I’m having a blasty-bast with Group Explorer today. How have I never played with this before!?
Complex analysis – the perfect subject for combining visual and analytic thinking. I just read “Visual Complex Analysis” by Tristan Needham, a basic book on complex analysis: lots of historical references, uncompromising explanations, lots of problem solving and plenty of beautiful illustrations!
I have this book. It is very good.
I can’t be the only one. Right?
Based on Joan Taylor’s wanderer tiling. http://tilings.math.uni-bielefeld.de/substitution/wanderer-refl/
Just some lines
A very famous open problem is the following.
How many colors are needed to color the plane so that no two points at
distance exactly 1 from each other are the same color? (This number is known as the chromatic number of the plane)
For example 2 colors are not enough. Consider 3 points that are the vertices of an equilateral triangle with side length 1. Clearly no two of these points can have the same color, so we need 3 colors just for these.
Until this week our best knowledge was that there is a possible coloring with 7 colors, and 3 colors are not enough but we didn’t know if the best answer is 4, 5, 6 or 7. It took more than 50 years to get something more, a couple of days ago AUBREY D.N.J. DE GREY presented a paper on arxiv, proving that 4 colors are not enough! See here: https://arxiv.org/pdf/1804.02385.pdf
Why is 7 color enough? Consider the following coloring:
Construct the hexagons such that the two opposite vertex of a hexagon are slightly closer to each other than 1. This way you cannot take two points from the same hexagon that are at distance one. If you take two points from different hexagons of the same color their distance will be at least bigger than 1. So this coloring is correct.
Why is 3 color not enough? Consider the following 7 points. I connected the ones that are at distance 1.
Can we color these points with 3 colors? Well, lets try! A,G and F must be all different, since they are connected. Also G, F and E must have three different colors. So if we color with 3 colors, A and E must have the same color. The same way we can see that A and D must have the same color. But this would mean that E and D have the same color which is not allowed! So 3 colors are not enough.
Booth of these were very easy to see, but until now we couldn’t do better.
AUBREY D.N.J. DE GREY proved that 4 colors are not enough. The proof is similar to the one above, but much more complicated. Some parts even used computer verification. On the following picture there are 1567 yellow points, and every pair is connected if their distance is 1. It is a lot of points, so it is not really visible :\
He showed that 4 color is not enough for these points, we need at least 5. This result doesn’t settle the problem, we still don’t know if 5 colors are enough for the whole plane, but we are closer to the answer.
There is a lot of interesting things connected to this problem maybe I will make some posts about it later.
If one remembers this particular episode from the popular sitcom ‘Friends’ where Ross is trying to carry a sofa to his apartment, it seems that moving a sofa up the stairs is ridiculously hard.
But life shouldn’t be that hard now should it?
The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what
is the shape of largest area in the plane that can be moved around a
right-angled corner in a two-dimensional hallway of width 1? This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked.
The most common shape to move around a tight right angled corner is a square.
And another common shape that would satisfy this criterion is a semi-circle.
what is the largest area that can be moved around?
Well, it has been
conjectured that the shape with the largest area that one can move around a corner is known as “Gerver’s
sofa”. And it looks like so:
Wait.. Hang on a second
sofa would only be effective for right handed turns. One can clearly
see that if we have to turn left somewhere we would be kind of in a tough
Prof.Romik from the University of California, Davis has
proposed this shape popularly know as Romik’s ambidextrous sofa that
solves this problem.
Although Prof.Romik’s sofa may/may not be the not the optimal solution, it is definitely is a breakthrough since this can pave the way for more complex ideas in mathematical analysis and more importantly sofa design.
Have a good one!