### Meta

### Categories

- algebra (46)
- arithmetic (80)
- books (30)
- calculus (7)
- challenges (57)
- combinatorics (26)
- complex numbers (6)
- computation (80)
- convergence (9)
- counting (36)
- famous numbers (48)
- fibonacci (18)
- fractals (13)
- games (34)
- geometry (71)
- golden ratio (8)
- group theory (28)
- humor (7)
- induction (8)
- infinity (19)
- iteration (24)
- links (76)
- logic (9)
- meta (43)
- modular arithmetic (30)
- number theory (107)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (103)
- people (21)
- pictures (71)
- posts without words (34)
- primes (55)
- probability (9)
- programming (20)
- proof (87)
- puzzles (18)
- recursion (16)
- review (21)
- sequences (28)
- solutions (31)
- teaching (14)
- trig (3)
- Uncategorized (6)
- video (19)

### Archives

- September 2019 (3)
- August 2019 (3)
- July 2019 (5)
- May 2019 (4)
- 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)

Impressive in deed.

It’s a geometric realization of . Equivalence classes of cells represent the field elements. Arithmetic in the field can be understood geometrically. For example, translating one cell to the left represents adding 1. Translating one cell along the lines heading northeast and northwest represent adding 4 and 5 respectively. Rotating clockwise about a 0 cell represents multiplication by 5, which is a primitive element of the field, and so on.

Thank you! I knew there had to be some algebra hiding in there but I wasn’t sure what it was. This is actually quite different from how I was thinking about creating the image in the first place.

I want to add that there’s some interesting finite geometry in this picture. Interpret equivalence classes of cells, that is, field elements, as “points”. Consider two adjacent cells in a row, 0 and 1 for example, and the cell directly below, 3 in this example. The centers of these three cells form a downward-pointing triangle. Call a set of three cells in such a configuration a “line”. There are seven “lines” in this figure: {0,1,3}, {1,2,4}, {2,3,5}, {3,4,6}, {0,4,5}, {1,5,6}, and {0,2,6}. By definition, every line contains three points, but since a given point can be in any of the three positions of a downward-pointing triangle, each point is contained in three lines. Furthermore, every cell has six neighbors, and with each of those it forms a distinct “line”. Hence two “points” determine a unique “line”. In addition, any two downward-pointing triangles share one cell, and therefore any two “lines” intersect in a unique “point”. We conclude that our system of equivalence classes of cells and downward-pointing triangles forms a finite projective plane of order 2.

Once also gets a finite projective plane of order 2 using upward-pointing triangles.

A second additional remark: if one uses a square tessellation instead of a hexagonal one, and labels consecutive squares in every row with the repeating pattern …, 4, 3, 2, 1, 0, 4, 3, 2, 1, 0, …, with 2 lying directly above 0 in the row above, then one gets a geometric realization of . (In the arithmetic of , multiplication by the primitive element 2 corresponds to clockwise rotation.)

The finite fields and are the only two that are realized in this way as periodic tessellations of the plane. One could say that they live on the torus. In contrast, lives on the sphere and corresponds to the tetrahedron. Finite fields with correspond to Platonic tessellations of the hyperbolic plane.

It’s a good step toward the chromatic number proof but I think there’s something missing. I’m tempted to blow off work for a bit this afternoon to provide a “response without words”. (Also this doesn’t seem to be tagged as a PWW.)

RWW: http://suberic.net/~dmm/graphics/sketches/Chrom7.png

Nice! And I completely agree re: something missing; I wasn’t intending for this PWW in and of itself to be a chromatic number proof. Your image gets closer but I think there’s still a lot missing. In particular, how do we know that this pattern can tile the whole plane? How do we know that every color is “the same” in the sense of there being a translation that will send any color to any other? “Just look at it, it’s obvious” somehow doesn’t sit well with me in this case.

Thanks for the kind words ðŸ™‚ I’m not sure what the best method of making it clear is. I feel like your image is a clearly repeating pattern and method to fully color the plane but I’m less connected to academic math so I don’t know what the proof requirements are.

(BTW: shared just my image with some friends as a puzzle and they got to “Something about the number of distinct colors needed so you can’t poke another white region with your 10-foot pole.” within 15 minutes. I then explained and pointed them to your blog.)

I suppose you’re right about the repeating pattern being convincing, especially if you show as many repetitions as I did (most of the pictures I see around the web relating to the Hadwiger-Nelson problem show just a few hexagons, hardly enough to get a general sense for the pattern). But for me this pattern raises more questions that I want to understand: is it possible to do something similar with other numbers of colors? What is special about 7 here? How would you communicate the pattern in a precise way to someone else (such as a computer program for creating a drawing!) without showing them a picture?

You have clearly found good friends if they are willing to think about a mathematical puzzle for 15 minutes. =)

Pingback: Some words on PWW #22 | The Math Less Traveled

I may have missed it, but was it specified how to colour the boundaries between the hexagons?

Ah, very good question! No, I didn’t specify. But if you’re considering it in a purely combinatorial sense it doesn’t matter. And if you want to use it as an construction showing that the plane can be colored with seven colors, there is plenty of “wiggle room” so it doesn’t matter there either. So it doesn’t really matter what you pick — just use one of the two/three adjacent colors for each boundary point.