DOCSLIB.ORG

Geometry Revealed

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry Revealed Geometry Revealed A Jacob's Ladder to Modern Higher Geometry Bearbeitet von Marcel Berger, Lester J. Senechal 1. Auflage 2010. Buch. xvi, 831 S. Hardcover ISBN 978 3 540 70996 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1430 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Chapter IX Lattices, packings and tilings in the plane Preliminary note. For the arrangement of Chaps. IX and X, we have adopted the following policy concerning the interplay between dimension 2 and dimensions 3 and higher. In the present chapter in spite of its title we will treat higher dimensions for those subjects that will not be treated in detail in the next chapter, either because we have chosen not to discuss them further or because we want to treat matters extremely fast. Finally, we will encounter practically no open problems in the case of the plane, contrary to the spirit of our book; this will be entirely different in Chap. X. IX.1. Lattices, a line in the standard lattice Z2 and the theory of continued fractions, an immensity of applications A lattice in the real affine plane A is a subset ƒ that can be written as the set of all a CmuCnv, where a is a point of A and having vectorized A where u, v are lineally independent vectors and where m, n range over the set Z of all integers. In the affine realm, there is essentially but a single lattice, i.e. Z2 R2, which is to say the set of pairs .m; n/ of integers. If we have in addition a Euclidean structure, then there will be different lattices; this will be the subject of Sect. IX.4, although we will encounter these “other” lattices starting with Sect. IX.3 below. The lattice Z2 and more generally the lattices of the Euclidean plane appear abundantly in mathematics: in number theory, which isn’t surprising; in analysis when there is anything to do with doubly periodic functions in two variables (Fourier series in two variables), but also in the theory of functions of a complex variable (elliptic functions, which we encountered through Poncelet’s theorem in Sect. IV.8); in “pure” geometry when we study compact locally Euclidean geometries (a par- ticular case of the space forms, see Sect. VI.XYZ), which arise naturally in bil- liards in Sect. XI.2.A; in physics, even though crystal structures are situated in three dimensional space, when these are cut by planes or we study their diffraction pat- terns by X-rays or other processes, we obtain planar lattice images (Brillouin zones, see for example 3.7 of Gruber and Wills (1993)). It is for readers to find still other resonances for lattices. Decimal calculation is a marvelous invention; in particular, the operations of addi- tion and multiplication are done “trivially”. But it is not the best adapted for certain problems involving rational numbers, since most of these have an infinite decimal 1 expansion ( 3 D 0;33333:::, etc.). Above all, it is not obvious from the decimal M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_9, 563 c Springer-Verlag Berlin Heidelberg 2010 564 CHAPTER IX. LATTICES, PACKINGS AND TILINGS IN THE PLANE expansion how an irrational number can be approximated by rationals other than the k crude 10n . Now such approximation is fundamental in number theory; the Greeks, 22 333 355 who approximated by 7 , then by 106 and 113 , understood this well. It involves the theory of continued fractions (a term that will be explained right away), that has been well understood since the middle of the twentieth Century. If we are to be re- proached for attaching so much importance to this part of number theory, numerous motivations arising from problems in geometry can be found below. The elementary theory of continued fractions, as simple as it may be, is always a bit subtle and hard to visualize. This is why, even though algebra is always necessary, it is agreeable here to present the beginnings of this theory in a geometrical fashion. Here we follow Chap. 7 of Stark (1970), but see also, for a summary, the Klein polygons of Erdös, Gruber and Hammer, (1989, p. 101). The basic idea for approximating a real by ra- tionals is surely that the approximation can be better when the denominator is larger. The other idea is to consider the reals ˛ of R2 as slopes of half lines emanating from the origin in the quadrant RC RC. We feel confident that the farther we go in the direction of ˛, since there are always points at a distance less than 1 from the given half line D of slope ˛>0, the more we can find rational slopes as close as we want to ˛. The whole interest of continued fractions is to end up doing this p both as systematically and as economically as possible; i.e. we would like that the q p p0 chosen at the stage considered minimizes j˛ q j among all the j˛ q0 j for