Four Is Not Enough — Quanta Magazine

My latest column for Quanta Magazine explores the elementary geometry underlying an open problem in mathematics that has been given new life thanks to a recent, surprising discovery.

The Hadwiger-Nelson problem, also known as finding the chromatic number of the plane, involves determining the minimum number of colors necessary to color every point of the plane subject to a specific restriction.

Consider the standard geometric plane, an infinite expanse of points in two dimensions. Your task is to color each of the infinitely many points in the plane. You might wish to color the entire plane red, or maybe half red and half blue, or maybe you’d splatter the color like in a Jackson Pollack painting. But there’s one rule in our plane coloring problem: If two points are exactly 1 unit apart, they cannot be the same color. Can you color every point in the plane without violating this rule?

“Of course!” you might say, “I’ll just use infinitely many colors.” There is a certain elegance to this sneaky approach (setting aside the philosophical question of whether infinitely many colors exist), but can you do it with finitely many colors? And if so, how many different colors would you need? 

Though studied for nearly 70 years, the Hadwiger-Nelson problem remains unsolved, but an unexpected discovery earlier this year has narrowed the possibilities. In my column, I explore elementary approaches to establishing both upper and lower bounds on the chromatic number of the plane, and discuss the exciting discovery that has re-energized the mathematical community around this problem. You can read my full article here.

By MrHonner, ago

MfA Muller Award

I am extremely proud to be an inaugural winner of the MfA Muller Award for Professional Influence in Education!

The MfA Muller Award was created to recognize and reward teachers who influence the profession of teaching through their work in and out of the classroom. This is from the official description of the award:

For teaching to be truly respected, it is essential for the voices of great teachers to be at the forefront of conversations about math and science education. We need their influence both locally and nationally. Teachers must be included in public discourse about educational policies, reforms, and initiatives that impact the profession of math and science teaching, as well as within their own school communities. Great teachers who use their professional influence to engage, encourage, and support others in their profession are an inspiration and warrant recognition.

The MfA Muller award carries a $20,000 honorarium for the recipient, and includes a $5,000 award to the nominating organization. This is all made possible through the generosity of MfA board member Pete Muller and his family, through their Dancing Tides foundation.

Being a part of the Math for America community is, itself, a tremendous honor. To be recognized as a standout member of that community is incredibly humbling. I am grateful to Pete Muller, Jim and Marilyn Simons, and the entire MfA community.

You can read the press release for inaugural MfA Muller Award here.

By MrHonner, ago

Visualizing Cantor’s Zig Zag

A famous and intriguing result in mathematics is that there are just as many points on a line as there are in a plane. This seems counterintuitive at first: planes contain infinitely many lines, so not only should a plane have infinitely many more points than a line, it should have infinitely times as many points as a line! But this is one of the many curious consequences of the mathematics of infinity.

Here, we’ll restrict ourselves to points in the plane with non-negative integer coordinates. Think about points of the form ( c, 0 ), where c is a non-negative integer. Since there are infinitely many integers, this set of points is infinite, and the points all lie on the line y = 0. The set of points of the form ( c, 1 ) is also infinite, and these points all lie on the line y = 1. Notice that, since these two lines are parallel, every point on one matches up perfectly with a point on the other: (0,0) with (0,1); (1,0) with (1,1); (2,0) with (2,1), and so on.

This matching offers a reasonable argument that the two sets have the same number of points: Every point in each set has a unique partner in the other, so counting the points in one is equivalent to counting the points in the other. In this case, we say that the two sets are in one-to-one correspondence. And if anything, this only seems to bolster the argument that there are more points in the plane than on a line: There are infinitely many lines of the form y = k in the plane, and each one contains as many points as the line y = 0. So the plane should contain infinitely times as many points as the line! But the mathematics of infinity is tricky business.

Even though it seems like there are far more points in the plane than on the line, it’s possible to match the two sets up in a one-to-one correspondence. It’s not obvious how to do that, but thanks to Georg Cantor and his famous zig zag, we know it can be done. Here’s a visualization I created in Desmos to demonstrate this matching.

This animation shows how each point in the quarter-plane can be paired up with exactly one point on the half-line, and vice versa. The zig-zag pattern enumerates the points in the plane, showing that they could be rightly imagined as though all lying in order on a straight line. This one-to-one correspondence shows that the sets are the same size. And while this demonstration is limited to only part of the plane, the argument can be extended: for example, skipping every other point on the line y = 0 would create space to accommodate the points with negative y-coordinates.

The animation above is an extension of an earlier version shared on Twitter.

Thanks to Chris Long (@octonion) for inspiring this journey into the infinite by linking to this great paper on Cantor packing polynomials, which I used to create the above Desmos demonstrations. And Kelsey Houston-Edwards also recently shared a fun and related problem. I guess infinity is in the air!

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By MrHonner, ago

Math Photo: Lego-Linear Approximations

This replica of The Thinker at Legoland got me thinking about linear approximations.

One of the fundamental ideas in Calculus is that certain kinds of curves can be very closely approximated by straight lines. In fact, when examined closely enough, these differentiable curves are essentially indistinguishable from straight lines. This is important because lines are easy to understand and analyze, whereas curves can be very complicated.

We see this phenomenon at play in Lego sculpture. Here, The Thinker’s curves are being approximated by rectangular Legos, and beautifully so. And scale plays an important role: a larger Thinker looks better in Lego than a smaller one, because the finer the approximation, the better the fit. This is something that any child who has ever tried to make a Thinker out of handful of Legos knows this firsthand.

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By MrHonner, ago
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