Formula for Calculating Components of Hypercubes
The number of 0-D components, i.e., points, is doubled each time a new dimension is added. This is because the 0-D unit is the most basic component, and there are no new 0-D components created by any trailings into a new dimension. Therefore, the number is doubled each time.
Working upwards from the number of 0-D components, which is immediately predictable for a basic unit of x dimensions, we can compute the number of successive y-D components within this x-D unit by the following method:
1. For 1-D components, multiply the number of 0-D components by x, then divide by 2. This is because there are x dimensions radiating out at right angles from each point, and a line segment is always shared by exactly two points. In other words, each point defines x new line segments, and by halving this number we determine the exact number of unique line segments.
2. For 2-D components, multiply the number of 1-D components by (x-1), then divide by 4. This is because there are x-1 planes/squares/faces that can be constructed at right angles from any one given line segment in an x-D cube, and each of these faces is shared by exactly 4 line segments.
3. For 3-D components, multiply the number of 2-D components by (x-2), then divide by 6. This is because there are x-2 cubes that can be constructed at right angles upon any one given 2-D face in an x-D cube, and each of these cubes is shared by exactly 6 faces.
By now it is readily apparent that for y-D components, multiply the number of (y-1) - D components by x - (y - 1), then divide by 2y.
This progression can be condensed into a single expression:
2^{(x-y) }x! | ||
n = | _________ | |
y!(x - y)! |
This expression represents the number of y-D components found in a basic unit of x dimensions.
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