The discussion here continues from last week, Part 4.
The 3-D cube in Figure 1 (of Part 4) demonstrates the idea of the projection of an object from a higher dimension to a lower dimension. The object as drawn is not, in fact, a 3-D cube, but the projection of such a cube into two dimensions. Even so, the reader immediately identifies it as a picture of a 3-D cube. Another way of displaying a higher-dimensional object in a lower dimension is by unfolding it into a lower dimension.
Figure 2: Unfolding of a square into one dimension
The unfolding of a 2-D object, a square, into one dimension is demonstrated in two stages in Figure 2. The unfolding of a 3-D cube into two dimensions is shown in Figure 3. The latter is obviously what results when a cardboard box is flattened.
Figure 3: Unfolded cube in two dimensions
An interesting feature of multidimensional geometry can be seen using the succession of “cubes” in Figure 1. Notice first that the 1-D cube can be generated by sliding the 0-D cube in a direction, or dimension, at right angles to itself. The beginning position defines one end of the line; the sliding action sweeps out the line; and the final position defines the other end of the line. This process is more easily seen when using the 1-D cube to generate a 2-D cube, or square. The beginning position of the line defines one side of the square; each end of the line sweeps out two sides of the square; and the final position of the line forms the fourth side of the square. Figure 4 demonstrates these two examples.
Figure 4: Generating higher-dimensional “cubes” by sliding lower dimensional “cubes”
Hang in there. There’s a point to all this. We’ll continue next week.
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