### Meta

### Categories

- algebra (46)
- arithmetic (70)
- books (30)
- calculus (7)
- challenges (54)
- combinatorics (21)
- complex numbers (6)
- computation (67)
- convergence (9)
- counting (32)
- famous numbers (48)
- fibonacci (18)
- fractals (13)
- games (34)
- geometry (71)
- golden ratio (8)
- group theory (27)
- humor (6)
- induction (7)
- infinity (19)
- iteration (24)
- links (76)
- logic (7)
- meta (43)
- modular arithmetic (24)
- number theory (91)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (94)
- people (21)
- pictures (70)
- posts without words (27)
- primes (48)
- probability (6)
- programming (18)
- proof (83)
- puzzles (16)
- recursion (16)
- review (21)
- sequences (28)
- solutions (28)
- teaching (14)
- trig (3)
- Uncategorized (6)
- video (19)

### Archives

- 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)

## Post without words #21

This entry was posted in combinatorics, geometry, posts without words and tagged enumeration, orthogonal, polygons. Bookmark the permalink.

OEIS knows the answer…

Yes, and there’s an interesting story behind that entry — more later!

I looked up the OEIS sequence. The relationship to sums of non-negative integers is fascinating!

And I just realized why its true.

Walking along the paths counterclockwise there is an constant number of right turns in each group of diagrams, increasing from group to group.

Indeed! What about the total number of turns, both right & left?

The number of edges increases by 2 each group.

That’s right. Now, how do these two observations relate?

To do a full turn I have to do 4 right turns, plus an equal number of additional left and right turns (assuming we don’t want loops.). So the number of right turns determines the number of left turns.

Took me a couple of looks but since the area and perimeter weren’t constant, there was a clear metric to check next. This would be an interesting challenge to automate, with removal of reflections and rotations and length variations and collision avoidance.

“This would be an interesting challenge to automate” — that’s a very perceptive comment. In fact, the image you see above *was* generated in an automated way, and indeed, it was quite interesting! In a sense, I have been trying (off and on) to figure out how to generate this image for more than a year. I’ll explain more in future posts.

Can’t wait! 🙂

Consider a regular 2n+2 gon with n-1 painted edges where n=1,2,3,4….

.

The various such painted polygons are represented by your shapes.

It’s just a very difficult game of Tetris 🙂

Start with a square. We bite off one of the corners to get an L shape (see second image). We can aslo bite into an edge to get a C shape (see third image). The corner biting is worth 1 bite and the edge biting is worth 2 bites. For any right angled polygon, edge biting and corner biting is defined the same way. Your shapes are the sets of n-bitten squares where n=0,1,2,3,4.

Pingback: Orthogonal polygons | The Math Less Traveled

Pingback: Efficiently listing orthobraces | The Math Less Traveled