Tesi di Laurea — Diplomarbeit The concept of dimension: a theoretical and an informal approach Ester Dalvit Relatore Betreuer Prof. Dr. Domenico Luminati Prof. Dr. J¨urgen Hausen Anno Accademico 2006–2007 Contents Introduction vii Introduzione xi Einleitung xv I A survey on dimension 1 1 Dimension of manifolds 3 1.1 Basicconcepts........................... 3 1.2 Tangentspace........................... 5 1.3 Sard’sTheorem .......................... 8 1.4 SomeresultsinAlgebraicTopology . 15 1.5 Invarianceofdomain . 16 2 Dimension of affine varieties 21 2.1 Basicconcepts........................... 22 2.2 Affinetangentspaces . 27 2.3 Derivations ............................ 29 2.4 Regularandsingularpoints . 36 2.5 Transcendencedegree. 38 2.6 Morphisms............................. 42 2.7 Krulldimension.......................... 45 2.8 Zariskidimension . .. .. .. 47 2.9 Krulldimensionforrings. 49 3 Dimension of metric spaces 53 3.1 Three concepts of dimension . 53 3.2 Somebasictheory......................... 57 3.3 The large inductive dimension . 60 3.4 SumtheoremforInd ....................... 66 3.5 Subspace and decomposition theorems for Ind . 69 3.6 ProducttheoremforInd . 70 3.7 Coveringdimension. 71 3.8 Coincidencetheorems. 80 3.9 The dimension of the Euclidean space . 85 II Informal learning of mathematics 89 4 An exhibition about the 4D space 91 4.1 Somebasicdefinitions . 92 4.2 Contentsoftheexhibition . 94 4.2.1 Introduction........................ 94 4.2.2 Models........................... 94 4.2.3 Symmetry ......................... 97 4.2.4 Slices............................ 97 4.2.5 Dice ............................ 98 4.2.6 Flipacube ........................ 98 4.2.7 Polytopes ......................... 98 4.3 Flipacube ............................ 99 4.3.1 Animationsandimages. .101 4.3.2 Flipasquare .......................103 4.3.3 Flipacube ........................106 4.4 Otheranimations . .110 4.4.1 Foldingupa(hyper)cube. 110 4.4.2 Projections ........................112 4.4.3 Hypersphere. .117 4.5 Futurework ............................120 Acknowledgments 121 Bibliography 123 List of Figures 1.1 Chartsandacharttransformation. 4 4.1 The unfoldings of a cube and a hypercube. 95 4.2 Parallel projections of a cube and a hypercube. 95 4.3 Schlegel diagram of a cube and a hypercube. 95 4.4 A model of the two tori of a hypercube (image by matematita). 96 4.5 The rotation axes of the cube in a kaleidoscope. 97 4.6 The cube present in the exhibition. 100 4.7 Howtoflipasquareonthesurfaceofacube. 101 4.8 Flippingasquareonacube. .103 4.9 Folding up the cube to see the square. 104 4.10 Folding up the cube to see the square. 105 4.11 A square seems a trapezium in the Schlegel diagram. 106 4.12 A square seem a line segment in the Schlegel diagram. 106 4.13 Flippingacubeonahypercube. 107 4.14 Changing the unfolding of the hypercube. 107 4.15 How to flip a cube on an (unfolded) hypercube. 108 4.16 How to flip a cube on a hypercube (Schegel diagram). 109 4.17 Foldingacubeandahypercube.. 110 4.18 Howtofoldacubeinthe3Dspace. 111 4.19 Howtofoldahypercubeinthe4Dspace.. 112 4.20 Projectionsofahypercube. 113 4.21 Projectionsofacube.. .114 4.22 Projectionsofahypercube. 115 4.23 A (deformed) sphere inside the solid tori of the hypersphere. 117 4.24 The sphere deformation in the hypersphere. 118 Introduction In this work we analyze the concept of dimension from different viewpoints and we describe the preparation of an exhibition about the four dimensional Euclidean space. The concept of dimension plays a central role in many fields of mathe- matics, especially in geometry, being one of the first tools used to classify objects. In general, it is important to find some properties that will be used to define and characterize the different classes of classes. The dimension is often one of the first properties that are investigasted. The most natural class of spaces one can refer to is the class of Euclidean spaces. When one gives a definition of dimension for a class of spaces which contains the Euclidean spaces, it is then important to verify that this definition coincides with the usual one on Euclidean spaces. The definitions of dimension are often given independently in the different fields of mathematics. It is important to verify that they coincide on spaces that can be considered as subjects of different fields. Nevertheless, even in the cases where the coincidence of different definitions is a well known fact, the proofs are often difficult to be found in the literature. Even if in mathematics one defines and studies spaces with an arbitrary number of dimensions — the dimension can be a natural or real number or even infinite —, most people have only an intuitive notion of dimension, which derives from the experience. It is then natural to think of objects as having length, width, and height and also distinguish the different nature of linear, plane and solid objects. The relevant dimension of the thread is its length, while to measure a sheet one needs length and height, and for a die also a third dimension is to be considered. In