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Sierpinski pyramid It is said a line drawn on paper is one-dimensional, a flat object in plane is two-dimensional, and a solid object in space is three-dimensional. Although one casually draws a line on paper, it isn't at all clear how thick or thin it should be. If drawn too thick, one might mistake it as a stripe having a width, which is surely two-dimensional. It is, however, safe to draw a two-dimensional object on paper. In fact, we use a paper or flat TV screen to display a three-dimensional object as well. Although this is not really possible, we can visualize it by perspective and shading to give depth perception in the third dimension. This notion of space filling is called the Euclidean dimension. In this way, the Cantor function has dealt with a one-dimensional line and Sierpinski triangle function a two-dimensional triangle. We now extend the Sierpinski function to a three-dimensional object of 3-sided pyramid, also known as the tetrahedron, which means an object of four faces. That is, the Sierpinski pyramid function breaks up the input pyramid into half-size pyramids and throws away all but the four corner ones, as shown in figure 3. Here, 4 half-size pyramids add up to a half the volume of the original pyramid. Applying the Sierpinski pyramid function again to each of the 4 half-size pyramids, there appear 16 quarter-size pyramids of the second generation. By Prog#5b we see how the Sierpinski pyramid function breaks up the input pyramid up to the third generation. (Project a - Classroom construction of Sierpinski pyramids)
Figure 3. Sierpinski pyramid
The Sierpinski triangle and pyramid suggest models to build frames and towers using the least amount of materials. For instance, instead of a solid triangle, one may use triangular frames to carry the same load and thereby resulting in a material saving, and further reinforcement can be achieved by adding smaller triangular stiffeners as well.
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