Geometry

Apple Pies are Delicious

“Cherry pie is delicious!” Nick said, with a big smile. “Apple pies are, too.” He was explaining his trick for remembering the formulas for the circumference and area of a circle. A bunch of his classmates nodded along, many who attended the same middle school as Nick. I didn’t quite get it.

Nick diagrammed it out for me.

“Cherry pie is delicious” –> $C \pi d$ –> $C = \pi d$

“Apples pies are, too” –> $A \pi r 2$ –> $A = \pi r^2$

Now, I don’t mind a good mnemonic now and then; I still sing the alphabet song, after all. But this struck me as extremely silly. These formulas get used all the time and they are deeply connected to many other important concepts. Relying on a memory trick creates a flimsy foundation for an important body of knowledge. I decided to show Nick just how flimsy.

The next day in class, I approached Nick. “You know, after thinking about it, I agree with you: apple pies are delicious.” He was pleased. But his smile quickly faded. He wrote something out in his notes. “Wait, that’s not right.”

“So apple pies are not delicious?” I asked.

“It’s ‘Cherry pie is delicious‘.” He showed me the formula.

“But apple pies are delicious, right?”

“Yeah, but that’s just not how it works.”

“This is kind of confusing”, I said. “Oh wait. Now I see. Apple pies are delicious too!” I wrote out $A \pi r d 2$ , followed by $A = \pi \frac{rd}{2}$ . “Perfect!”

“What?”

“See here,” I said. I wrote out $A = \pi \frac{rd}{2} = \pi \frac{r2r}{2} = \pi r^2$ . “You’re method works perfectly!”

Nick started scribbling more in his notebook. I quietly walked away.

Over the next few days I continued my demonstration. “Cherry pies are delicious, too!” I’d say. Or, “Apple pies are really, really delicious!” I might have even said something like “Some apple pies are to die for.”

My demonstration was successful. Maybe too successful. Nick got the area of circle wrong on the next test.

When I handed it back, he acknowledged my point with a combination of irritation and admiration. Nick never got the area of a circle wrong again. And we never had to talk about his Dear Aunt Sally again, either.

[No mathematical understanding was harmed in this story.]

By MrHonner, 8 yearsFebruary 4, 2018 ago

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.

By MrHonner, 8 yearsDecember 12, 2017 ago

Geometry Photography

Math Photo: Spiky Symmetry

These cacti caught my. I can see both a dodecagon and a star in the 12-fold symmetry of the cactus in front. And to my surprise, the cactus behind it has thirteen sections!

I wonder about the range, and deviation, of the number of sections of these cacti. And what are the biological principles that govern these mathematical characteristics?

By MrHonner, 8 years ago

Geometry Testing

Regents Recap, August 2017: How Do You Explain that Two Things are Equal?

Sue believes these two cylinders from the August, 2017 New York Regents Geometry exam have equal volumes. Is Sue correct? Explain why.

Yes, Sue, you are correct: the two cylinders have equal volumes. I computed both volumes and clearly indicated that they are the same. Take a look!

Wait. Why did I only get half-credit? What’s the problem, Sue? You don’t think this is an “explanation”? The two volumes are equal. The explanation for why they are equal is that I computed both volumes and got the same number. I don’t know of any better explanation for two things being equal than that.

What’s that? You wanted me to say “Cavalieri’s Principle”? But if I compute the two volumes and show that they are equal, why would I need to say they are equal because of some other reason? Oh, never mind, Sue. See you in Algebra 2.

Geometry Resources

Symmetry, Algebra and the Monster — Quanta Magazine

I’m excited to announce the launch of my column for Quanta Magazine! In Quantized Academy I’ll be writing about the fundamental mathematical ideas that underlie Quanta’s stories on cutting edge science and research. Quanta consistently produces exciting, high-quality science journalism, and it’s a tremendous honor to be a part of it.

My debut column, Symmetry, Algebra and the Monster, uses the symmetries of the square to explore the basic group theory that connects algebra and geometry.

You could forgive mathematicians for being drawn to the monster group, an algebraic object so enormous and mysterious that it took them nearly a decade to prove it exists. Now, 30 years later, string theorists — physicists studying how all fundamental forces and particles might be explained by tiny strings vibrating in hidden dimensions — are looking to connect the monster to their physical questions. What is it about this collection of more than 10^53 elements that excites both mathematicians and physicists?