are uncertain if this is the onl zipping to a flat polhedron among the zipping unfoldings. Future Work As is evident from the foregoing, there is little theor behind the unfoldings detailed here. The central open problem is to gain more insight into which polhedra are net pairs, or more specificall, zipper pairs. Perhaps intuition can be strengthened b tackling specific subquestions that fall under this general umbrella. It is eas to list such questions, all of are open because of the lack of a general theor. For eample, the Hamiltonian unfoldings of the Archimedean solids detailed in [LDD + 0] could be eplored. An interesting specific but tangential question raised b this work is to determine the eact number of geometricall distinct Hamiltonian paths on a regular icosahedron. Acknowledgments. I thank Stephanie Annessi and Katherine Lipow for help in enumerating and folding the icosahedron Hamiltonian unfoldings. References [DDLO00] Erik D. Demaine, Martin L. Demaine, Anna Lubiw, and Joseph O Rourke. Eamples, countereamples, and enumeration re-
14 (c) Figure : Hamiltonian cut path on an icosahedron. Its corresponding Hamiltonian unfolding. (c) Rezipping folds it to a flat doubl covered parallelogram of side lengths and.
15 sults for foldings and unfoldings between polgons and poltopes. Technical Report 09, Smith College, Northampton, Jul 000. arxiv:cs.cg/ [DDLO0] Erik D. Demaine, Martin L. Demaine, Anna Lubiw, and Joseph O Rourke. Enumerating foldings and unfoldings between polgons and poltopes. Graphs and Combin., 8():9 0, 00. See also [DDLO00]. [DO07] Erik D. Demaine and Joseph O Rourke. Geometric Folding Algorithms: Linkages, Origami, Polhedra. Cambridge Universit Press, Jul [LDD + 0] Anna Lubiw, Erik Demaine, Martin Demaine, Arlo Shallit, and Jonah Shallit. Zipper unfoldings of polhedral complees. In Proc. nd Canad. Conf. Comput. Geom., pages 9, August 00. [LO9] [O R0] Anna Lubiw and Joseph O Rourke. When can a polgon fold to a poltope? Technical Report 08, Dept. Comput. Sci., Smith College, June 99. Presented at Amer. Math. Soc. Conf., Oct. 99. Joseph O Rourke. On flat polhedra deriving from aleandrov s theorem. Jul 00. [Pak0] Igor Pak. Lectures on discrete and polhedral geometr. http: // 00. [She7] Geoffre C. Shephard. Conve poltopes with conve nets. Math. Proc. Camb. Phil. Soc., 78:89 0, 97.
Zipper Unfolding of Domes and Prismoids
CCCG 2013, Waterloo, Ontario, August 8 10, 2013 Zipper Unfolding of Domes and Prismoids Erik D. Demaine Martin L. Demaine Ryuhei Uehara Abstract We study Hamiltonian unfolding cutting a convex polyhedron
More information8. BOOLEAN ALGEBRAS x x
8. BOOLEAN ALGEBRAS 8.1. Definition of a Boolean Algebra There are man sstems of interest to computing scientists that have a common underling structure. It makes sense to describe such a mathematical
More information11.4 Polar Coordinates
11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.
More informationHamiltonicity and Fault Tolerance
Hamiltonicit and Fault Tolerance in the k-ar n-cube B Clifford R. Haithcock Portland State Universit Department of Mathematics and Statistics 006 In partial fulfillment of the requirements of the degree
More informationLinear Equation Theory - 2
Algebra Module A46 Linear Equation Theor - Copright This publication The Northern Alberta Institute of Technolog 00. All Rights Reserved. LAST REVISED June., 009 Linear Equation Theor - Statement of Prerequisite
More informationApplications. 12 The Shapes of Algebra. 1. a. Write an equation that relates the coordinates x and y for points on the circle.
