The Shape of Spacetime
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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). -
Art Concepts for Kids: SHAPE & FORM!
Art Concepts for Kids: SHAPE & FORM! Our online art class explores “shape” and “form” . Shapes are spaces that are created when a line reconnects with itself. Forms are three dimensional and they have length, width and depth. This sweet intro song teaches kids all about shape! Scratch Garden Value Song (all ages) https://www.youtube.com/watch?v=coZfbTIzS5I Another great shape jam! https://www.youtube.com/watch?v=6hFTUk8XqEc Cookie Monster eats his shapes! https://www.youtube.com/watch?v=gfNalVIrdOw&list=PLWrCWNvzT_lpGOvVQCdt7CXL ggL9-xpZj&index=10&t=0s Now for the activities!! Click on the links in the descriptions below for the step by step details. Ages 2-4 Shape Match https://busytoddler.com/2017/01/giant-shape-match-activity/ Shapes are an easy art concept for the littles. Lots of their environment gives play and art opportunities to learn about shape. This Matching activity involves tracing the outside shape of blocks and having littles match the block to the outline. It can be enhanced by letting your little ones color the shapes emphasizing inside and outside the shape lines. Block Print Shapes https://thepinterestedparent.com/2017/08/paul-klee-inspired-block-printing/ Printing is a great technique for young kids to learn. The motion is like stamping so you teach little ones to press and pull rather than rub or paint. This project requires washable paint and paper and block that are not natural wood (which would stain) you want blocks that are painted or sealed. Kids can look at Paul Klee art for inspiration in stacking their shapes into buildings, or they can chose their own design Ages 4-6 Lois Ehlert Shape Animals http://www.momto2poshlildivas.com/2012/09/exploring-shapes-and-colors-with-color.html This great project allows kids to play with shapes to make animals in the style of the artist Lois Ehlert. -
Algebraic Geometry Michael Stoll
Introductory Geometry Course No. 100 351 Fall 2005 Second Part: Algebraic Geometry Michael Stoll Contents 1. What Is Algebraic Geometry? 2 2. Affine Spaces and Algebraic Sets 3 3. Projective Spaces and Algebraic Sets 6 4. Projective Closure and Affine Patches 9 5. Morphisms and Rational Maps 11 6. Curves — Local Properties 14 7. B´ezout’sTheorem 18 2 1. What Is Algebraic Geometry? Linear Algebra can be seen (in parts at least) as the study of systems of linear equations. In geometric terms, this can be interpreted as the study of linear (or affine) subspaces of Cn (say). Algebraic Geometry generalizes this in a natural way be looking at systems of polynomial equations. Their geometric realizations (their solution sets in Cn, say) are called algebraic varieties. Many questions one can study in various parts of mathematics lead in a natural way to (systems of) polynomial equations, to which the methods of Algebraic Geometry can be applied. Algebraic Geometry provides a translation between algebra (solutions of equations) and geometry (points on algebraic varieties). The methods are mostly algebraic, but the geometry provides the intuition. Compared to Differential Geometry, in Algebraic Geometry we consider a rather restricted class of “manifolds” — those given by polynomial equations (we can allow “singularities”, however). For example, y = cos x defines a perfectly nice differentiable curve in the plane, but not an algebraic curve. In return, we can get stronger results, for example a criterion for the existence of solutions (in the complex numbers), or statements on the number of solutions (for example when intersecting two curves), or classification results. -
Geometric Modeling in Shape Space
Geometric Modeling in Shape Space Martin Kilian Niloy J. Mitra Helmut Pottmann Vienna University of Technology Figure 1: Geodesic interpolation and extrapolation. The blue input poses of the elephant are geodesically interpolated in an as-isometric- as-possible fashion (shown in green), and the resulting path is geodesically continued (shown in purple) to naturally extend the sequence. No semantic information, segmentation, or knowledge of articulated components is used. Abstract space, line geometry interprets straight lines as points on a quadratic surface [Berger 1987], and the various types of sphere geometries We present a novel framework to treat shapes in the setting of Rie- model spheres as points in higher dimensional space [Cecil 1992]. mannian geometry. Shapes – triangular meshes or more generally Other examples concern