For many centuries before people considered the concept of higher-dimensional space, they recognized numerical and algebraic patterns in plane and solid geometry. Artisans and scientists and mathematicians developed formulas to describe the regularities that they observed in their measurements, and they knew that the coefficients and exponents that appeared in these formulas were related to the dimensions of the space in which they were working. In a real sense, the concept of dimension became identified with the exponents, with exponent 2 appearing in the formulas of plane geometry and exponent 3 in the formulas of solid geometry. But some of the algebraic patterns that had analogous forms in two- and three-dimensional geometry also had analogues with exponent 4 or higher. What sort of geometry might correspond to these new relationships? The examination of the way numerical and algebraic patterns work in plane and solid geometry. paved the way for the appearance of higher-dimensional geometry.
Differences in formulas appear when we consider measurements of analogous objects in different dimensions, We can see the characteristic features of dimensions most clearlv when we scale an object up or down. Consider the problem of preparing a photograph for mailing. A square photograph requires a certain amount of string and a certain amount of wrapping paper. If we double the size of the square photograph, we need twice as much string and four times as much paper. Doubling the size of a cubical box requires twice as much string, four times as much paper, and eight times as much packing material. Similarly, if we double the size of an entrance hall, then all linear quantities, like the length of wiring, are doubled. But the quantities involving area, like the amount of paint for the walls and the square feet of carpeting for the floor, are multiplied by four, and quantities involving volume, like the cubic feet of space to be handled by the air conditioning units, increase by a factor of eight.
Enlarging a photograph by doubling all dimensions multiplies the area by four. |
The quantities length, area, and volume express "the amount of material" for objects in different dimensions. The significant fact is that we can determine the dimension simply bv looking at the power of two by which the quantity is multiplied when the size is doubled. A quantity like volume is called three-dimensional if it is multiplied bv two to the third power when the size of the object is doubled. The two-dimensional quantity, area, is multiplied by two to the second power when the size is doubled, while the one-dimensional quantity, length, is multiplied by just two (or "two to the first power") when the size is doubled. If we encounter a quantity that it is multiplied by 16, two to the fourth power, when the size is doubled, then we would say that the quantity is four-dimensional. A five-dimensional quantity would be multiplied by 32 if the size were doubled. [an error occurred while processing this directive]