Over one-hundred years ago Edwin Abbott Abbott used slicing to illustrate the dimensional analogy in his classic satire Flatland.^{1} It is a great exercise to try to take on the viewpoint of A Square, living in a two-dimensional universe, especially when he is visited by a sphere from a higher dimension. The frustrated attempts of the sphere to teach A Square about the third dimension give wonderful insights into the challenges of communication and visualization in geometry. (Early parts of Flatland may be difficult for some students, and some of the social satire may be skipped over at first reading. Abbott was an active education reformer and worker for equality who was satirizing the narrow-minded attitudes of Victorian England with respect to class society and particularly with respect to women. Only at the end does A Square begin to gain a more enlightened view of his society.)
What would happen if we were visited by a sphere from a dimension higher than our own? Instead of growing and changing circles in a plane, we would see growing and changing spheres in space. We would be inclined to interpret such an event as the inflation and deflation of a balloon, but the point of the exercise is that such a phenomenon could be interpreted equally well as the slices of a hypersphere penetrating our three-dimensional universe.
If A Square were visited by a cube from the third dimension, he would see a variety of polygons, depending on the position of the cube as it passed different water levels. What would be the analogous three-dimensional slices of a four-dimensional hypercube? This is one place where computer graphics can be of great help (as in the film The Hypercube: Projections and Slicing).^{3}
Slicing techniques are important in many modern scientific applications, especially since the development of computer graphics. X-ray tomography uses computer graphics in the reconstruction of three-dimensional objects from planar sections. Topographers and geologists construct and analyze contour maps showing the elevations of different configurations above and below the surface of the earth. Similar slicing methods are used by biologists, while researchers in materials science use computer graphics to show the parts of a three-dimensional surface with a given temperature or density. Exploratory data analysis uses techniques of projections and slicing to investigate high-dimensional data sets from social sciences as well as from the physical and biological sciences.
Students of calculus will appreciate the power of slicing techniques — for example, in relating the volume of a surface of revolution to the changing areas of its circular cross sections or in finding the contour lines on the surface of a graph in three-space. Long before students are introduced to the notions of critical point theory, they can already understand and appreciate slicing phenomena that relate different dimensions. What happens if we slice a doughnut or a bagel in different directions? It is easy to carry out the actual experiments and see that there are positions where the slice yields a pair of circles. Less obvious is the slice that consists of two interlocked circles. Again, a good way to see this would be to experiment with a transparent inner tube filled halfway with colored liquid. Geometry can be a surprising observational science.
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. |