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I think the first blue circle shouldn’t have a blue dot.

Why do you think that?

Hmm.. I think I’ve changed my mind again since I said that. What I was thinking is that the definition of cyclic groups is that they’re generated my a single element (and these elements are the blue dots). But the interesting thing about the trivial group is that it’s generated by even fewer elements than that! That is to say the one element group is also generated by the subset . Therefore it seemed a bit wrong to colour in , since it’s unnecessary to use it to generate .

However other considerations seem to favour colouring it in. For example and are definitely coprime.

As the number of radii continue to grow by one, blue dots of intersections with the circle appear on the new intersections, never repeating if they had appeared earlier.

Not sure if you want us to guess in the comments. I’m pretty sure I know what it is and I wonder if the inverse images would actually be more patternful.

Sure, guessing in the comments is fine. I could certainly make the inverse image, but actually I think this one is already quite patternful!

Well, it looks like gcd == 1. In the inverse, there are many multiplicative relations. If X is in there, so is n*X, you could assign a colour to each prime and colour it’s spoke based on it’s factorization, with the lower primes closer to the centre (maybe excluding primes not in the main number).

Right, but there are lots of multiplicative relations in this version too! Look carefully.

So there’s the cyclic group structure but maybe rather than multiplicative I should have said periodic, additive.

Some observations,

1. The dots seem to exhibit rotational symmetry whenever the number of spokes can be written as n^m (for m > 1). See 4,8,9,16.

2. There are n-1 dots when a circle has n spokes for n prime

Nice observations! Re: #1, are there other numbers that also exhibit rotational symmetry?

I can’t be sure from the few samples there but it looks like if the number of spokes n is divisible by n^m (m>1) (point 1) then it also has rotational symmetry. Eg. 12 = 2^2 * 3, 18 = 3^3 * 2, 20 = 2^2 * 5

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