I am not sure how many of you are convinced by the idea that the area is made of parallel line segments while the volume is made of parallel planes. Personally I am a little confused by this.
First of all, Cavalieri's principle does not implies this result. The statement of the principle is that "If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal" (http://mathworld.wolfram.com/CavalierisPrinciple.html). Personally I think it is only looking at the cross-sections, but it does not imply that the solids are made of the thin slices. Similar argument applies to the area as well.
Regarding the activity that the parallel line segments shade area, personally I find it is a bit flawed. The line segments we have drawn in this activity, are not really line segments but thin rectangle blocks. If we are looking at the line segment itself, by applying a similar argument as the density of the real numbers, we can show that between any pairs of parallel line segments we have drawn, we can always find a space between these two line segments. Thus we cannot completely fill the area of the rectangle block with parallel line segments. Similarly in the activity with A4 papers, we have already assumed that each A4 paper has a specific volume. Thus we can then sum up the volume.
I am not sure how helpful it would be to the students at their age. However, I just find it would be a bit dangerous to let them carry this idea with them especially when they are working on problems at a higher level. Till now, I finally understand why some students have some difficulties in differentiating between projections and cross-sections, between point-groups and space-groups, when they are working with 3D chemical models.
Nevertheless, Cavalieri's principle is quite useful in deriving area or volume formula of figures/solids with special shapes. I have tried to look for a formal proof for this principle but so far I cannot find any. Shall we accept it as an inductive observation or a hypothesis? :)
"Personally I think it is only looking at the cross-sections, but it does not imply that the solids are made of the thin slices. Similar argument applies to the area as well."
ReplyDeleteThat is one of the key point of his principle. You can look at cross section plane as a thin slice where the slice is infinitely thin and almost becoming 2-dimension. Summing them up will then give u the same volume when the 2 objects u are comparing when same height and same cross sectional areas.
Good observation. Area and Volume can be viewed as approximately made up of very fine line segments and thin slices respectively.
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