15 Responses to Post without words #10

1. Macbi says:

   August 26, 2016 at 2:01 pm

   I think the first blue circle shouldn’t have a blue dot.
   • Brent says:

     August 27, 2016 at 1:27 pm

     Why do you think that?
     • Macbi says:

       August 28, 2016 at 10:35 am

       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.
2. dan kearney says:

   August 27, 2016 at 10:16 pm

   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.
3. Fergal Daly says:

   August 27, 2016 at 10:55 pm

   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.
   • Brent says:

     August 28, 2016 at 7:29 am

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

       August 28, 2016 at 8:40 am

       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).
       • Brent says:

         August 28, 2016 at 10:16 am

         Right, but there are lots of multiplicative relations in this version too! Look carefully.
       • Fergal Daly says:

         August 29, 2016 at 8:19 am

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

   August 29, 2016 at 6:34 am

   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
   • Brent says:

     August 29, 2016 at 4:12 pm

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

       August 29, 2016 at 9:32 pm

       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

5. Pingback: A few words about PWW #10 | The Math Less Traveled

6. Pingback: Totient sums | The Math Less Traveled

7. Pingback: Primitive roots of unity | The Math Less Traveled

Comments are closed.