which the denominator q0 is 6 q. It turns out that we can do both things simultaneously. In all that follows, we assume that ˛ is irrational, i.e. the half line D does not contain any point of Z2 other than the origin. We now proceed by recurrence. It is nice to use vectorial language and to consider the vectors Vn D .pnI qn/, where the pn=qn denote the desired approximations for ˛. The first vector V0 D .1I Œ˛/ is of course that given by the integer part Œ˛ of ˛. As a point it is situated below D, but as close as possible among the vectors of the form .1; q/. The next vector will be V1 D .0; 1/ C a1V0, where a1 is the last integer such that V1 is above D. Then V2 will be V0 C a2V1, which is the last of this form below D, etc. We show (however we do it, it is never obvious) that the sequence fVn D Vn2 C anVn1 D .pn;qn/g thus defined by recurrence is indeed automatically the best approximation possible in the sense of minimizing with bounded denominator. We have thus an alternating sandwiching: p p p p p 0 < 2 < 4 < ˛ < 3 < 1 q0 q2 q4 q3 q1 with lim pn D ˛. By construction n!1 qn ˇ ˇ ˇ ˇ ˇ p ˇ ˇ pn ˇ ˇ˛ ˇ > ˇ˛ ˇ if q 6 qn; q qn IX.1. LATTICES, LINES IN Z2 AND CONTINUED FRACTIONS 565 Fig. IX.1.1. Fig. IX.1.2. but then the sandwiching is also good in an unexpected way, as we always have ˇ ˇ ˇ p ˇ 1 ˇ n ˇ 6 ˛ 2 : qn qn See Sect. IX.3 for a lightning fast and optimal proof! This provides an explanation of the term “continued fraction”, as the preceding shows (proceed with gcd’s) that we can simply define the pairs .pnI qn/ with the aid of the sequence of integers fang that are defined by recurrence by an integer part condition on the sequences f˛n;ang, where we have already seen that a0 D Œ˛, and further: 1 1 ˛1 D ;a1 D Œ˛1; ˛2 D ;a2 D Œ˛2;::: ˛ a0 ˛1 a1 Then ˛ is perfectly defined by the infinite sequence fa0;a1;a2;:::g and the conver- gent sequence of successive fractions 1 1 a0;a0 C ;a0 C ; etc., a1 1 a1 C a2 566 CHAPTER IX. LATTICES, PACKINGS AND TILINGS IN THE PLANE whence the expression continued fraction, wanting in fact to say “infinite fraction”. We write by notational convention: ˛ Dha0;a1;a2;:::i: For the details of the proofs, see Stark (1970). We now owe our readers some examples. Here are two extremes: the first is Dh3; 7; 15; 1; 292; : : :i for which the associated continued fractions are thus: 1 22 1 333 355 103 993 3; 3 C D ;3C D ; ; ; ::: 7 7 1 106 113 33 102 7 C 15 p Then 3 Dh1; 1; 2; 1; 2; 1; 2; : : :i for the fractions 5 7 19 26 71 97 265 1; 2; ; ; ; ; ; ; ; :::I 3 4 11 15 41 56 153 p the continued fraction that represents 3 is truly periodic (to infinity). These examples call for remarks, for questions. The obvious fact that never ceases to surprise the author is that , which is supposed to be a very bad irrational since it is even transcendental it is never a root of a polynomial with integer co- efficients is very well approximated by the rationals, and very quickly! However, it can’t be so very well approximated, since we know from Mahler in 1953 that p > 1 j q j q42 for each q. In contrast, it is an open problem to know the best (small- k p > 1 q est) power such that j q j qk for each integer . Presently it is known through k D 7; that is, is surely a transcendental number, but not by so much! Also known c p 6 c is the best constant for approximation in the form j q j qk . For references about the number , the bible is Berggren, Borwein and Bortwein (2000), but it is necessary to be immersed in it and it is perhaps better to first read the informal exposition (van der Poorten, 1979).
Load more

Recommended publications
  • Geometry by Its History
    Undergraduate Texts in Mathematics Geometry by Its History Bearbeitet von Alexander Ostermann, Gerhard Wanner 1. Auflage 2012. Buch. xii, 440 S. Hardcover ISBN 978 3 642 29162 3 Format (B x L): 15,5 x 23,5 cm Gewicht: 836 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 2 The Elements of Euclid “At age eleven, I began Euclid, with my brother as my tutor. This was one of the greatest events of my life, as dazzling as first love. I had not imagined that there was anything as delicious in the world.” (B. Russell, quoted from K. Hoechsmann, Editorial, π in the Sky, Issue 9, Dec. 2005. A few paragraphs later K. H. added: An innocent look at a page of contemporary the- orems is no doubt less likely to evoke feelings of “first love”.) “At the age of 16, Abel’s genius suddenly became apparent. Mr. Holmbo¨e, then professor in his school, gave him private lessons. Having quickly absorbed the Elements, he went through the In- troductio and the Institutiones calculi differentialis and integralis of Euler. From here on, he progressed alone.” (Obituary for Abel by Crelle, J. Reine Angew. Math. 4 (1829) p. 402; transl. from the French) “The year 1868 must be characterised as [Sophus Lie’s] break- through year.
