Although one could easily understand the logic as expressed by the wikipedia’s coin analogy of the principle, our group actually faced difficulties in trying to understand the principle analytically from the theory statements.
Initially, we had the interpretation that as long as we have 2 regions bounded between two parallel lines, any other line parallel to these lines would cut the regions in equal lengths and the area would subsequently be the same, as show in Figure 1 below. (i.e. EF = GH, and the purple and orange areas would be the same).
Figure 1
Subsequently, we thought of several counter examples (one of which shown in Figure 2) that this would not be the case. By having an oval stretched horizontally, we see that the shape is still bounded by the two parallel lines, but the observation of AB = CD no longer holds, and it is obvious that the areas of these two figures are definitely not the same.
Figure 2
Next, we also observed that the areas of two regions bounded by the two parallel lines could be identical without having the two lengths being the same – as shown in Figure 3.
Figure 3
Conclusively, based on these observations, we decided that the restrictions for the Cavalieri’s principles are:
- The shape need not be the same
- Individual Lengths need not be the same
- But the sum of the lengths need to be the same = > area will be the same
1) If the sum of lengths are the same => the area will be the same. (2D case)
2) If the sum of areas are the same => the volume will be the same. (3D case)
As such, after much struggle in understanding the point of the principle, we decided that the visualising it with a rope works best for our group. A long rope can be cut into several pieces of shorter ropes and no matter how it is rearranged, the sum of lengths would be the same => resulting in the sum of areas being the same (2D) => the volume being the same. (3D)
Figure 4
Group A
Straight from wiki:
ReplyDelete2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.
I think the key-point here is intersecting "both regions in line segments of equal length". For your Figure 2, the red line doesn't intersect regions in line segments of equal length. AB is shorter than CD. That is why Cavalieri's Principle doesn't hold. Therefore it's not a counter-example to disprove the principle but more of an example that doesn't show Cavalieri's Principle (if I'm making sense).
I think the idea is not to look at the area first but to look at the line segments intersecting first. And if all line segments are of the same length, then the area would be equal. In that way, the confusion from Figure 3 can be addressed too.
I hope I make sense and that this helps.
Sincerely,
Muhammad Na'im
Thanks Na'im, that makes sense to me.
ReplyDeleteJust a thought, what if you could match every line segment to that of another line segment not necessarily on the same red line, but say on another line. Like in Figure 3 say there was a blue line and the line segment which cuts the green triangle is the same length as that of JK. Wouldn't that give the same surface area? So if you just looked at line segments on the same red line and they were of different length, you would conclude that the areas were different.
ReplyDeleteDina