This is Part 4 of a discussion on higher dimensionality. In Part 1 I mentioned a story by Robert A. Heinlein in which a four-dimensional cube (hypercube) played a role in the story. I promised there to provide some discussion of what was claimed to be true of hypercubes in the story. Well, here it begins.
Is there any simple way for the layperson to think about spatial dimensions higher than the three the mind is capable of? Can enough insight be gained to allow the average person to evaluate the merit of the “higher-dimensional” approach? The answer to these questions seems to be yes. However, the purpose of the short discussion that follows is not to be exhaustive, but simply to provide some examples in the realm of geometry that the reader can easily visualize. It should be noted that adding a new dimension to the treatment of a problem increases the complexity of possible solutions in ways far exceeding geometry. But, for anyone interested in examining the potential of this approach, geometry most easily provides an avenue for understanding a few of the added properties.
One of the unspoken axioms of mathematics holds that it is impossible for the human mind to visualize an object with dimensions higher than the three in which the mind resides. While it is possible, with mathematics, to analyze objects in higher dimensions, the mind cannot intuitively grasp their properties. So, for example, the author of the “house” story mentioned in Part 1 could use the properties of a four-dimensional cube to tell a good tale, but his architect could build only an “unfolded” cube. The only way to visualize objects in higher dimensions is to project (or alternately, unfold) those objects into a lower dimension.
I’ve drawn some figures (below) to help “see” the effects that come out of higher dimensions. In Figure 1 the first (leftmost) object is a single point. This represents a zero-dimensional (0-D) object because it has no length, width, or height (of course to draw it, as well as the other objects in the diagram, it is necessary to give it a little of each dimension.) The next object in the figure is a line, and is one-dimensional (1-D). It has length, but no other property. The third object is a square, and is obviously two-dimensional (2-D) in that it has both length and width. Finally, the fourth is a three-dimensional (3-D) cube that is projected into two dimensions. Sometimes a mathematician will refer to all four objects as “cubes” of different dimensions. So the square would be called a cube in two dimensions, while the line would be called a cube in one dimension. The point could be called a cube in zero dimensions, but laypeople may not find such a description meaningful. Perhaps that’s enough geometry for one day.
Figure 1: Objects in 0, 1, 2, and 3 dimensions
To be continued next week.
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