some sense, we consider the thread as one dimensional and the sheet as two dimensional, even if both objects are three dimensional. This is a first “intuitive and informal formalization” of the concept of dimension and is already clear to children. We can exploit this notion to give an idea of the properties of four dimensional objects and, reflecting on similarities and differences between linear, plane and solid objects, it is possible to “jump” to the four dimensions. viii Introduction The concept of n-dimensional space (n 4) is not easy to be illustrated, ≥ especially because people often try to find representations through drawings, but it can be very hard to find efficacious representations of objects in a higher dimension and also to interpret and understand them. Nevertheless, imagination and fantasy of many authors have been fascinated by the four dimensional space and the literature has shown interest towards the concept of dimension since the late 19th century. One of the first works about the concept of dimension and perhaps the most famous, which has inspired many other literary works, is Flatland: A Romance of Many Dimensions, published by Edwin A. Abbott in 1884. The author uses the novel to criticize the society of his time, but this work is very interesting also from the mathematical point of view. The protagonist is a square, living in a two dimensional world, which gets in touch with a sphere and makes a journey in the three dimensional world. In this manner the author presents the three dimensional objects from the point of view of the square, which is at first sceptical about the existence of a third dimension. This is perhaps the first time that the general public is invited to reflect on the concept of dimension and on the possibility that other universes with a different number of dimensions can exist or at least can be imagined. In the same period Charles H. Hinton published two “scientific romances”, as he called them, What is the Fourth Dimension? and A Plane World, where he presents some ideas about the concept of dimension. He also invented the words ana and kata (from the Greek “up” and “down”), to indicate the two opposite directions in the fourth dimension, the equivalents of left and right, forwards and backwards, and up and down. Vinn and vout are other words with the same meaning of ana and kata, coined by the science fiction author Rudy Rucker in his 2002 book Spaceland, a story in which a man meets a four dimensional being. Henry More, an English philosopher who lived in the 17th century, had already given a name to the fourth spatial dimension, calling it spissitude. He thought that the spiritual beings are four dimensional and they can reduce their extension in the first three dimensions to become larger in the fourth dimension. Many other short stories and novels dealing with the concept of di- mensions have been written. Ian Stewart indagates various concepts of dimensions in his 2001 book Flatterland. The Planiverse is a novel by Alexander K. Dewdney, written in 1984, which describes a two dimensional world, its physics, biology and chemistry. The novel Sphereland was written in 1965 by Dionys Burger as a sequel of Flatland and indagates the spherical geometry, to arrive to the concept that our universe is the hypersurface of a hypersphere. Robert A. Heinlein’s novel “–And He Built a Crooked House–” was written in 1941 and talks about an adventure in a hypercubic house. Introduction ix The science fiction, both in films and books, often uses the concept of “hyperspace” to give an atmosphere of mistery and suggestion. Exploiting the people’s curiosity for these topics makes perhaps possible to draw the attention of the general public to an exhibition on four dimensional objects and different ways of representing and visualizing them. Suggestive ideas like hyperspace are powerful tools to make the visitor be interested in the exhibition and many related topics can be presented and explained, for example projections, symmetries, rotations, polyhedra and polytopes, sections, knots and links. This is the aim of an exhibition which is being prepared by the interuniversitary research centre for the communication and informal learning of mathematics matematita. A part of the work for this thesis was done within the project of the exhibition. Many ideas of this work were inspired by the work of Thomas Banchoff, in particular the animations about the hypercube and the hypersphere. Banchoff uses electronic media to support his mathematical researches and as a powerful tool in mathematical educational. He developed many web-based projects1 to use computer graphics for education and informal learning of mathematics. Also his book Beyond the Third Dimension and the web pages2 with the same title, which contain a virtual exhibition about mathemtical objects, recall and explain the concept of dimension. The present work is divided into two parts: in the first one, consisting of three chapters, we present various definitions of dimension, giving attention to different fields of mathematics.