Beyond the Third Dimension
Date: 5/16/96 at 17:23:49 From: Anonymous Subject: Beyond the Third Dimension I am searching for information on beyond the third dimension. Everywhere I go I get technical information that I don't understand. I am in high school and I have to write a five page paper on this. I need as much information as well as pictures or diagrams on this subject as I can find. Every source I get has a different explanation and I am very confused at this point. Any help would be incredibly appreciated. Thanks in advance, Rob
Date: 5/16/96 at 21:0:40 From: Doctor Jodi Subject: Re: Beyond the Third Dimension Hi there! Actually, the subject line (Beyond the third dimension) is also the title of a book by Thomas Banchoff. This is a great book designed to introduce the fourth dimension - you might give it a read if you have a chance. Here's something to start with: From: Bob Hesse Subject: Viewing Four-dimensional Objects in Three Dimensions Date: Tue, 6 Sep 1994 15:54:12 GMT Viewing four-dimensional objects in three dimensions: Given that humans only visualize three dimensions, how is it possible to visualize four-dimensional, or higher, objects? This question is the underlying idea of a short novel written over a hundred years ago by Edwin A. Abbot called FLATLAND. FLATLAND is a story about two-dimensional creatures--triangles, squares, circles and other polygons--that live on a plane. The story contains a section where one of the squares is visited by a three-dimensional object, a sphere. The sphere explains to the square the existence of higher dimensional objects like itself, and ways in which the square can understand the form of such objects. The method the sphere gives to the square can be generalized so that the form of four-dimensional objects can be seen in three dimensions. This method of viewing higher-dimensional objects as well as others is one way people can understand the shape of higher- dimensional space. Before attempting to view four-dimensional objects in three-dimensional space, let us consider viewing a three dimensional object in two-dimensional space. In FLATLAND, the method in which the sphere showed its form to square was by raising its body through the Flatland surface. The square saw at first a point that quickly grew to a circle, which continued increasing in size, and then started decreasing in size until it became a point, and then it disappeared. So the square perceived the sphere to be an infinite collection of circles pieced together... Before continuing further, it should be mentioned that for simplicity's sake and for aesthetic purposes, the forms which we will consider viewing are polytopes, the generalized term for polyhedra and polygons. A second way to view three-dimensional polytopes in two dimensions is by means of a projection. Projection is a popular method for Cartographers to create maps of the world from a globe. For instance the United Nations flag is created by a projection of the globe about the south pole. One especially useful type of projection in mathematics is called stereographic projection. Stereographic projection takes a sphere and maps it over the entire plane in the following manner. If one lays a sphere on a plane, the point of the sphere touching the plane stays fixed while the point directly opposite it, i.e. "the North Pole" gets sent to infinity. Any other point on the sphere is sent to the unique point on the plane found by intersecting the plane with a line made from the point at the north pole and the point on the sphere... A third way to view polyhedra in two dimensions is through a method defined by Barbara Hausmann and Hans-Peter Seidel as "Cut-Throughs" and "Fold-Downs." Since polyhedra have as faces regular polygons, one could cut a polyhedra on the edges and fold it in a way so that all the faces are lying on the plane... As you may have already surmised, all of the above methods can be used to visualize four-dimensional polytopes in three dimensions. But before showing these different ways of viewing polytopes, an explanation of how these polytopes are constructed is in order. As regular polyhedra are constructed from regular polygons, so are regular 4-dimensional polytopes constructed from regular polyhedra. Recall that there are only five regular polyhedra: 1. The tetrahedron, constructed from four equilateral triangles. 2. The cube, constructed from six squares. 3. The octahedron, constructed from eight equilateral triangles. 4. The dodecahedron, constructed from 12 regular pentagons. 