Applications 1. a. Write an equation that relates the coordinates and for points on the circle. 1 8 (, ) 1 8 O 8 1 8 1 (13, 0) b. Find the missing coordinates for each of these points on the circle. If
More informationModule 3, Section 4 Analytic Geometry II
Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationUnit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents
Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE
More information2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
More informationReview Topics for MATH 1400 Elements of Calculus Table of Contents
Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical
More informationOn Range and Reflecting Functions About the Line y = mx
On Range and Reflecting Functions About the Line = m Scott J. Beslin Brian K. Heck Jerem J. Becnel Dept.of Mathematics and Dept. of Mathematics and Dept. of Mathematics and Computer Science Computer Science
More information10.3 Solving Nonlinear Systems of Equations
60 CHAPTER 0 Conic Sections Identif whether each equation, when graphed, will be a parabola, circle, ellipse, or hperbola. Then graph each equation.. - 7 + - =. = +. = + + 6. + 9 =. 9-9 = 6. 6 - = 7. 6
More informationQuick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)
Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,
More information11.1 Double Riemann Sums and Double Integrals over Rectangles
Chapter 11 Multiple Integrals 11.1 ouble Riemann Sums and ouble Integrals over Rectangles Motivating Questions In this section, we strive to understand the ideas generated b the following important questions:
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationStrain Transformation and Rosette Gage Theory
Strain Transformation and Rosette Gage Theor It is often desired to measure the full state of strain on the surface of a part, that is to measure not onl the two etensional strains, and, but also the shear
More informationGeneralized Least-Squares Regressions III: Further Theory and Classication
Generalized Least-Squares Regressions III: Further Theor Classication NATANIEL GREENE Department of Mathematics Computer Science Kingsborough Communit College, CUNY Oriental Boulevard, Brookln, NY UNITED
More informationAnalytic Geometry in Three Dimensions
Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used
More information2009 Math Olympics Level I
Saginaw Valle State Universit 009 Math Olmpics Level I. A man and his wife take a trip that usuall takes three hours if the drive at an average speed of 60 mi/h. After an hour and a half of driving at
More informationFIRST- AND SECOND-ORDER IVPS The problem given in (1) is also called an nth-order initial-value problem. For example, Solve: Solve:
.2 INITIAL-VALUE PROBLEMS 3.2 INITIAL-VALUE PROBLEMS REVIEW MATERIAL Normal form of a DE Solution of a DE Famil of solutions INTRODUCTION We are often interested in problems in which we seek a solution
More informationGlossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
More informationChapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
More information8.1 Exponents and Roots
Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents
More informationUNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
More informationMultiple Choice. 3. The polygons are similar, but not necessarily drawn to scale. Find the values of x and y.
Accelerated Coordinate Algebra/Analtic Geometr answers the question. Page 1 of 5 Multiple Choice 1. The dashed triangle is an image of the solid triangle. What is the scale factor of the image?. The polgons
More information3.7 InveRSe FUnCTIOnS
CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
More informationIntroduction to Differential Equations
Introduction to Differential Equations. Definitions and Terminolog.2 Initial-Value Problems.3 Differential Equations as Mathematical Models Chapter in Review The words differential and equations certainl
More informationQ.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or
STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R
More informationNo. For example, f(0) = 3, but f 1 (3) 0. Kent did not follow the order of operations when undoing. The correct inverse is f 1 (x) = x 3
Lesson 10.1.1 10-6. a: Each laer has 7 cubes, so the volume is 42 cubic units. b: 14 6 + 2 7 = 98 square units c: (1) V = 20 units 3, SA = 58 units 2 (2) V = 24 units 3, SA = 60 units 2 (3) V = 60 units
More informationFundamentals of Algebra, Geometry, and Trigonometry. (Self-Study Course)
Fundamentals of Algebra, Geometry, and Trigonometry (Self-Study Course) This training is offered eclusively through the Pennsylvania Department of Transportation, Business Leadership Office, Technical
More informationMath 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals
Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (
More informationDiagnostic Assessment Number and Quantitative Reasoning
Number and Quantitative Reasoning Select the best answer.. Which list contains the first four multiples of 3? A 3, 30, 300, 3000 B 3, 6, 9, 22 C 3, 4, 5, 6 D 3, 26, 39, 52 2. Which pair of numbers has
More informationabsolute value The distance of a number from zero on a real number line.
