Shape summary

If you’ve spent any time on the internet, you may have stumbled onto an argument over the question How many holes does a straw have?

You’re probably already leaning toward an answer that feels intuitive. It’s obvious a straw has two holes –⁠ just look at it! But wait –⁠ where does one hole start and the other end? Maybe that means a straw has just one hole that goes all the way through. Or does it have zero holes, since it’s just a rolled-up rectangle and rectangles don’t have holes? 

Picture a diagram of five concentric circles. A mosquito is born on the outermost ring. What’s the probability of finding the mosquito’s dead body, at the end of its life, anywhere else on the diagram?

This is the question that Sir Ronald Ross aimed to answer at the beginning of the twentieth century. Ross’s study of a mosquito’s flight path gave birth to a geometric theory that eventually permeated all sorts of other fields, from physics to finance and even to poetry. That theory is called the random walk.

Sir Ronald Ross wasn’t the only early twentieth-century mathematician spending time thinking about mosquitoes. Halfway across the world, a Russian named Andrei Markov was doing the same. 

Markov wasn’t thinking about mosquitoes whose every decision was random. Instead, he considered a mosquito whose decision on where to fly was dependent on its decision the previous day. 

Let’s go back to Sir Ronald Ross. Ross was the person who aimed to describe the probability of a mosquito ending up a particular distance from wherever it was born. 

We’ve left out part of the story, because in reality, Ross’s goals went much further. Ultimately, he wanted to quantify disease itself – and, furthermore, to quantify the spread of just about anything, from religions to military enlistments. In the course of that effort, he, along with a mathematician named Hilda Hudson, determined that pandemics spread in a sequence called a geometric progression, sometimes called exponential growth.

What do games like checkers, Connect Four, and chess have in common? Well, geometrically speaking, they’re all considered trees.

Games are considered trees when they observe a few parameters. Those are: they involve two players taking turns; the outcomes aren’t determined by random chance –⁠ that means no coin flips, spinners, or card draws;⁠ and every game ends after a finite number of steps. 

Picture yourself as a mountaineer trying to reach a high summit. You lost your map at some point along the way. Now how do you get to the top?

One way might be to look down at your feet and assess the slope. When you step northward, you see that the grade goes up a little. As you inch toward the south, the grade goes down. Then, you try turning northeast, and the grade suddenly goes up by a lot. Having identified the direction with the steepest upward slope, you walk a few steps northeast and then repeat the process over and over until you reach the top.

In November 2018, Democrats in the US state of Wisconsin were celebrating. Their gubernatorial candidate had won out over the Republican incumbent, they’d taken over the attorney general and state treasurer positions, and a Democratic senator had won reelection.

But it wasn’t quite a victory across the board. They’d picked up just one seat in the Wisconsin State Assembly –⁠ the lower house of the state legislature –⁠ allowing the Republicans to keep their 63–⁠36 majority. Meanwhile, the Republicans actually gained a seat in the state senate. 

What does a gerrymandered district map look like? Sometimes, it’s pretty obvious. 

Take, for instance, Pennsylvania’s Seventh District, which looked like a mangled depiction of the cartoon character Goofy kicking Donald Duck. The district was so obviously drawn to capture specific voters from specific regions that the courts threw it out in 2018.