Chapter 11: the Mathematics of Symmetry
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Snakes in the Plane
Snakes in the Plane by Paul Church A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2008 c 2008 Paul Church I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Recent developments in tiling theory, primarily in the study of anisohedral shapes, have been the product of exhaustive computer searches through various classes of poly- gons. I present a brief background of tiling theory and past work, with particular empha- sis on isohedral numbers, aperiodicity, Heesch numbers, criteria to characterize isohedral tilings, and various details that have arisen in past computer searches. I then develop and implement a new “boundary-based” technique, characterizing shapes as a sequence of characters representing unit length steps taken from a finite lan- guage of directions, to replace the “area-based” approaches of past work, which treated the Euclidean plane as a regular lattice of cells manipulated like a bitmap. The new technique allows me to reproduce and verify past results on polyforms (edge-to-edge as- semblies of unit squares, regular hexagons, or equilateral triangles) and then generalize to a new class of shapes dubbed polysnakes, which past approaches could not describe. My implementation enumerates polyforms using Redelmeier’s recursive generation algo- rithm, and enumerates polysnakes using a novel approach. -
Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing by ⌬ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding fixed, then increasing by (1). -
Foundations of Newtonian Dynamics: an Axiomatic Approach For
Foundations of Newtonian Dynamics: 1 An Axiomatic Approach for the Thinking Student C. J. Papachristou 2 Department of Physical Sciences, Hellenic Naval Academy, Piraeus 18539, Greece Abstract. Despite its apparent simplicity, Newtonian mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtle- ties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and deriva- tive theorems. This article attempts to clarify these issues for the benefit of the stu- dent by revisiting the foundations of Newtonian dynamics and by proposing a rigor- ous axiomatic approach to the subject. This theoretical scheme is built upon two fun- damental postulates, namely, conservation of momentum and superposition property for interactions. Newton’s laws, as well as all familiar theorems of mechanics, are shown to follow from these basic principles. 1. Introduction Teaching introductory mechanics can be a major challenge, especially in a class of students that are not willing to take anything for granted! The problem is that, even some of the most prestigious textbooks on the subject may leave the student with some degree of confusion, which manifests itself in questions like the following: • Is the law of inertia (Newton’s first law) a law of motion (of free bodies) or is it a statement of existence (of inertial reference frames)? • Are the first two of Newton’s laws independent of each other? It appears that -
Enhancing Self-Reflection and Mathematics Achievement of At-Risk Urban Technical College Students
Psychological Test and Assessment Modeling, Volume 53, 2011 (1), 108-127 Enhancing self-reflection and mathematics achievement of at-risk urban technical college students Barry J. Zimmerman1, Adam Moylan2, John Hudesman3, Niesha White3, & Bert Flugman3 Abstract A classroom-based intervention study sought to help struggling learners respond to their academic grades in math as sources of self-regulated learning (SRL) rather than as indices of personal limita- tion. Technical college students (N = 496) in developmental (remedial) math or introductory col- lege-level math courses were randomly assigned to receive SRL instruction or conventional in- struction (control) in their respective courses. SRL instruction was hypothesized to improve stu- dents’ math achievement by showing them how to self-reflect (i.e., self-assess and adapt to aca- demic quiz outcomes) more effectively. The results indicated that students receiving self-reflection training outperformed students in the control group on instructor-developed examinations and were better calibrated in their task-specific self-efficacy beliefs before solving problems and in their self- evaluative judgments after solving problems. Self-reflection training also increased students’ pass- rate on a national gateway examination in mathematics by 25% in comparison to that of control students. Key words: self-regulation, self-reflection, math instruction 1 Correspondence concerning this article should be addressed to: Barry Zimmerman, PhD, Graduate Center of the City University of New York and Center for Advanced Study in Education, 365 Fifth Ave- nue, New York, NY 10016, USA; email: [email protected] 2 Now affiliated with the University of California, San Francisco, School of Medicine 3 Graduate Center of the City University of New York and Center for Advanced Study in Education Enhancing self-reflection and math achievement 109 Across America, faculty and policy makers at two-year and technical colleges have been deeply troubled by the low academic achievement and high attrition rate of at-risk stu- dents. -
