Comments on: Subdividing the plane http://www.tangentspace.net/cz/archives/2006/10/602/ somewhere near the beginning. Mon, 01 Dec 2008 23:06:25 +0000 http://wordpress.org/?v=2.5 By: Alex http://www.tangentspace.net/cz/archives/2006/10/602/#comment-65044 Alex Sun, 15 Oct 2006 20:57:49 +0000 http://www.tangentspace.net/cz/archives/2006/10/602#comment-65044 I agree with the sentiment-- those words are abused and misused much too often for anyone to be completely comfortable even when they're employed correctly-- but you have to admit that somethings *are* obvious. (just not in this case :)) I agree with the sentiment– those words are abused and misused much too often for anyone to be completely comfortable even when they’re employed correctly– but you have to admit that somethings *are* obvious. (just not in this case :))

]]> By: ObsessiveMathsFreak http://www.tangentspace.net/cz/archives/2006/10/602/#comment-64986 ObsessiveMathsFreak Sun, 15 Oct 2006 14:36:38 +0000 http://www.tangentspace.net/cz/archives/2006/10/602#comment-64986 The words "evidently", "obvious", "immediate" and their ilk may be translated for, "I am too lazy to do this" at best, and "I am unable to do this" at worst. In either case, they have no place in any professional mathematics text. The words “evidently”, “obvious”, “immediate” and their ilk may be translated for, “I am too lazy to do this” at best, and “I am unable to do this” at worst. In either case, they have no place in any professional mathematics text.

]]> By: Chris Willmore http://www.tangentspace.net/cz/archives/2006/10/602/#comment-64838 Chris Willmore Sat, 14 Oct 2006 00:16:52 +0000 http://www.tangentspace.net/cz/archives/2006/10/602#comment-64838 Consider adding each line in turn to the plane. The first line divides the plane into two regions. Each successive line crosses, at a maximum, all of the lines that come before it once; each of these intersections divides the new line into n segments (if we're laying the nth line down). For the sake of simplicity, we consider the two rays that result on either end of the new line to be segments as well. Each of those n segments of the new line divides a different region in two, so each segment adds 1 to the total count of regions in the plane. Therefore, the nth line added to the drawing adds a maximum of n new regions into the plane, by dividing n regions into 2n regions; the maximum number of regions after adding n lines is therefore 1 (1 2 ... n), or (n^2 n 2)/2. If you draw it out on paper, you can see that this is the case... if you add each line such that it doesn't run through any preexisting intersection points, and it crosses every line on the page already, you get 2 regions after one line, 4 regions after two lines, 7 regions after three lines, etc. Hope this helps! Great blog, btw. Consider adding each line in turn to the plane. The first line divides the plane into two regions. Each successive line crosses, at a maximum, all of the lines that come before it once; each of these intersections divides the new line into n segments (if we’re laying the nth line down). For the sake of simplicity, we consider the two rays that result on either end of the new line to be segments as well. Each of those n segments of the new line divides a different region in two, so each segment adds 1 to the total count of regions in the plane.

Therefore, the nth line added to the drawing adds a maximum of n new regions into the plane, by dividing n regions into 2n regions; the maximum number of regions after adding n lines is therefore 1 (1 2 … n), or (n^2 n 2)/2.

If you draw it out on paper, you can see that this is the case… if you add each line such that it doesn’t run through any preexisting intersection points, and it crosses every line on the page already, you get 2 regions after one line, 4 regions after two lines, 7 regions after three lines, etc.

Hope this helps! Great blog, btw.

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