G L O S S A R Y A absolute value The distance of a number from zero on a real number line. acute angle An angle whose measure is less than 90. acute triangle A triangle in which each of the three interior
More informationINTRODUCTION TO DIFFERENTIAL EQUATIONS
INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminolog. Initial-Value Problems.3 Differential Equations as Mathematical Models CHAPTER IN REVIEW The words differential and equations certainl
More information1.3 LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION
. Limits at Infinit; End Behavior of a Function 89. LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION Up to now we have been concerned with its that describe the behavior of a function f) as approaches some
More information9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson
Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationChapter 8: More on Limits
Chapter 8: More on Limits Lesson 8.. 8-. a. 000 lim a() = lim = 0 b. c. lim c() = lim 3 +7 = 3 +000 lim b( ) 3 lim( 0000 ) = # = " 8-. a. lim 0 = " b. lim (#0.5 ) = # lim c. lim 4 = lim 4(/ ) = " d. lim
More informationInheritance of smmetr for positive solutions of semilinear elliptic boundar value problems Bernd Kawohl Λ Mathematical Institute Universit of Cologne D 5923 Cologne German Guido Sweers Λ Applied Math.
More informationOrdinary Differential Equations
58229_CH0_00_03.indd Page 6/6/6 2:48 PM F-007 /202/JB0027/work/indd & Bartlett Learning LLC, an Ascend Learning Compan.. PART Ordinar Differential Equations. Introduction to Differential Equations 2. First-Order
More informationSECTION 8-7 De Moivre s Theorem. De Moivre s Theorem, n a Natural Number nth-roots of z
8-7 De Moivre s Theorem 635 B eactl; compute the modulus and argument for part C to two decimal places. 9. (A) 3 i (B) 1 i (C) 5 6i 10. (A) 1 i 3 (B) 3i (C) 7 4i 11. (A) i 3 (B) 3 i (C) 8 5i 12. (A) 3
More informationAlgebra 1B Assignments Exponential Functions (All graphs must be drawn on graph paper!)
Name Score Algebra 1B Assignments Eponential Functions (All graphs must be drawn on graph paper!) 8-6 Pages 463-465: #1-17 odd, 35, 37-40, 43, 45-47, 50, 51, 54, 55-61 odd 8-7 Pages 470-473: #1-11 odd,
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationLESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II
LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Student Name: School Name:
INTEGRATED ALGEBRA The Universit of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Wednesda, August 18, 2010 8:30 to 11:30 a.m., onl Student Name: School Name: Print our name
More information1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs
0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals
More informationAre You Ready? Find Area in the Coordinate Plane
SKILL 38 Are You Read? Find Area in the Coordinate Plane Teaching Skill 38 Objective Find the areas of figures in the coordinate plane. Review with students the definition of area. Ask: Is the definition
More informationChapter One. Chapter One
Chapter One Chapter One CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons ConcepTests and Answers and Comments for Section.. Which of the following functions has its domain identical with its range?
More informationThe Steiner Ratio for Obstacle-Avoiding Rectilinear Steiner Trees
The Steiner Ratio for Obstacle-Avoiding Rectilinear Steiner Trees Anna Lubiw Mina Razaghpour Abstract We consider the problem of finding a shortest rectilinear Steiner tree for a given set of pointn the
More informationConstant 2-labelling of a graph
Constant 2-labelling of a graph S. Gravier, and E. Vandomme June 18, 2012 Abstract We introduce the concept of constant 2-labelling of a graph and show how it can be used to obtain periodic sphere packing.
