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Class 1 Polygons
Nov. 22nd, 2004 04:42 pmFor the last couple of weeks, I've been hung up trying to write the introduction to a research paper, getting nowhere. It just wouldn't gel for me. I decided to set aside a chunk of time during the break, to devote to getting the bloody thing done.
So I spent a few hours on campus this morning and finally broke the logjam. I've written the intro and the first (not very interesting) part; parts II and III are the interesting stuff, and should flow fairly easily. With any luck, I should have the first draft done by Wednesday afternoon. It'll still need editing - a general cleanup, plus fitting it into whatever format the journal I send it to wants - but things are looking good now.
My research has also been kickstarted. Part II of the paper will be devoted to what I call "polygons of Fourier class 1" - which is too much of a mouthful, so I usually just say "class 1 polygons". Last week I started thinking about them again from a different angle, and a lot of intriguing stuff has popped up. Some of it will go into this paper, but a lot of it depends on ideas I'll be developing in my next paper. (It won't fit in that one either, but could form the core of a third article.) Here's the scoop (in nontechnical terms):
1) All triangles are class 1. They have a bunch of interesting properties, not shared with most polygons. It turns out that any class 1 polygon with an odd number of sides shares some of those properties. (I've known this for a while, and most of it will go into Part II.)
2) A quadrilateral is class 1 if and only if it's a parallelogram. Parallelograms have a different list of special properties, not shared with most polygons. But if the number of sides of a class 1 polygon is a multiple of 4, it shares some of those properties. (I've known a little of this for a while, but a lot of it's new. Some of it goes into part II.)
3) If the number of sides of a class 1 polygon is even, but not a multiple of 4, its properties seem to be somewhere in between. (This is all new, and will take more investigation - I'm not at all sure just what's going on.)
So. A sense of completion, and a set of new vistas. I can't ask for more.
So I spent a few hours on campus this morning and finally broke the logjam. I've written the intro and the first (not very interesting) part; parts II and III are the interesting stuff, and should flow fairly easily. With any luck, I should have the first draft done by Wednesday afternoon. It'll still need editing - a general cleanup, plus fitting it into whatever format the journal I send it to wants - but things are looking good now.
My research has also been kickstarted. Part II of the paper will be devoted to what I call "polygons of Fourier class 1" - which is too much of a mouthful, so I usually just say "class 1 polygons". Last week I started thinking about them again from a different angle, and a lot of intriguing stuff has popped up. Some of it will go into this paper, but a lot of it depends on ideas I'll be developing in my next paper. (It won't fit in that one either, but could form the core of a third article.) Here's the scoop (in nontechnical terms):
1) All triangles are class 1. They have a bunch of interesting properties, not shared with most polygons. It turns out that any class 1 polygon with an odd number of sides shares some of those properties. (I've known this for a while, and most of it will go into Part II.)
2) A quadrilateral is class 1 if and only if it's a parallelogram. Parallelograms have a different list of special properties, not shared with most polygons. But if the number of sides of a class 1 polygon is a multiple of 4, it shares some of those properties. (I've known a little of this for a while, but a lot of it's new. Some of it goes into part II.)
3) If the number of sides of a class 1 polygon is even, but not a multiple of 4, its properties seem to be somewhere in between. (This is all new, and will take more investigation - I'm not at all sure just what's going on.)
So. A sense of completion, and a set of new vistas. I can't ask for more.