Reflection Invariant and Symmetry Detection
1 Reflection Invariant and Symmetry Detection Erbo Li and Hua Li Abstract—Symmetry detection and discrimination are of fundamental meaning in science, technology, and engineering. This paper introduces reflection invariants and defines the directional moments(DMs) to detect symmetry for shape analysis and object recognition. And it demonstrates that detection of reflection symmetry can be done in a simple way by solving a trigonometric system derived from the DMs, and discrimination of reflection symmetry can be achieved by application of the reflection invariants in 2D and 3D. Rotation symmetry can also be determined based on that. Also, if none of reflection invariants is equal to zero, then there is no symmetry. And the experiments in 2D and 3D show that all the reflection lines or planes can be deterministically found using DMs up to order six. This result can be used to simplify the efforts of symmetry detection in research areas,such as protein structure, model retrieval, reverse engineering, and machine vision etc. Index Terms—symmetry detection, shape analysis, object recognition, directional moment, moment invariant, isometry, congruent, reflection, chirality, rotation F 1 INTRODUCTION Kazhdan et al. [1] developed a continuous measure and dis- The essence of geometric symmetry is self-evident, which cussed the properties of the reflective symmetry descriptor, can be found everywhere in nature and social lives, as which was expanded to 3D by [2] and was augmented in shown in Figure 1. It is true that we are living in a spatial distribution of the objects asymmetry by [3] . For symmetric world. Pursuing the explanation of symmetry symmetry discrimination [4] defined a symmetry distance will provide better understanding to the surrounding world of shapes. -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
Phoronomy: Space, Construction, and Mathematizing Motion Marius Stan
To appear in Bennett McNulty (ed.), Kant’s Metaphysical Foundations of Natural Science: A Critical Guide. Cambridge University Press. Phoronomy: space, construction, and mathematizing motion Marius Stan With his chapter, Phoronomy, Kant defies even the seasoned interpreter of his philosophy of physics.1 Exegetes have given it little attention, and un- derstandably so: his aims are opaque, his turns in argument little motivated, and his context mysterious, which makes his project there look alienating. I seek to illuminate here some of the darker corners in that chapter. Specifi- cally, I aim to clarify three notions in it: his concepts of velocity, of compo- site motion, and of the construction required to compose motions. I defend three theses about Kant. 1) his choice of velocity concept is ul- timately insufficient. 2) he sided with the rationalist faction in the early- modern debate on directed quantities. 3) it remains an open question if his algebra of motion is a priori, though he believed it was. I begin in § 1 by explaining Kant’s notion of phoronomy and its argu- ment structure in his chapter. In § 2, I present four pictures of velocity cur- rent in Kant’s century, and I assess the one he chose. My § 3 is in three parts: a historical account of why algebra of motion became a topic of early modern debate; a synopsis of the two sides that emerged then; and a brief account of his contribution to the debate. Finally, § 4 assesses how general his account of composite motion is, and if it counts as a priori knowledge. -
Symmetry and Beauty in the Living World I Thank the Governing Body and the Director of the G.B
