“Boom!” said the Mathematical Ninja, as the smoke cleared and the student jumped. Shell-shocked, he looked again at the whiteboard. “Wh-what just happened?”

The Mathematical Ninja sighed. “OK, one more time. We’re trying to estimate the size of the shapes you’d need to cover a sphere with $n$ patches, which - for reasons that may never become clear - we’re going to approximate as regular hexagons.”

“But Euler…”

“Never mind Euler. My classroom, I can bend the rules of geometry if I want to.”

“Hmph. OK, so we’re covering the sphere with flat, regular hexagons in a way that’s clearly impossible.”

The Mathematical Ninja nodded. “The surface area of a sphere is…”

“$4\pi r^2$!”

“And a hexagon…?”

“… I don’t know! But I can work it out. Six equilateral triangles, each with side length $x$, so $\frac{6}{2}x^2 \sin(60º)$, or $\frac{3}{2}\sqrt{3}x^2$!”

The Mathematical Ninja allowed their brows to furrow at the use of degrees, but continued.

“And we’ve got $n$ of those, so $\frac{3}{2}n\sqrt{3}x^2 = 4\pi r^2$.”

“Agreed,” said the student. He twisted his shoulders a fraction of a radian as if to go for his calculator, but caught himself in time.

“So $x$ is about $2.2 \frac{r}{\sqrt n}$.”

“Seriously?” said the student. “May I, uh, check it?”

“You dare to doubt? Actually, never mind, go ahead.”

While the student tapped away, the Mathematical Ninja turned to the camera and addressed the audience directly. “What I’ve done here is simply said $x^2 = \frac{8\pi}{3\sqrt{3}} \frac{r^2}{n}$. Obviously, $8\pi$ is about $\frac{176}{7}$ and $3\sqrt{3}$ is about $\frac{26}{5}$. Dividing those gives $\frac{880}{182} = \frac{440}{91}$. I’m going to need to square root that, and say $\sqrt{440} \simeq 21$, or a tiny bit less; $\sqrt{91} \simeq 9.5$, or a tiny bit more. And $21 \div 9.5$ is about 5% more than 2.1, which is about $2.2$.” They turned back to the student.

“Two point… 19927,” he said.

“Boom,” said the Mathematical Ninja again, and let off another smoke bomb.

* Edited July 14th to correct a missing square root sign. Apologies, Mathematical Ninja, for misquoting you!

* Edited 2020-12-28 to correctly gender TMN. Sorry, sensei!