More informationHigher. Functions and Graphs. Functions and Graphs 15
Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 5 Set Theor 5 Functions 6 Inverse Functions 9 4 Eponential Functions 0 5 Introduction to Logarithms 0 6 Radians 7 Eact Values
More informationTrigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
More information2 3 x = 6 4. (x 1) 6
Solutions to Math 201 Final Eam from spring 2007 p. 1 of 16 (some of these problem solutions are out of order, in the interest of saving paper) 1. given equation: 1 2 ( 1) 1 3 = 4 both sides 6: 6 1 1 (
More informationf x, y x 2 y 2 2x 6y 14. Then
SECTION 11.7 MAXIMUM AND MINIMUM VALUES 645 absolute minimum FIGURE 1 local maimum local minimum absolute maimum Look at the hills and valles in the graph of f shown in Figure 1. There are two points a,
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
8 CHAPTER Limits and Their Properties Section Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach Learn different was that a it can fail to eist Stud and use
More informationFunctions. Introduction CHAPTER OUTLINE
Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of
More informationUnit 10 - Graphing Quadratic Functions
Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
More information9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b
vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component
More informationEigenvectors and Eigenvalues 1
Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and
More information10.5 Graphs of the Trigonometric Functions
790 Foundations of Trigonometr 0.5 Graphs of the Trigonometric Functions In this section, we return to our discussion of the circular (trigonometric functions as functions of real numbers and pick up where
More information. This is the Basic Chain Rule. x dt y dt z dt Chain Rule in this context.
Math 18.0A Gradients, Chain Rule, Implicit Dierentiation, igher Order Derivatives These notes ocus on our things: (a) the application o gradients to ind normal vectors to curves suraces; (b) the generaliation
More informationThe Coordinate Plane. Circles and Polygons on the Coordinate Plane. LESSON 13.1 Skills Practice. Problem Set
LESSON.1 Skills Practice Name Date The Coordinate Plane Circles and Polgons on the Coordinate Plane Problem Set Use the given information to show that each statement is true. Justif our answers b using
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationPREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared by IITians.
www. Class XI TARGET : JEE Main/Adv PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) ALP ADVANCED LEVEL PROBLEMS Straight Lines 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared b IITians.
More information10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.
Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + =
More informationSection 7.3: Parabolas, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons
Section 7.: Parabolas, 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. 0, Carl Stitz.
More informationMath 2201 Review (2013 Sample/2013 Exam)
Math 01 Review (013 Sample/013 Eam) 013 Sample Eam Selected Response: Choose the appropriate response on the answer sheet or SCANTRON. 1. Lisa draws four parallelograms measures all sides. She writes the
More informationAnswers Investigation 3
Answers Investigation Applications. a., b. s = n c. The numbers seem to be increasing b a greater amount each time. The square number increases b consecutive odd integers:,, 7,, c X X=. a.,,, b., X 7 X=
More informationRational Numbers and Exponents
Rational and Exponents Math 7 Topic 4 Math 7 Topic 5 Math 8 - Topic 1 4-2: Adding Integers 4-3: Adding Rational 4-4: Subtracting Integers 4-5: Subtracting Rational 4-6: Distance on a Number Line 5-1: Multiplying
More informationSolutions to Two Interesting Problems
Solutions to Two Interesting Problems Save the Lemming On each square of an n n chessboard is an arrow pointing to one of its eight neighbors (or off the board, if it s an edge square). However, arrows
More informationFunctions. Introduction
Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For
More informationToda s Theorem: PH P #P
CS254: Computational Compleit Theor Prof. Luca Trevisan Final Project Ananth Raghunathan 1 Introduction Toda s Theorem: PH P #P The class NP captures the difficult of finding certificates. However, in
More informationSection 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
More informationLaurie s Notes. Overview of Section 3.5
Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More informationRolle s Theorem. THEOREM 3 Rolle s Theorem. x x. then there is at least one number c in (a, b) at which ƒ scd = 0.
4.2 The Mean Value Theorem 255 4.2 The Mean Value Theorem f '(c) 0 f() We know that constant functions have zero derivatives, ut could there e a complicated function, with man terms, the derivatives of
More informationOptimization Which point on the line y = 1 2x. is closest to the origin? MATH 1380 Lecture 18 1 of 15 Ronald Brent 2018 All rights reserved.
