### Meta

### Categories

- algebra (46)
- arithmetic (74)
- books (30)
- calculus (7)
- challenges (56)
- combinatorics (21)
- complex numbers (6)
- computation (76)
- convergence (9)
- counting (32)
- famous numbers (48)
- fibonacci (18)
- fractals (13)
- games (34)
- geometry (71)
- golden ratio (8)
- group theory (28)
- humor (7)
- induction (7)
- infinity (19)
- iteration (24)
- links (76)
- logic (9)
- meta (43)
- modular arithmetic (29)
- number theory (104)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (98)
- people (21)
- pictures (71)
- posts without words (33)
- primes (55)
- probability (6)
- programming (20)
- proof (83)
- puzzles (18)
- recursion (16)
- review (21)
- sequences (28)
- solutions (30)
- teaching (14)
- trig (3)
- Uncategorized (6)
- video (19)

### Archives

- April 2019 (2)
- March 2019 (3)
- February 2019 (3)
- January 2019 (4)
- November 2018 (3)
- October 2018 (4)
- September 2018 (4)
- August 2018 (6)
- July 2018 (2)
- June 2018 (5)
- May 2018 (3)
- April 2018 (5)
- March 2018 (4)
- February 2018 (3)
- January 2018 (4)
- December 2017 (3)
- November 2017 (3)
- October 2017 (1)
- September 2017 (1)
- July 2017 (4)
- June 2017 (4)
- May 2017 (9)
- April 2017 (7)
- March 2017 (5)
- February 2017 (4)
- January 2017 (3)
- December 2016 (4)
- November 2016 (6)
- October 2016 (6)
- September 2016 (2)
- August 2016 (5)
- July 2016 (2)
- June 2016 (4)
- May 2016 (4)
- April 2016 (2)
- March 2016 (3)
- February 2016 (9)
- January 2016 (8)
- December 2015 (5)
- November 2015 (29)
- August 2015 (3)
- June 2015 (2)
- April 2015 (1)
- May 2014 (1)
- December 2013 (1)
- October 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (3)
- March 2013 (3)
- February 2013 (2)
- January 2013 (5)
- December 2012 (3)
- November 2012 (4)
- October 2012 (5)
- September 2012 (1)
- August 2012 (4)
- July 2012 (1)
- June 2012 (6)
- May 2012 (2)
- April 2012 (3)
- March 2012 (1)
- February 2012 (4)
- January 2012 (5)
- December 2011 (1)
- November 2011 (7)
- October 2011 (4)
- September 2011 (6)
- July 2011 (2)
- June 2011 (4)
- May 2011 (5)
- April 2011 (2)
- March 2011 (4)
- February 2011 (1)
- January 2011 (1)
- December 2010 (1)
- November 2010 (4)
- October 2010 (2)
- September 2010 (1)
- August 2010 (1)
- July 2010 (1)
- June 2010 (2)
- May 2010 (3)
- April 2010 (1)
- February 2010 (6)
- January 2010 (3)
- December 2009 (8)
- November 2009 (7)
- October 2009 (3)
- September 2009 (3)
- August 2009 (1)
- June 2009 (4)
- May 2009 (5)
- April 2009 (4)
- March 2009 (2)
- February 2009 (1)
- January 2009 (7)
- December 2008 (1)
- October 2008 (2)
- September 2008 (7)
- August 2008 (1)
- July 2008 (1)
- June 2008 (1)
- April 2008 (5)
- February 2008 (4)
- January 2008 (4)
- December 2007 (3)
- November 2007 (12)
- October 2007 (2)
- September 2007 (4)
- August 2007 (3)
- July 2007 (1)
- June 2007 (3)
- May 2007 (1)
- April 2007 (4)
- March 2007 (3)
- February 2007 (7)
- January 2007 (1)
- December 2006 (2)
- October 2006 (2)
- September 2006 (6)
- July 2006 (4)
- June 2006 (2)
- May 2006 (6)
- April 2006 (3)
- March 2006 (6)

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

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

Pingback: Totient sums | The Math Less Traveled

Pingback: Primitive roots of unity | The Math Less Traveled