kinematic spaces and Lie groups which straight line graphs in Euclidean space – are treated as points in are convenient for handling congruent shapes and motion design. a shape space. We introduce useful Riemannian metrics in this These examples show that it is often beneficial and insightful to space to aid the user in design and modeling tasks, especially to endow the set of objects under consideration with additional struc- explore the space of (approximately) isometric deformations of a ture and to work in more abstract spaces. We will show that many given shape. Much of the work relies on an efficient algorithm to geometry processing tasks can be solved by endowing the set of compute geodesics in shape spaces; to this end, we present a multi- closed orientable surfaces – called shapes henceforth – with a Rie- resolution framework to solve the interpolation problem – which mannian structure. -
Shape Synthesis Notes
SHAPE SYNTHESIS OF HIGH-PERFORMANCE MACHINE PARTS AND JOINTS By John M. Starkey Based on notes from Walter L. Starkey Written 1997 Updated Summer 2010 2 SHAPE SYNTHESIS OF HIGH-PERFORMANCE MACHINE PARTS AND JOINTS Much of the activity that takes place during the design process focuses on analysis of existing parts and existing machinery. There is very little attention focused on the synthesis of parts and joints and certainly there is very little information available in the literature addressing the shape synthesis of parts and joints. The purpose of this document is to provide guidelines for the shape synthesis of high-performance machine parts and of joints. Although these rules represent good design practice for all machinery, they especially apply to high performance machines, which require high strength-to-weight ratios, and machines for which manufacturing cost is not an overriding consideration. Examples will be given throughout this document to illustrate this. Two terms which will be used are part and joint. Part refers to individual components manufactured from a single block of raw material or a single molding. The main body of the part transfers loads between two or more joint areas on the part. A joint is a location on a machine at which two or more parts are fastened together. 1.0 General Synthesis Goals Two primary principles which govern the shape synthesis of a part assert that (1) the size and shape should be chosen to induce a uniform stress or load distribution pattern over as much of the body as possible, and (2) the weight or volume of material used should be a minimum, consistent with cost, manufacturing processes, and other constraints. -
Geometry and Art LACMA | | April 5, 2011 Evenings for Educators
Geometry and Art LACMA | Evenings for Educators | April 5, 2011 ALEXANDER CALDER (United States, 1898–1976) Hello Girls, 1964 Painted metal, mobile, overall: 275 x 288 in., Art Museum Council Fund (M.65.10) © Alexander Calder Estate/Artists Rights Society (ARS), New York/ADAGP, Paris EOMETRY IS EVERYWHERE. WE CAN TRAIN OURSELVES TO FIND THE GEOMETRY in everyday objects and in works of art. Look carefully at the image above and identify the different, lines, shapes, and forms of both GAlexander Calder’s sculpture and the architecture of LACMA’s built environ- ment. What is the proportion of the artwork to the buildings? What types of balance do you see? Following are images of artworks from LACMA’s collection. As you explore these works, look for the lines, seek the shapes, find the patterns, and exercise your problem-solving skills. Use or adapt the discussion questions to your students’ learning styles and abilities. 1 Language of the Visual Arts and Geometry __________________________________________________________________________________________________ LINE, SHAPE, FORM, PATTERN, SYMMETRY, SCALE, AND PROPORTION ARE THE BUILDING blocks of both art and math. Geometry offers the most obvious connection between the two disciplines. Both art and math involve drawing and the use of shapes and forms, as well as an understanding of spatial concepts, two and three dimensions, measurement, estimation, and pattern. Many of these concepts are evident in an artwork’s composition, how the artist uses the elements of art and applies the principles of design. Problem-solving skills such as visualization and spatial reasoning are also important for artists and professionals in math, science, and technology. -
Topics in Complex Analytic Geometry