    [Show full text]
  • Six Mathematical Gems from the History of Distance Geometry
    Six mathematical gems from the history of Distance Geometry Leo Liberti1, Carlile Lavor2 1 CNRS LIX, Ecole´ Polytechnique, F-91128 Palaiseau, France Email:[email protected] 2 IMECC, University of Campinas, 13081-970, Campinas-SP, Brazil Email:[email protected] February 28, 2015 Abstract This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron’s formula, Cauchy’s theorem on the rigidity of polyhedra, Cayley’s generalization of Heron’s formula to higher dimensions, Menger’s characterization of abstract semi-metric spaces, a result of G¨odel on metric spaces on the sphere, and Schoenberg’s equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling. Keywords: Euler’s conjecture, Cayley-Menger determinants, Multidimensional scaling, Euclidean Distance Matrix 1 Introduction Distance Geometry (DG) is the study of geometry with the basic entity being distance (instead of lines, planes, circles, polyhedra, conics, surfaces and varieties). As did much of Mathematics, it all began with the Greeks: specifically Heron, or Hero, of Alexandria, sometime between 150BC and 250AD, who showed how to compute the area of a triangle given its side lengths [36]. After a hiatus of almost two thousand years, we reach Arthur Cayley’s: the first paper of volume I of his Collected Papers, dated 1841, is about the relationships between the distances of five points in space [7]. The gist of what he showed is that a tetrahedron can only exist in a plane if it is flat (in fact, he discussed the situation in one more dimension).
    [Show full text]
  • CHAPTER 5Morphology of Permanent Molars
    CHAPTER Morphology of Permanent Molars Topics5 covered within the four sections of this chapter B. Type traits of maxillary molars from the lingual include the following: view I. Overview of molars C. Type traits of maxillary molars from the A. General description of molars proximal views B. Functions of molars D. Type traits of maxillary molars from the C. Class traits for molars occlusal view D. Arch traits that differentiate maxillary from IV. Maxillary and mandibular third molar type traits mandibular molars A. Type traits of all third molars (different from II. Type traits that differentiate mandibular second first and second molars) molars from mandibular first molars B. Size and shape of third molars A. Type traits of mandibular molars from the buc- C. Similarities and differences of third molar cal view crowns compared with first and second molars B. Type traits of mandibular molars from the in the same arch lingual view D. Similarities and differences of third molar roots C. Type traits of mandibular molars from the compared with first and second molars in the proximal views same arch D. Type traits of mandibular molars from the V. Interesting variations and ethnic differences in occlusal view molars III. Type traits that differentiate maxillary second molars from maxillary first molars A. Type traits of the maxillary first and second molars from the buccal view hroughout this chapter, “Appendix” followed Also, remember that statistics obtained from by a number and letter (e.g., Appendix 7a) is Dr. Woelfel’s original research on teeth have been used used within the text to denote reference to to draw conclusions throughout this chapter and are the page (number 7) and item (letter a) being referenced with superscript letters like this (dataA) that Treferred to on that appendix page.
    [Show full text]
  • Students' Understanding of Axial and Central Symmetry
    Full research paper STUDENTS’ UNDERSTANDING OF Vlasta Moravcová* Jarmila Robová AXIAL AND CENTRAL SYMMETRY Jana Hromadová Zdeněk Halas Department of Mathematics ABSTRACT Education, Faculty of Mathematics The paper focuses on students’ understanding of the concepts of axial and central symmetries in and Physics, Charles University, Czech a plane. Attention is paid to whether students of various ages identify a non-model of an axially Republic symmetrical figure, know that a line segment has two axes of symmetry and a circle has an infinite number of symmetry axes, and are able to construct an image of a given figure in central symmetry. * [email protected] The results presented here were obtained by a quantitative analysis of tests given to nearly 1,500 Czech students, including pre-service mathematics teachers. The paper presents the statistics of the students’ answers, discusses the students’ thought processes and presents some of the students’ original solutions. The data obtained are also analysed with regard to gender differences and to the type of school that students attend. The results show that students have two principal misconceptions: that a rhomboid is an axially symmetrical figure and that a line segment has just one axis of symmetry. Moreover, many of the tested students confused axial and central symmetry. Finally, the possible causes of these errors are considered and recommendations for preventing these errors are given. Article history KEYWORDS Received Axial symmetry, central symmetry, circle, geometrical concepts, line segment, rhomboid October 22, 2020 Received in revised form January 19, 2021 HOW TO CITE Accepted Moravcová V., Robová J., Hromadová J., Halas Z.
