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I Discovered a New Shape July 25, 2015

Posted by Sobek in News.
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Well, it’s new to me, anyway.

I was working through some trigonometry problems, and there were a whole bunch that boiled down to “you have a seven-sided polygon, each side is x cm, find the area of the polygon.”  I didn’t do very many of those problems before I decided to make a short cut and just work out a formula:

A=(n [d/2]^2)/tan (180/n)

Where n is the number of sides, and d is the length of one side.

Figuring out that formula saved me some time, but as far as I know isn’t super useful.

At one point, I wondered what would happen if I did some weird stuff with the formula. Like what’s the area of a polygon with two sides? That produces a zero in the denominator, so it’s undefined. That makes sense. But what about a polygon with 1.9 sides?  That actually does produce an answer, and so does 1.8. If you set the side length to some constant (I picked 2 so the calculations would be easier), you end up with a dampened tangent/cotangent curve. Even more interesting is that you get holes in the curve at regular intervals.

After all that, I don’t know how to draw a regular polygon with 1.9 sides, but I can tell you how much area it has.

Comments»

1. geoff - July 25, 2015

Well, this is a bit of synchronicity. I was just reading about this uber-mathemetician, and then his name popped up in this thread that talks about your non-integer polygons:

http://mathforum.org/kb/message.jspa?messageID=1068891

2. Sobek - July 25, 2015

I’m not alone!

3. Sobek - July 26, 2015

The first thing that really got me interested in this mind of abstract mathematics is the book Flatland. That was mind blowing.

4. geoff - July 27, 2015

I’ve got a copy of Planiverse that I’ll send you (if I can dig up your address). It’s a follow-on book written by A. K. Dewdney, which figures out a lot of the practical things you’d need to live in a 2D world. Like how does a door work, and how do hinges work, etc.

It sounds dry, but it’s actually amusing.

5. Sobek - July 27, 2015

It looks like there are a bunch of “sequels” out there. I have Flatterland, which is nowhere near as good.


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