SYMMETRY AND BEAUTY IN THE LIVING WORLD I thank the Governing Body and the Director of the G.B. Pant Institute of Himalayan Environment & Development for providing me this opportunity to deliver the 17th Govind Ballabh Pant Memorial Lecture. Pt. Pant, as I have understood, was amongst those who contributed in multiple ways to shape and nurture the nation in general and the Himalayan area in particular. Established to honour this great ‘Son of the Mountains’, the Institute carries enormous responsibilities and expectations from millions of people across the region and outside. Undoubtedly the multidisciplinary skills and interdisciplinary approach of the Institute and the zeal of its members to work in remote areas and harsh Himalayan conditions will succeed in achieving the long term vision of Pt. Pant for the overall development of the region. My talk ‘Symmetry and Beauty in the Living World’ attempts to discuss aspects of symmetry and beauty in nature and their evolutionary explanations. I shall explain how these elements have helped developmental and evolutionary biologists to frame and answer research questions. INTRODUCTION Symmetry is an objective feature of the living world and also of some non-living entities. It forms an essential element of the laws of nature; it is often sought by human beings when they create artefacts. Beauty has to do with a subjective assessment of the extent to which something or someone has a pleasing appearance. It is something that people aspire to, whether in ideas, creations or people. Evolutionary biology tells us that it is useful to look for an evolutionary explanation of anything to do with life. -
Isometries and the Plane
Chapter 1 Isometries of the Plane \For geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child. (W. K. Clifford) The focus of this first chapter is the 2-dimensional real plane R2, in which a point P can be described by its coordinates: 2 P 2 R ;P = (x; y); x 2 R; y 2 R: Alternatively, we can describe P as a complex number by writing P = (x; y) = x + iy 2 C: 2 The plane R comes with a usual distance. If P1 = (x1; y1);P2 = (x2; y2) 2 R2 are two points in the plane, then p 2 2 d(P1;P2) = (x2 − x1) + (y2 − y1) : Note that this is consistent withp the complex notation. For P = x + iy 2 C, p 2 2 recall that jP j = x + y = P P , thus for two complex points P1 = x1 + iy1;P2 = x2 + iy2 2 C, we have q d(P1;P2) = jP2 − P1j = (P2 − P1)(P2 − P1) p 2 2 = j(x2 − x1) + i(y2 − y1)j = (x2 − x1) + (y2 − y1) ; where ( ) denotes the complex conjugation, i.e. x + iy = x − iy. We are now interested in planar transformations (that is, maps from R2 to R2) that preserve distances. 1 2 CHAPTER 1. ISOMETRIES OF THE PLANE Points in the Plane • A point P in the plane is a pair of real numbers P=(x,y). d(0,P)2 = x2+y2. • A point P=(x,y) in the plane can be seen as a complex number x+iy. -
Chapter 1 – Symmetry of Molecules – P. 1
Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules. -
Symmetry of Graphs. Circles
Symmetry of graphs. Circles Symmetry of graphs. Circles 1 / 10 Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Symmetry of graphs. Circles 2 / 10 Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. -
Lesson 4: Definition of Reflection and Basic Properties
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 8•2 Lesson 4: Definition of Reflection and Basic Properties Student Outcomes . Students know the definition of reflection and perform reflections across a line using a transparency. Students show that reflections share some of the same fundamental properties with translations (e.g., lines map to lines, angle- and distance-preserving motion). Students know that reflections map parallel lines to parallel lines. Students know that for the reflection across a line 퐿 and for every point 푃 not on 퐿, 퐿 is the bisector of the segment joining 푃 to its reflected image 푃′. Classwork Example 1 (5 minutes) The reflection across a line 퐿 is defined by using the following example. MP.6 . Let 퐿 be a vertical line, and let 푃 and 퐴 be two points not on 퐿, as shown below. Also, let 푄 be a point on 퐿. (The black rectangle indicates the border of the paper.) . The following is a description of how the reflection moves the points 푃, 푄, and 퐴 by making use of the transparency. Trace the line 퐿 and three points onto the transparency exactly, using red. (Be sure to use a transparency that is the same size as the paper.) . Keeping the paper fixed, flip the transparency across the vertical line (interchanging left and right) while keeping the vertical line and point 푄 on top of their black images. The position of the red figures on the transparency now represents the Scaffolding: reflection of the original figure. 푅푒푓푙푒푐푡푖표푛(푃) is the point represented by the There are manipulatives, such red dot to the left of 퐿, 푅푒푓푙푒푐푡푖표푛(퐴) is the red dot to the right of 퐿, and point as MIRA and Georeflector, 푅푒푓푙푒푐푡푖표푛(푄) is point 푄 itself.