5. The icosahedron, constructed from twenty equilateral triangles. There are only six four-dimensional polytopes. They are the following: 1. The 4-simplex, constructed from five tetrahedra, three tetrahedra meeting at an edge. 2. The hypercube, constructed from eight cubes meeting three per edge. 3. The 16-cell, constructed from sixteen tetrahedra, with four tetrahedra meeting at an edge. 4. The 120-cell, constructed from 120 dodecahedra, with three dodecahedra meeting per edge. 5. The monstrous 600-cell, constructed from 600 tetrahedra, with five tetrahedra meeting at an edge. Since the above examples of viewing three-dimensional polytopes in two dimensions all contain the cube, let us continue viewing in the fourth dimension by looking at the hypercube. First, let us look at some projections of the hypercube. Figure 5 is a series of diagrams created by rotating the hypercube about a plane in four dimensions, or a combination of plane rotations. Figure 6 is a sequence of pictures of a slicing of the hypercube into three dimensions. Note that as the hypercube passes through our three-dimensional space, it is growing and then shrinking from various polyhedra shapes. This is analogous to the slicing of the cube in the plane shown earlier. Figure 7 is a set of pictures showing the step-by-step approach of the hypercube being cut-through and folded-down. Note that in each stage of the process a cube pops out of the hypercube, which is analogous to a square coming out of a cube as one dissects a cube. In FLATLAND, the square discovered that the sphere was an infinite collection of circles. However the square was unable ever to actually view the sphere in the same way we three-dimensional beings are able. Similarly, we can discover what some four-dimensional objects look like by viewing aspects of them in three dimensions. But like the square, we are limited in understanding the whole nature of these objects. This article is based on an interview and a seminar given by Barbara Hausmann at the Geometry Center... [see] http://www.geom.uiuc.edu/docs/forum/polytope/ __________ As far as reading about the fourth dimension goes, there are quite a few good books around. BOOKS - Flatland by Edwin Abbott This classic novel focuses on the life of our 2D hero, A. Square, who is declared mad and imprisoned for discussing his travels into a 1D world and his encounter with a 3D sphere. *GREAT INTRO - Planiverse Talks about lots of sticky details about life in 2D--from zippers and doors to playing volleyball. It's an awesome story and again should give you ideas about 4D. - Sphereland - Shape of Space by Jeff Weeks Asks the question, "What is the shape of the universe?" and explores the mathematics of 4D by introducing elementary topology. No college math needed. (also a Geometry Center video) ON THE WEB http://www.geom.uiuc.edu/docs/forum/polytope/ - an illustrated article by Bob Hess called "Viewing four-dimensional objects in three dimensions" http://mathforum.org/dr.math/problems/dixon24.html - a simple answer to the question "What is the fourth dimension mathematically?" http://www.uccs.edu/~eswab/hyprspac.htm - 4D polytopes, the 4D equivalent of polygons and polyhedra *GREAT pictures http://www.graphics.cornell.edu/~gordon/peek/peek.html - description of Peek, an interactive software program to look at projections and cross-sections of higher dimensional objects http://mathforum.org/~sarah/HTMLthreads/articletocs/4d.visualization.html - interviews with mathematicians: visualizing 4D in their work OTHER A MATHEMATICS SAMPLER, by W. Berlinghoff and K. Grant Chapter 8, "Mathematics of Space and Time," includes some detailed explanations of hypercubes, hyperspheres, hypercylinders, etc., how to examine them by looking at cross sections, and the like. It's written for a first-year college liberal-arts audience, so the mathematics is not very sophisticated. However, the reading level might be a little above the comfort range of some high school students. In any event, the material is easily adaptable to high-school mathematics. The book is published by Ardsley House (New York); 3rd Edition, 1992. The final section of the chapter is a 6-page overview of "4-Space in Fiction and in Art" which supplies some connections between this branch of mathematics and some well-known artistic and literary works of the past 100 years or so (including Dali's "Corpus Hypercubus" cited in Heidi B.'s earlier posting.) Good luck with your project and let us know how it goes! Write back if you need more help! -Doctor Jodi, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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