    [Show full text]
  • Quadrilateral Lace Susan Happersett Beacon, NY, USA; [email protected] Abstract
    Bridges 2020 Conference Proceedings Quadrilateral Lace Susan Happersett Beacon, NY, USA; [email protected] Abstract This paper will describe the exact procedures I use to draw algorithmically generated mapping patterns in the framework of quadrilaterals. I began with a square format, then rhomboid and trapezoidal. Layering concentric self-similar shapes creates the illusion of 3-dimensional space. Shifting the self-similar shapes out of the concentric order creates the sense of movement across the plane. The development of my lace drawings has been a multi-year process. By a lace pattern, or point-to-point mapping pattern, I mean carefully laying out a collection of points in the plane according to a pattern, and then joining certain pairs of points with line segments, where the pairs to be joined are specified by a definite procedure. The first lace drawings I algorithmically generated were restricted to points on the Cartesian Coordinate system: the x-axis, the y-axis and the lines y = x and y = −x [1]. Expanding on the types of point-to-point mappings I could produce, I decided to use the perimeters of quadrilaterals to assign my points. I began by working with a square with 7 equally distant points on each side. I picked an odd number of points for aesthetic reasons. Four sides on the square, 7 points per side, produces 24 points in total. I wanted a point on the center of each side. Once the parts were defined, I needed to write a rule to define the mapping process to be carried out for each point.
    [Show full text]
  • Geometry for Drafting Section 5.1 Applied Geometry for Board Drafting Section 5.2 Applied Geometry for CAD Systems
    5 Geometry for Drafting Section 5.1 Applied Geometry for Board Drafting Section 5.2 Applied Geometry for CAD Systems Chapter Objectives • Identify geometric shapes and construc- tions used by drafters. • Construct various geometric shapes. • Solve technical and mathematical prob- lems through geomet- ric constructions using drafting instruments. • Solve technical and mathematical prob- lems through geomet- ric constructions using a CAD system. • Use geometry to reduce or enlarge a drawing or to change its proportions. Defying Convention It has been said that Zaha Hadid has built a career on defying convention—conventional ideas of architectural space, and of construction. What do you see in the building shown here that defi es convention? 132 Drafting Career Zaha Hadid, Architect Architect Zaha Hadid’s designs for the Cincinnati Contemporary Art Center were “like a rollercoaster, a little scary, but exhilarating,” says Center direc- tor Charles Desmarais. Critics said “she was a paper architect, someone who had great respect as a theo- rist and as a thinker about architecture but who hadn't had the opportunity to build.” “She totally got what we were trying to do,” said Desmarais, “which was to try and bridge that sort of gap between the inside and the outside, between the world and the museum.” She certainly did. Zaha Hadid is the fi rst woman in the world to design a museum and to win the prestigious Pritzker Architec- ture Prize. Academic Skills and Abilities • Math • Computer sciences • Business management skills • Verbal and written communication skills • Organizing and planning skills Career Pathways There is a wealth of opportunities outside the classroom for expanding your drafting knowledge.
