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

Story Collider Performance

In April I was privileged to participate in a Story Collider event on the Lower East side, where I and five other Math for America teachers told stories about our lives in and out of the classroom.

The full video of my story is now available. Watch it to find out why I consider teaching to be a party at the edge of human knowledge.

It was humbling to take part along side my MfA colleagues, whose stories were courageous, personal, and deeply moving. You can find links to all the stories here.

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By MrHonner, ago

Teaching With the Data of Economic Mobility

My latest piece for the New York Times Learning Network was inspired by some amazing data visualizations from The Upshot.

These animations show trends in economic mobility gathered from a landmark study of 20 million Americans. In my lesson, students use the Upshot’s customizable tools to collect and analyze data from the study to determine which groups of Americans have the best chance of improving their economic standing.

Here’s the introduction:

America is often referred to as the land of opportunity. But are all opportunities created equal? Do all Americans have the same chance of achieving the American dream?

A groundbreaking study of United States census data examined how the economic status of 20 million Americans changed from childhood to adulthood, and while the data has a lot to tell us about economic opportunity in the United States, it is likely to raise more questions than it answers.

In this lesson, students use tools created by The New York Times to explore data from the study on economic mobility. They will analyze and categorize economic outcomes, compare and contrast statistics for different demographic groups, and pose and explore their own questions about what this data has to say about economic opportunity.

Does everyone in America have the same chance at success? Let’s see what the data says.

The full lesson is freely available here.

By MrHonner, ago

Thanks, James

I was recently surprised by a visit from a student I hadn’t seen in eight years. Though he had often come to mind, I never expected to see him again.

Eight school years ago, James and I got off to a good start. But in the spring term, expectations were not being met. On a bad day, I communicated this to James in a way that negatively impacted our relationship. He stopped coming to class.

I didn’t see him for a week or so. When I tracked him down, he told me he had lost respect for me because of the way I had treated him. It was hard to hear; the truth often is. I apologized. He resumed coming to class and finished the year, but our relationship was never quite right again.

The falling out was on both our minds when he visited. “I still think about it,” I said. “Yeah, I wanted to apologize for that,” James replied. I told him no apology was necessary. In fact, I appreciated his courage and maturity in calling me out. I had good intentions, but my actions made the situation worse. It happens sometimes: Often enough, in my career, that I feel like I’ve finally learned my lesson.

Unfortunately I won’t get the opportunity to resolve all those issues from the past. But I’m glad I got that chance here. James is doing well, figuring out who he is, finding success. I’m not sure exactly why, but there was some comfort in knowing that our falling out affected him, too. I was grateful that we could find some positive resolution, even after eight years.

Teaching is an incredibly challenging job. It is a constant struggle to find balance: the balance between expectations and patience; between being tough and being understanding; between pushing a young person and letting them be. As teachers, we are challenged to find that balance a hundred times a day. Somedays we come up short, and we have to live with the consequences. This is the emotional toll of the job.

But that emotional investment also means a brief, unexpected visit years later can make a world of difference. Thanks, James.

By MrHonner, ago
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