# 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?