    [Show full text]
  • 21. Concepts and Categories
    Concepts and Categories Lecture 21 1 Learning, Perception, and Memory Rely on Thinking • Learning – Classical Conditioning • How can I predict some event? – Instrumental Conditioning • How can I control that event? • Perception – What is out there? Where is it? What is it doing? • Memory – What happened in the past? 2 “Every act of perception is an act of categorization” Bruner (1957) [paraphrase] • Fundamental Cognitive Process – Perceptual Identification... • Of Individual Object – Categorization... • As Belonging in Same Class as Other Objects • Categorical Knowledge is Part of Semantic Memory 3 Categories and Concepts • Enumeration • Rule • Attributes – Perceptual – Functional – Relational 4 Classical View of Categorization Aristotle, Categories (in the Organon, 4th C. BCE) Categories are Proper Sets • Defining Features – Singly Necessary – Jointly Sufficient 5 Defining Features • Geometrical Figures – Triangles • 2 Dimensions, 3 Sides, and 3 Angles – Quadrilaterals • 2 Dimensions, 4 Sides, and 4 Angles • Animals – Birds • Vertebrate, Warm-Blooded, Feathers, Wings – Fish • Vertebrate, Cold-Blooded, Scales, Fins 6 Categories as Proper Sets Aristotle, On Categories, etc. • Defining Features • Vertical Arrangement into Hierarchies – Perfect Nesting • Superordinate (Supersets) • Subordinate (Subsets) 7 Geometric Figures 8 Subcategories of Triangles • Classified by Length of Sides – Equilateral – Isosceles – Scalene • Classified by Internal Angles – Right – Oblique • Obtuse • Acute 9 Subcategories of Quadrilaterals • Trapeziums • Trapezoids
    [Show full text]
  • The Apical Angle: a Mathematical Analysis of the Ellipse
    The Apical Angle: A Mathematical Analysis of the Ellipse Brent R. Moody, MD,* John E. McCarthy, PhD,† and Roberta D. Sengelmann, MD*‡ *Division of Dermatology, Washington University School of Medicine, †Washington University College of Arts and Sciences, and ‡Cutaneous Surgery Center, Barnes-Jewish Hospital, St. Louis, Missouri background. The elliptical excision is a common surgical sions as presented in the dermatologic literature. We analyzed procedure. The dermatologic literature predominantly describes the geometry of the excisions and defined it mathematically. an excisional geometry with a 3:1 length:width ratio and an results. The apical angle of a 3:1 elliptical excision is not 30Њ. apical angle of 30Њ. The true apical angle varies from 37Њ to 74Њ depending on exci- objective. To analyze the elliptical excision by applying sional geometry. mathematical principles and define the apical angle and its rela- conclusion. The commonly presented apical angle of 30Њ is tionship to the length:width ratio. incorrect and does not reflect the true apical angle of elliptical methods. We examined numerous examples of elliptical exci- excisions. THE APICAL ANGLE is the angle created at the ends of Therefore, for the purposes of this study, tissue characteris- an elliptical excision. The elliptical excision, more prop- tics may be discounted entirely. erly termed a fusiform excision, and its variants are com- Using the assumption that the surgeon creates the excision monly utilized for excisional surgery. Many texts have by incising two arcs from a circle of radius r, we derived the described the geometry of the ellipse.1–7 We present a de- relationship between the apical angle and the length:width ra- tailed mathematical analysis of the apical angle and de- tio.
    [Show full text]
  • Mcmorran, Ciaran (2016) Geometry and Topography in James Joyce's Ulysses and Finnegans Wake
    McMorran, Ciaran (2016) Geometry and topography in James Joyce's Ulysses and Finnegans Wake. PhD thesis. http://theses.gla.ac.uk/7385/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Glasgow Theses Service http://theses.gla.ac.uk/ [email protected] Geometry and Topography in James Joyce’s Ulysses and Finnegans Wake Ciaran McMorran M.A., M.Litt. Submitted in fulfilment of the requirements for the Degree of Ph.D. in English Literature School of Critical Studies College of Arts University of Glasgow Supervised by Dr. John Coyle and Dr. Matthew Creasy Examined by Prof. Finn Fordham and Dr. Maria-Daniella Dick 2016 1 Abstract Following the development of non-Euclidean geometries from the mid-nineteenth century onwards, Euclid’s system had come to be re-conceived as a language for describing reality rather than a set of transcendental laws. As Henri Poincaré famously put it, ‘[i]f several geometries are possible, is it certain that our geometry [...] is true?’. By examining Joyce’s linguistic play and conceptual engagement with ground-breaking geometric constructs in Ulysses and Finnegans Wake, this thesis explores how his topographical writing of place encapsulates a common crisis between geometric and linguistic modes of representation within the context of modernity.
    [Show full text]
  • Euclid's Definitions, Postulates, and the First 30 Propositions of Book I*
    APPENDIX A Euclid's Definitions, Postulates, and the First 30 Propositions of Book I* Definitions 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 9. And when the lines containing the angle are straight, the angle is called rectilineal. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called perpendicular to that on which it stands. 11. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less than a right angle. 13. A boundary is that which is an extremity of anything. 14. A figure is that which is contained by any boundary or boundaries. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. 16. And the point is called the centre of the circle.
