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Sierpinski pyramid: In Project a of Lesson 5, a large Sierpinski pyramid is assembled from basic building-block pyramids that are made of cardboard paper. We propose here an alternate construction with candle wax. To do this requires a mold for the basic pyramid with three equilateral triangle sides of 1 inch, which we assume to hold 1 oz of candle wax. As shown in figure 9, there are 4 basic (building block) wax pyramids in the Sierpinski pyramid of side 2 inches and 16 basic wax pyramids in the Sierpinski pyramid of side 4 inches, as listed in table 6.
Figure 9. Candle wax Sierpinski Pyramids
Table 6. Making the Sierpinski pyramids with candle wax
From the straight-line plot of x versus y in figure 10, we find dimension
Figure 10. Log-log plot of table 6 You can now estimate fractal dimensions of the Cantor gasket (Project a) and Menger sponge (Project b) by following the line of reasoning we have presented thus far. |
. It is interesting to point out that the fractal dimension turns out an integer d = 2. Although the Sierpinski pyramid appears a three-dimensional object, it only has dimension 2. This means that, theoretically speaking, if one were to flatten out a wax Sierpinski pyramid into a very (infinitely) thin sheet, it will cover the outer triangular surfaces of the Sierpinski pyramid.