Optimization Which point on the line y = 1 is closest to the origin? y 1 - -1 0 1-1 - MATH 1380 Lecture 18 1 of 15 Ronald Brent 018 All rights reserved. Recall the distance between a point (, y) and (0,
More information7-6. nth Roots. Vocabulary. Geometric Sequences in Music. Lesson. Mental Math
Lesson 7-6 nth Roots Vocabular cube root n th root BIG IDEA If is the nth power of, then is an nth root of. Real numbers ma have 0, 1, or 2 real nth roots. Geometric Sequences in Music A piano tuner adjusts
More informationexample can be used to refute a conjecture, it cannot be used to prove one is always true.] propositions or conjectures
Task Model 1 Task Expectations: The student is asked to give an example that refutes a proposition or conjecture; or DOK Level 2 The student is asked to give an example that supports a proposition or conjecture.
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised June, 011 When ou come to New Meico State Universit, ou ma be asked to take the Mathematics Placement Eamination (MPE) Your inital placement
More informationInternational Examinations. Advanced Level Mathematics Pure Mathematics 1 Hugh Neill and Douglas Quadling
International Eaminations Advanced Level Mathematics Pure Mathematics Hugh Neill and Douglas Quadling PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street,
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationName: Richard Montgomery High School Department of Mathematics. Summer Math Packet. for students entering. Algebra 2/Trig*
Name: Richard Montgomer High School Department of Mathematics Summer Math Packet for students entering Algebra 2/Trig* For the following courses: AAF, Honors Algebra 2, Algebra 2 (Please go the RM website
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationIntroduction to Differential Equations. National Chiao Tung University Chun-Jen Tsai 9/14/2011
Introduction to Differential Equations National Chiao Tung Universit Chun-Jen Tsai 9/14/011 Differential Equations Definition: An equation containing the derivatives of one or more dependent variables,
More informationUnit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
More informationMathematics 2201 Common Mathematics Assessment Sample 2013
Common Mathematics Assessment Sample 2013 Name: Mathematics Teacher: 28 Selected Response 28 marks 13 Constructed Response 42 marks FINAL 70 Marks TIME: 2 HOURS NOTE Diagrams are not necessarily drawn
More informationALGEBRA 1 CP FINAL EXAM REVIEW
ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8
More informationAlgebra 2 Unit 2 Practice
Algebra Unit Practice LESSON 7-1 1. Consider a rectangle that has a perimeter of 80 cm. a. Write a function A(l) that represents the area of the rectangle with length l.. A rectangle has a perimeter of
More informationWe have examined power functions like f (x) = x 2. Interchanging x
CHAPTER 5 Eponential and Logarithmic Functions We have eamined power functions like f =. Interchanging and ields a different function f =. This new function is radicall different from a power function
More informationARCH 614 Note Set 2 S2011abn. Forces and Vectors
orces and Vectors Notation: = name for force vectors, as is A, B, C, T and P = force component in the direction = force component in the direction h = cable sag height L = span length = name for resultant
More informationP.4 Lines in the Plane
28 CHAPTER P Prerequisites P.4 Lines in the Plane What ou ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables
More informationFunctions. Introduction
Functions,00 P,000 00 0 70 7 80 8 0 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of work b MeasuringWorth)
More informationVerifying Properties of Quadrilaterals
Verifing roperties of uadrilaterals We can use the tools we have developed to find, classif, or verif properties of various shapes made b plotting coordinates on a Cartesian plane. Depending on the problem,
More informationDuality, Geometry, and Support Vector Regression
ualit, Geometr, and Support Vector Regression Jinbo Bi and Kristin P. Bennett epartment of Mathematical Sciences Rensselaer Poltechnic Institute Tro, NY 80 bij@rpi.edu, bennek@rpi.edu Abstract We develop
More informationMath Wrangle Practice Problems
Math Wrangle Practice Problems American Mathematics Competitions December 22, 2011 ((3!)!)! 1. Given that, = k. n!, where k and n are positive integers and n 3. is as large as possible, find k + n. 2.
More information