version: February 24, 2012 (revised and corrected) Topics in Complex Analytic Geometry by Janusz Adamus Lecture Notes PART II Department of Mathematics The University of Western Ontario c Copyright by Janusz Adamus (2007-2012) 2 Janusz Adamus Contents 1 Analytic tensor product and fibre product of analytic spaces 5 2 Rank and fibre dimension of analytic mappings 8 3 Vertical components and effective openness criterion 17 4 Flatness in complex analytic geometry 24 5 Auslander-type effective flatness criterion 31 This document was typeset using AMS-LATEX. Topics in Complex Analytic Geometry - Math 9607/9608 3 References [I] J. Adamus, Complex analytic geometry, Lecture notes Part I (2008). [A1] J. Adamus, Natural bound in Kwieci´nski'scriterion for flatness, Proc. Amer. Math. Soc. 130, No.11 (2002), 3165{3170. [A2] J. Adamus, Vertical components in fibre powers of analytic spaces, J. Algebra 272 (2004), no. 1, 394{403. [A3] J. Adamus, Vertical components and flatness of Nash mappings, J. Pure Appl. Algebra 193 (2004), 1{9. [A4] J. Adamus, Flatness testing and torsion freeness of analytic tensor powers, J. Algebra 289 (2005), no. 1, 148{160. [ABM1] J. Adamus, E. Bierstone, P. D. Milman, Uniform linear bound in Chevalley's lemma, Canad. J. Math. 60 (2008), no.4, 721{733. [ABM2] J. Adamus, E. Bierstone, P. D. Milman, Geometric Auslander criterion for flatness, to appear in Amer. J. Math. [ABM3] J. Adamus, E. Bierstone, P. D. Milman, Geometric Auslander criterion for openness of an algebraic morphism, preprint (arXiv:1006.1872v1). [Au] M. Auslander, Modules over unramified regular local rings, Illinois J. -
Phylogenetic Algebraic Geometry
PHYLOGENETIC ALGEBRAIC GEOMETRY NICHOLAS ERIKSSON, KRISTIAN RANESTAD, BERND STURMFELS, AND SETH SULLIVANT Abstract. Phylogenetic algebraic geometry is concerned with certain complex projec- tive algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers. 1. Introduction Our title is meant as a reference to the existing branch of mathematical biology which is known as phylogenetic combinatorics. By “phylogenetic algebraic geometry” we mean the study of algebraic varieties which represent statistical models of evolution. For general background reading on phylogenetics we recommend the books by Felsenstein [11] and Semple-Steel [21]. They provide an excellent introduction to evolutionary trees, from the perspectives of biology, computer science, statistics and mathematics. They also offer numerous references to relevant papers, in addition to the more recent ones listed below. Phylogenetic algebraic geometry furnishes a rich source of interesting varieties, includ- ing familiar ones such as toric varieties, secant varieties and determinantal varieties. But these are very special cases, and one quickly encounters a cornucopia of new varieties. The objective of this paper is to give an introduction to this subject area, aimed at students and researchers in algebraic geometry, and to suggest some concrete research problems. The basic object in a phylogenetic model is a tree T which is rooted and has n labeled leaves. -
CURVATURE E. L. Lady the Curvature of a Curve Is, Roughly Speaking, the Rate at Which That Curve Is Turning. Since the Tangent L
1 CURVATURE E. L. Lady The curvature of a curve is, roughly speaking, the rate at which that curve is turning. Since the tangent line or the velocity vector shows the direction of the curve, this means that the curvature is, roughly, the rate at which the tangent line or velocity vector is turning. There are two refinements needed for this definition. First, the rate at which the tangent line of a curve is turning will depend on how fast one is moving along the curve. But curvature should be a geometric property of the curve and not be changed by the way one moves along it. Thus we define curvature to be the absolute value of the rate at which the tangent line is turning when one moves along the curve at a speed of one unit per second. At first, remembering the determination in Calculus I of whether a curve is curving upwards or downwards (“concave up or concave down”) it may seem that curvature should be a signed quantity. However a little thought shows that this would be undesirable. If one looks at a circle, for instance, the top is concave down and the bottom is concave up, but clearly one wants the curvature of a circle to be positive all the way round. Negative curvature simply doesn’t make sense for curves. The second problem with defining curvature to be the rate at which the tangent line is turning is that one has to figure out what this means. The Curvature of a Graph in the Plane. -
Chapter 13 Curvature in Riemannian Manifolds
