Stars – part (iii) Some Glimpses of Daylight

So far I’ve mentioned drawing stars with 5, 6, 7, and 8 points.  Sometimes you can compose a stars by overlapping shapes; other times you draw a single continuous zigzagging line.

Most of this was pretty new to me, and I figured that if I was going to get any insight into things I’d have to collect as much data as possible, and do so in a systematic manner.

So for the 9-point situation I sketched out 9 points equally spaced round a circle. Joining each to the next one gave me a regular nonagon, which isn’t a star at all. But when I joined each point to the next-but-one point, i.e. missing out one point each time, I got a genuine star. (A)

What if I miss two points each time? I get an equilateral triangle, and if I make all the three possible equilateral triangles I’ve got a new star – an overlap. (B)

And if I miss three points each time there’s a second zig-zag star. (C)

There aren’t going to be any more, because I can see that missing out four points each time (or five, or six, …) simply gives me shapes I’ve already discovered.

STARS 9

So my table of result now looks like this:

Points Zig-zag Overlap Total
5 1 1
6 1 1
7 2 2
8 1 1 2
9 2 1 3

 

It had been a long time coming, but I was now indeed beginning to get some insights into what was happening. I still had plenty of questions –

are all stars either overlaps or zig-zags, or is there another type?

are there any numbers which will give no stars at all?

can I predict which types of stars, and how many, we’ll get for any number?

does every number after 6 give at least one zig-zag star?

can I predict how many zig-zags there’ll be for each number?

are there any numbers which will give more overlaps than zig-zags?

….

 

If you want some practical help, you can find circles with every number of dots up to 24 at http://nrich.maths.org/8506

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