Gary
GaryVasco
Posts: 3,352 
|
Post by Gary on Apr 30, 2017 22:03:49 GMT
Vasco, Here is a note with plot and question regarding the famous pizza: nh.ch7.notes.p353.pdf (144.27 KB) Gary |
|
|
|
Post by Admin on May 1, 2017 6:16:10 GMT
Gary
From reading your document it seems to me that the following comments may help to clarify the discussion in the book starting in paragraph 4 on page 352 and continuing to the end of the subsection on page 353.
1) I think it is better in this example, (as Needham does), to talk about loops rather than circles because the mapping process stretches and compresses the dough and circles do not maintain their shape.
2) Any $p$-point in the green area of the final pizza has two preimage points: one is in the brown central area of the original pizza and the other is in the black ring of the original pizza. These two points are superimposed on the final pizza to become the single point $p$.
3) Now imagine a separate little loop around each of the two preimage points on the original pizza and follow what happens to these two loops as the final pizza is created. (Note that these two loops maintain their separate identities in the final pizza.)
I think this line of thought should enable you to find the 'geometric rationale' associated with the multiplicities.
Vasco
|
|
Gary
GaryVasco
Posts: 3,352
|
Post by Gary on May 1, 2017 13:27:47 GMT
Gary From reading your document it seems to me that the following comments may help to clarify the discussion in the book starting in paragraph 4 on page 352 and continuing to the end of the subsection on page 353. 1) I think it is better in this example, (as Needham does), to talk about loops rather than circles because the mapping process stretches and compresses the dough and circles do not maintain their shape. 2) Any $p$-point in the green area of the final pizza has two preimage points: one is in the brown central area of the original pizza and the other is in the black ring of the original pizza. These two points are superimposed on the final pizza to become the single point $p$. 3) Now imagine a separate little loop around each of the two preimage points on the original pizza and follow what happens to these two loops as the final pizza is created. (Note that these two loops maintain their separate identities in the final pizza.) I think this line of thought should enable you to find the 'geometric rationale' associated with the multiplicities. Vasco Vasco, Thank you. Yes, I woke up a 5 am with the thought that the wall was extruded from a thin (but not infinitely thin) ring outside the core. Then drawing a tiny simple loop with positive rotation around a point in that ring and stretching the loop vertically would produce a ring on the wall that would appear to have positive rotation. Then the wall is tipped outwards and pressed downwards, so, seen from above, a loop in this layer would have negative rotation. I think viewpoint must figure into the function. I suppose we have to regard the view from above as implicit in the construction of the pizza. I had the same thought about 1) when I began to produce the figure. I decided to let the pizza chef be creative and stretch it with Moebius transformations. Gary |
|
