Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Mar 12, 2017 15:47:53 GMT
Vasco,
Regarding [4], do you agree that the numbers "n", "n+2", "n+6", and "n+4" should be written "|n|", "|n|+2", "|n|+6", and "|n|+4" to indicate the number of crossings?
Gary
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Post by Admin on Mar 12, 2017 16:49:37 GMT
Vasco, Regarding [4], do you agree that the numbers "n", "n+2", "n+6", and "n+4" should be written "|n|", "|n|+2", "|n|+6", and "|n|+4" to indicate the number of crossings? Gary Gary No, I think it is OK as it is, because the last sentence of subsection 3 starts by saying "Figure [4] illustrates these possibilities for a case in which $n=2$,...", and so we know that $n=2$. Prior to that last sentence Needham is talking about loops generally and so he uses $|n|$, but then figure [4] illustrates a specific case where we know that $n$ is positive. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Mar 12, 2017 19:17:50 GMT
Vasco, Regarding [4], do you agree that the numbers "n", "n+2", "n+6", and "n+4" should be written "|n|", "|n|+2", "|n|+6", and "|n|+4" to indicate the number of crossings? Gary Gary No, I think it is OK as it is, because the last sentence of subsection 3 starts by saying "Figure [4] illustrates these possibilities for a case in which $n=2$,...", and so we know that $n=2$. Prior to that last sentence Needham is talking about loops generally and so he uses $|n|$, but then figure [4] illustrates a specific case where we know that $n$ is positive. Vasco Vasco, If you reversed the orientation, reversed the signs of the intersection points, and plugged in "-2" for $n$ into Figure 4, would you get the correct number of intersection points? I accept your reasoning that [4] illustrates a special case, but I think the absolute brackets would clarify it, especially given the preceding sentence. Gary |
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Post by Admin on Mar 13, 2017 17:00:18 GMT
Gary No, I think it is OK as it is, because the last sentence of subsection 3 starts by saying "Figure [4] illustrates these possibilities for a case in which $n=2$,...", and so we know that $n=2$. Prior to that last sentence Needham is talking about loops generally and so he uses $|n|$, but then figure [4] illustrates a specific case where we know that $n$ is positive. Vasco Vasco, If you reversed the orientation, reversed the signs of the intersection points, and plugged in "-2" for $n$ into Figure 4, would you get the correct number of intersection points? I accept your reasoning that [4] illustrates a special case, but I think the absolute brackets would clarify it, especially given the preceding sentence. Gary Gary The answer to your question is no in my opinion, because $n,n+2,n+6,n+4$ would be equal to $-2,0,4,2$ which are not equal to the number of intersection points. If you wanted to make figure [4] suitable as a general case then you would have to leave out the arrows and the little circles with pluses and minuses and replace $n$ by $|n|$. I think it would be better to draw two diagrams one for each orientation of the loop. I think what Needham decided to do was to show the diagram for positive orientation and then let the reader work out how it would look for negative orientation. It's not really possible to draw one diagram for all cases while maintaining arrows and pluses and minuses, without it being very messy. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Mar 14, 2017 20:20:50 GMT
Vasco, If you reversed the orientation, reversed the signs of the intersection points, and plugged in "-2" for $n$ into Figure 4, would you get the correct number of intersection points? I accept your reasoning that [4] illustrates a special case, but I think the absolute brackets would clarify it, especially given the preceding sentence. Gary Gary The answer to your question is no in my opinion, because $n,n+2,n+6,n+4$ would be equal to $-2,0,4,2$ which are not equal to the number of intersection points. If you wanted to make figure [4] suitable as a general case then you would have to leave out the arrows and the little circles with pluses and minuses and replace $n$ by $|n|$. I think it would be better to draw two diagrams one for each orientation of the loop. I think what Needham decided to do was to show the diagram for positive orientation and then let the reader work out how it would look for negative orientation. It's not really possible to draw one diagram for all cases while maintaining arrows and pluses and minuses, without it being very messy. Vasco Vasco, Yes, I see your point now. He needed to accommodate the circles indicating sign. Gary |
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