Archive of posts filed under the Resources category.

The (Math) Problem with Pentagons — Quanta Magazine

My latest column for Quanta Magazine is about the recent classification of pentagonal tilings of the plane. Tilings involving triangles, quadrilaterals, and more have been well-understood for over a thousand years, but it wasn’t until 2017 that the question of which pentagons tile the plane was completely settled.

Here’s an excerpt.

People have been studying how to fit shapes together to make toys, floors, walls and art — and to understand the mathematics behind such patterns — for thousands of years. But it was only this year that we finally settled the question of how five-sided polygons “tile the plane.” Why did pentagons pose such a big problem for so long?

In my column I explore some of the reasons that certain kinds of pentagons might, or might not, tile the plane. It’s a fun exercise in elementary geometry, and a glimpse into a complex world of geometric relationships.

The full article is freely available here.

Investigating the Math Behind Biased Maps

My latest piece for the New York Times Learning Network gets students investigating the mathematics of gerrymandering.  Through applying geometry, proportionality, and the efficiency gap, students explore the notion of a “workable standard” for identifying and evaluating biased electoral maps.

Here is an excerpt:

Math lies at the heart of gerrymandering, in which the shapes of voting districts and distributions of voters are manipulated to preserve and expand political power.

The strategy of gerrymandering is not new… However, new, sophisticated mathematical and computer mapping tools have made gerrymandering an even more powerful way to tilt the playing field. In many states, where the majority party has the authority to rewrite the electoral map, legislators essentially have the power to choose their voters — to create districts in any shape or size that will weaken their opponents and increase their dominance.

In this lesson, we help students uncover the mathematics behind these biased electoral maps. And, we help them apply their mathematical knowledge to identify and address the problem.

In fact, the questions students will work through are similar to those the Supreme Court is now considering on whether gerrymandering can ever be declared unconstitutional.

The article was co-authored with Michael Gonchar of the NYT Learning Network, and is freely available here.

Related Posts

The Math Behind Gerrymandering and Wasted Votes — Quanta Magazine

The U.S. Supreme Court is currently considering a case about partisan gerrymandering in Wisconsin and Texas. One of the keys to the case is the “efficiency gap”, an attempt quantify the partisan bias in a given electoral map. For my latest article in Quanta Magazine, I explain and explore the efficiency gap using simple examples, and talk about some of the implications of this particular measurement.

Imagine fighting a war on 10 battlefields. You and your opponent each have 200 soldiers, and your aim is to win as many battles as possible. How would you deploy your troops? If you spread them out evenly, sending 20 to each battlefield, your opponent could concentrate their own troops and easily win a majority of the fights. You could try to overwhelm several locations yourself, but there’s no guarantee you’ll win, and you’ll leave the remaining battlefields poorly defended. Devising a winning strategy isn’t easy, but as long as neither side knows the other’s plan in advance, it’s a fair fight.

Now imagine your opponent has the power to deploy your troops as well as their own. Even if you get more troops, you can’t win.

The full article is freely available here.

Global Math Week Symposium

James Tanton is on a mission to bring joyous mathematics to the world.  His Global Math Project is about to launch Global Math Week:  during the week of October 9th, over 600,000 students from around the world will enjoy a shared mathematical experience based on Tanton’s Exploding Dots, a wonderful, surprising, and awe-inspiring take on place value.

James has been traveling the world for the past year spreading the good word about mathematics and his exploding dots.  If you haven’t yet signed up, I encourage you to do so.  The mathematics is wonderful, relevant, and inspired, and the Global Math Project has lots of resources at their homepage.

To kick off Global Math Week, the Global Math Project together with the Museum of Mathematics will be hosting a symposium at NYU’s Courant Institute.  Mathematical luminaries like Po-Shen Loh, Henry Segerman, and many others will be on hand to celebrate.  And I’m honored to be participating in a panel discussion on Uplifting Mathematics for All, where we will discuss how to make mathematics meaningful, fun, and coherent in and out of the classroom.

So get ready for Global Math Week!  Hopefully this is the first of many to come.

AMS — Math in the Media

The debut of my column in Quanta Magazine was recently featured by the American Mathematical Society’s Math in the Media!

In addition to a nice review of my first Quantized Academy column, “Symmetry, Algebra, and the Monster“, I was also interviewed by Math in the Media’s Rachel Crowell.  Here’s an excerpt:

AMS: What excites you most about Quanta’s addition of the Quantized Academy series?

PH: Quanta does a wonderful job showing how mathematics and science are vibrant, active endeavors.  The writers bring math and science alive, telling exciting stories of mathematicians, scientists and their work. Quantized Academy can help connect students, teachers, and other lifelong learners to those stories and the math behind them.

You can read the entire article here.  Thanks to the AMS, and to Rachel Crowell, for taking an interest and helping to spread the word!