Graphs in Higher Dimensions

B3D, chapter 4, includes lengthy discussions of two topics in the realm of higher-dimensionality: calculations of the numbers of vertices, sides, etc. of an n-simplex or n-cube, and the representation of data with more than two or three variables. Seeing these topics covered in the same chapter prompted me to wonder if there could possibly be a relation between them. I recall learning in precalculus that the highest exponent of a function greatly influences the shape of its graph. With the pollen example, it is, however, a different issue; in one example, the higher dimension is that of the exponent, while in the other, the higher dimension corresponds to the number of variables. How are these attributes of a function related? Clearly, both are important in dealing with dimensionality; as we have seen, there is a significant link between exponents and dimensions, hence the nomenclature wherein a quantity to the second power is "squared," and the third power is a "cube". (Of course there is no corresponding nickname for the fourth power... how archaic!) The number of variables also codes for aspects of higher dimensions; a cube needs three variables (x,y,z) to code for the locations of it points. What, then, is the relationship between the functions (x,y)^3 and (x,y,z)^2? And what of (x,y,z)^3?

While on the subject of graphs, I am also reminded of another curiosity of precal: the polar graph. I remember clearly the day that those wacky circluar pages in the backs of our graph paper notebooks became more than a place to draw make-believe radar screens. The polar graph has a special significance, because sinusoidal graphs make pretty heart shapes when plotted on them, which has the advantage of demonstrating the periodic and cyclical nature of such functions. In three-space, a polar graph would then take the form of a sphere (or a ball or whatever). What sort of graphs would benefit from this sort of representation? Perhaps something like sin x = cos y? Would a torus-shaped graph have similar properties? And what about a hyper-sphere or hyper-torus?

Professor Banchoff's Comments

Dear Dave,

You are right that the two main topics in this chapter seems rather disparate at first, but they do fit together, as you indicate. Somehow there are these data, easy enough to describe on a chart but not that easy to visualize. Graphing is one of the key elements, and when I teach the calculus course in functions of two variables, we spend a great deal of time trying to extend the graphing concepts from beginning calculus into at least the third dimension. Somehow the richness of those polar coordinate graphs in the plane is not captured quite as well in spherical coordinates, though some of it is. The physicists love to talk about "Spherical Harmonics", that extend those sine and cosine graphs and the floral patterns they can generate in the plane. You are right that the torus shows up in really nice guises as we play with different ways of setting up coordinates in four-space, similar to but not identical to the way we put latitude and longitude on the sphere (and then look at equations such as the one you mentioned: sin x = cos y).

Maybe we can have a demo sometime of the graphing routines we used in Math 35. The shapes there are quite fascinating, and the techniques are great for trying to analyze real-world data sets.

Prof. B.

-david stanke