Class 1 Polygons
Nov. 22nd, 2004 04:42 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowFor 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.
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Date: 2004-11-23 12:33 am (UTC)no subject
Date: 2004-11-23 10:40 am (UTC)Wow, that sounds really interesting!! And understandable, thanks! :-)
Congratulations on getting through the block!!!
And also on being inspired, work is much more fun that way, isn't it? I need some of that myself. :-)
It all sounds really neat, so if you have the urge, do post it. Actually, if you've got some stuff published that I might be able to access, I'd like to see it. Just to look at and admire, but it does all sound very interesting, and I'd like to see more.
Again, Good Job!! :-D