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The way this is presented in the blog post is as if this is a really insightful but now under-appreciated gem of mathematics. But all that description is sayin

The way this is presented in the blog post is as if this is a really insightful but now under-appreciated gem of mathematics. But all that description is saying is, when a 2D shape is made by rotation, its area is the multiple of the 1D generator and the 1D path it takes? You don't say! And then volumes! when a 3D volume is made, its a 2D shape going through 1D path ...wow. Forgive me if I am not impressed. This is not an unknown theorem but a trivial mathematical fact that one learns somewhere around Grade 7 in school. Of course, Pappus deserves credit for discovering in 300 AD, and the paper by the Goodmans is nice to have a general formal proof of. But even that paper concedes that they are simply proving a general proof for completeness.

lucasvb 12y ago
But it isn't that simple. The insight is in the importance of using the centroid of the shape and the path it traces. It can't be any other point for the simpler form to work. That's the cleverness, and the reason the centroid has that property is interesting enough, and not immediately obvious. But by understanding why it works, we can see we can apply it for much more general cases than surfaces of revolution, which is usually the only treatment the theorem gets out there. The purpose of the post was to illustrate the generality of the theorem.
- drvortex 12y ago
  The centroid is the average of all points of a shape. It is therefore the only point that contains information about the entire shape. Any other point doesn't. Naturally, since paths are distances and can also be averaged in the same way, it is actually quite obvious that the path travelled by the centroid of a shape is also the average path of all the points on that shape. In fact, it directly follows from the definition of a centroid consider that the path of a shape is the addition of one more dimensional coordinate to each of the points on that shape. Sorry,I still do not see what makes this so special.

lucasvb 12y ago

But it isn't that simple. The insight is in the importance of using the centroid of the shape and the path it traces. It can't be any other point for the simpler form to work. That's the cleverness, and the reason the centroid has that property is interesting enough, and not immediately obvious. But by understanding why it works, we can see we can apply it for much more general cases than surfaces of revolution, which is usually the only treatment the theorem gets out there. The purpose of the post was to illustrate the generality of the theorem.

drvortex 12y ago
The centroid is the average of all points of a shape. It is therefore the only point that contains information about the entire shape. Any other point doesn't. Naturally, since paths are distances and can also be averaged in the same way, it is actually quite obvious that the path travelled by the centroid of a shape is also the average path of all the points on that shape. In fact, it directly follows from the definition of a centroid consider that the path of a shape is the addition of one more dimensional coordinate to each of the points on that shape. Sorry,I still do not see what makes this so special.