Chapter 13 Curvature in Riemannian Manifolds 13.1 The Curvature Tensor If (M, , )isaRiemannianmanifoldand is a connection on M (that is, a connection on TM−), we− saw in Section 11.2 (Proposition 11.8)∇ that the curvature induced by is given by ∇ R(X, Y )= , ∇X ◦∇Y −∇Y ◦∇X −∇[X,Y ] for all X, Y X(M), with R(X, Y ) Γ( om(TM,TM)) = Hom (Γ(TM), Γ(TM)). ∈ ∈ H ∼ C∞(M) Since sections of the tangent bundle are vector fields (Γ(TM)=X(M)), R defines a map R: X(M) X(M) X(M) X(M), × × −→ and, as we observed just after stating Proposition 11.8, R(X, Y )Z is C∞(M)-linear in X, Y, Z and skew-symmetric in X and Y .ItfollowsthatR defines a (1, 3)-tensor, also denoted R, with R : T M T M T M T M. p p × p × p −→ p Experience shows that it is useful to consider the (0, 4)-tensor, also denoted R,givenby R (x, y, z, w)= R (x, y)z,w p p p as well as the expression R(x, y, y, x), which, for an orthonormal pair, of vectors (x, y), is known as the sectional curvature, K(x, y). This last expression brings up a dilemma regarding the choice for the sign of R. With our present choice, the sectional curvature, K(x, y), is given by K(x, y)=R(x, y, y, x)but many authors define K as K(x, y)=R(x, y, x, y). Since R(x, y)isskew-symmetricinx, y, the latter choice corresponds to using R(x, y)insteadofR(x, y), that is, to define R(X, Y ) by − R(X, Y )= + . -
Math, Physics, and Calabi–Yau Manifolds
Math, Physics, and Calabi–Yau Manifolds Shing-Tung Yau Harvard University October 2011 Introduction I’d like to talk about how mathematics and physics can come together to the benefit of both fields, particularly in the case of Calabi-Yau spaces and string theory. This happens to be the subject of the new book I coauthored, THE SHAPE OF INNER SPACE It also tells some of my own story and a bit of the history of geometry as well. 2 In that spirit, I’m going to back up and talk about my personal introduction to geometry and how I ended up spending much of my career working at the interface between math and physics. Along the way, I hope to give people a sense of how mathematicians think and approach the world. I also want people to realize that mathematics does not have to be a wholly abstract discipline, disconnected from everyday phenomena, but is instead crucial to our understanding of the physical world. 3 There are several major contributions of mathematicians to fundamental physics in 20th century: 1. Poincar´eand Minkowski contribution to special relativity. (The book of Pais on the biography of Einstein explained this clearly.) 2. Contributions of Grossmann and Hilbert to general relativity: Marcel Grossmann (1878-1936) was a classmate with Einstein from 1898 to 1900. he was professor of geometry at ETH, Switzerland at 1907. In 1912, Einstein came to ETH to be professor where they started to work together. Grossmann suggested tensor calculus, as was proposed by Elwin Bruno Christoffel in 1868 (Crelle journal) and developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita (1901). -
Round Table Talk: Conversation with Nathan Seiberg
Round Table Talk: Conversation with Nathan Seiberg Nathan Seiberg Professor, the School of Natural Sciences, The Institute for Advanced Study Hirosi Ooguri Kavli IPMU Principal Investigator Yuji Tachikawa Kavli IPMU Professor Ooguri: Over the past few decades, there have been remarkable developments in quantum eld theory and string theory, and you have made signicant contributions to them. There are many ideas and techniques that have been named Hirosi Ooguri Nathan Seiberg Yuji Tachikawa after you, such as the Seiberg duality in 4d N=1 theories, the two of you, the Director, the rest of about supersymmetry. You started Seiberg-Witten solutions to 4d N=2 the faculty and postdocs, and the to work on supersymmetry almost theories, the Seiberg-Witten map administrative staff have gone out immediately or maybe a year after of noncommutative gauge theories, of their way to help me and to make you went to the Institute, is that right? the Seiberg bound in the Liouville the visit successful and productive – Seiberg: Almost immediately. I theory, the Moore-Seiberg equations it is quite amazing. I don’t remember remember studying supersymmetry in conformal eld theory, the Afeck- being treated like this, so I’m very during the 1982/83 Christmas break. Dine-Seiberg superpotential, the thankful and embarrassed. Ooguri: So, you changed the direction Intriligator-Seiberg-Shih metastable Ooguri: Thank you for your kind of your research completely after supersymmetry breaking, and many words. arriving the Institute. I understand more. Each one of them has marked You received your Ph.D. at the that, at the Weizmann, you were important steps in our progress.