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Math Expands

Already in the nineteenth century, people realized that time could be treated as a dimension with many similarities to the three dimensions of space, and this theme was taken up and developed by twentieth-century relativity physicists. These scientists studied events, which can be specified by four coordinates, three of space and one of time. One might make an appointment to meet someone at a building on the corner of Third Avenue and Fourth Street on the fifth floor, and (3,4,5) would be a good abbreviation for that location, but the appointment requires a fourth coordinate, say 10 a.m., to specify the time as well. Thus the event could be recorded as (3,4,5,10). Describing the geometry of such events uses a mathematics of four-dimensional objects.

There are other fourth dimensions besides time, however, and we have to learn how to operate in those as well. In order to present a good representation of a point cloud with four numbers for each observation, we need to know how to draw a four- dimensional box and either project it into three-space or slice it by a three-space. We are in a challenging position quite analogous to that of the narrator of trying to think of ways to represent in his two-dimensional world an image of a cube from three-space.

Four-space can be rotated about the origin by specifying the new positions of the unit vectors along the four coordinate axes. An interactive demo is available for you to experiment with this yourself.

Modern physics goes well beyond four-space to study objects that require ten or twenty-six dimensions for their effective representation. See the work of Michio Kaku for a good introduction to these notions, referring back to the analogy. Jeff Weeks uses dimensional analogies to discuss cosmology, in particular the possible geometric models for the shape of our universe. At this site you will find links to many activities that give insight into two-dimensional surfaces in complicated three- and four-dimensional spaces, and generalizations of surfaces called three-manifolds.

Madeleine L'Engle is another author who has used concepts from four dimensions and higher in her work. Her novel "A Wrinkle in Time" describes events in a world of five dimensions, the usual four of ordinary space-time plus a fifth number giving an additional coordinate in space.

The mathematics of computer animation uses many different coordinates to represent the position and the movement of characters on the plane, simulating action in three-dimensional space. Like the choreographer Julie Strandberg, the animators Edwin Catmull and Tony DeRose must identify the position of each character using many numbers for each part of the body, and then tell how each part moves relative to the other parts. To prepare for a career in animation, or in the design of simulations of video games, you need all the math you can get.

An interesting site that combines art and mathematics is "Surfaces Beyond the Third Dimension", originally a physical exhibition at the Providence Art Club, but continuing now as a virtual experience on the Web. It provides images and movie clips of various surfaces that exist in four dimensions, and includes descriptions of the mathematics, as well as the artistic motivations for some of these images.

Here is another page that includes a gallery of art influenced by the fourth dimension, plus a list of references to books about the fourth dimension.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.