    [Show full text]
  • MATH 205 [email protected] This Form Is a Review Sheet for MATH 090 Students Based Primarily Around the Teaching & Instructions of Professor V
    Created by: Jasmine E. Aue MATH 205 [email protected] This form is a review sheet for MATH 090 students based primarily around the teaching & instructions of Professor V. Smith of ESU. HELPFUL TIPS FOR COURSE CONTENT (​ie. not formulas but important​) FORMULA LIST ℉ = ((℃/5)(9)) + 32 | Lines, Arrows, Points & Planes Fahrenheit to Celsius formula This section is a review of lines, arrows, points & planes. How they are defined, look and interact with each other. ℃ = ((℉ − 32)/9)(5) ​ |​ Celsius to ❏ Collinear​: same line Fahrenheit formula ❏ 2 Coplanar​: same plane a2 + b = c2 |​ Pythagorean ❏ ● ● Closed​: a line with an endpoint on each end, kind of like this Theorem ❏ Half​: a line with only one endpoint, kind of like this ● d = (x − x ) + (y − y ) ❏ Open​: a line with no endpoints, like this √ 2 1 2 1 | ❏ → ⋃ → = ↔ Diameter of a Circle ❏ → ⋂ → = r2 = (x − h)2 + (y − k)2 |​ Radius ❏ ●● ⋃ ●→ = → Formula ❏ FE￮ ⋂ ￮ ￮ ￮ → → = a2 + b2 = c2 |​ Pythagorean ❏ Intersecting Lines​: lines that have only one point in common ❏ Intersecting Planes​: planes that have only one line in common Theorem ❏ Skew​: cannot be coplanar y = mx + b ​ |​ y is from (x,y), m ❏ Concurrent​: 3 or more intersections at the same point is the slope, x is from (x,y) & b ❏ Parallel​: same slope, non-intersecting is the y-intercept m = (y2 − y1) / (x2 − x1) |​ this is ❏ There is exactly one line containing any two distinct points the point-slope formula, use ❏ If two points are in a plane then the line of points are in a plane this to find the slope of a line ❏ If two
    [Show full text]
  • The Use of Kinesio Tape, with a Strengthening Protocol, in Aiding Scapular Retraction Through Facilitation of the Rhomboids Elen
    The use of kinesio tape, with a strengthening protocol, in aiding scapular retraction through facilitation of the rhomboids Elena A. Robinson This thesis is submitted in partial fulfillment of the requirements of the Research Honors Program in the Department of Sports Medicine Marietta College Marietta, Ohio April 27, 2018 This Research Honors Thesis has been approved for the Department of Chemistry and the Honors and Investigative Studies Committee by Chyrsten Gessel Faculty Thesis Advisor Jaclyn Schwieterman Thesis Committee Member Dr. Alicia Doerflinger Thesis Committee Member Acknowledgements I would like to thank Chyrsten Gessel, Jaclyn Schwieterman, Dr. Alicia Doerflinger, and Dr. Brown. I would also like to thank my family, friends and classmates for their support. Table of Contents Abstract 1 Introduction 2 Literature Review 3 Anatomy 3 Biomechanics/Pathomechanics 7 Previous Research 9 Kinesio Tape (KT) 11 Muscle Facilitation Using KT 12 Kinesio Tape and RSP 15 KT with Strength Protocols 16 RSP and Rhomboid Muscles 18 Methods 18 Participants 18 Instruments 19 Kinesio Tape Protocol 19 Placebo Tape Protocol 20 Static Postural Assessment 20 Dynamic Postural Assessment 21 Strength Protocol 22 Overall Procedure 22 Statistical Analysis 24 Results 24 Discussion 25 Limitations 29 Conclusion 30 Future Research 30 Works Cited 32 Appendix A 34 Appendix B 38 Appendix C 39 Appendix D 40 Appendix E 41 Appendix F 42 Appendix G 45 1 The use of kinesio tape, with a strengthening protocol, in aiding scapular retraction through facilitation of the rhomboids CONTEXT: Rounded shoulder posture (RSP) is a type of poor upper body posture that can lead to other biomechanical changes of the body, as well as pathological issues.
    [Show full text]