PBS

When you think about math, what do you think of? Numbers? Equations? Patterns maybe? How about… knots? As in, actual tangles and knots?

Set theory is supposed to be a foundation of all of mathematics. How does it handle infinity?

Imagine you have four cubes, whose faces are colored red, blue, yellow, and green. Can you stack these cubes so that each color appears exactly once on each of the four sides of the stack?

Imagine you have a square-shaped room, and inside there is an assassin and a target. And suppose that any shot that the assassin takes can ricochet off the walls of the room, just like a ball on a billiard table. Is it possible to position a finite number of security guards inside the square so that they block every possible shot from the assassin to the target?

Cho Cho Cho is the first song and music video produced and written by the youth of Goma, Democratic Republic of Congo. Created in a Beat Making Lab, the song was inspired by a popular call-and-response chant in Goma, and the video was filmed during Yole!Africa's SKIFF Festival dance competition.

A Beat Making Lab is an electronic music studio small enough to fit in a backpack. We build Labs in collaboration with communities all around the world; donating laptops, microphones, software and other equipment. From our first Beat Making Lab in DR-Congo, producer Apple Juice Kid reaches into the backpack and reveals some of the tools that we use.

Why do women shave their legs and armpit hair, but men don’t? And what does World War II have to do with it?

Why were most slaves in America from West Africa? Slavery has existed throughout history in various forms across the globe, but who became enslaved was almost always based on military conquest. So why did Europeans travel thousands of miles to enslave people from a particular geographic region? Watch the episode to find out.

Why do all strip malls look the same, even across different states? And what do zoning laws, tax breaks, and a guy from Vienna have to do with it?

Money isn’t actually valuable. It’s just pieces of metal, pieces of paper, or more recently, computer code. So how did money come to be such a driving force of modern life?

Clothing is ultimately just fabric designed to cover our bodies. So why was it illegal for women to wear men’s clothing and for men to wear women’s clothing? And why is it still illegal in some parts of the world?

If you take it as a given that extinct dinosaurs were all weird and wonderful, then you gotta at least consider that Stegosaurus was one of the weirdest and wonderfulest.

We know a lot about dinosaurs but there’s one question that has plagued paleontologists for decades: what color were they?

Smilodon was a fearsome Ice Age cat, the size of a modern-day tiger, that had a pair of fangs nearly 18 centimeters long. But it was only the last and largest of the great sabertooths: ridiculously long canines had already been a trend for millions of years by the time Smilodon was prowling around. And you know what? Those giant teeth just might make a comeback.

What if we told you that there was a time when oxygen almost wiped out all life on Earth? 3 billion years ago, when the world was a place you’d never recognize, too much of a good thing almost ruined everything for everybody.

Today, we’re familiar with two types of flying vertebrates -- birds and bats. But over 66 million years ago, there was a giraffe-sized reptile that soared through the sky.

With its lizard-like appearance and that distinctive sail on it back, Dimetrodon is practically the mascot of the Palaeozoic Era, a time before flowers, birds, mammals, and even crocodiles. But if you take a close look at this sail-backed animal, you might see a little bit of yourself.

Natural history is full of living things that were long thought to have gone extinct only to show up again, alive and well. Paleontologists have a word for these kinds of organisms: They call them Lazarus taxa.

There are many fossils that challenge our ability to form even the most basic idea of how a living thing looked, or lived, or functioned. One of the longest-running of these mysteries involved a 270-million-year-old sea creature called Helicoprion that once swam the seas around the supercontinent of Pangea.

Insects outnumber humans by a lot and we only like to think we're in charge because we're bigger than they are. But insects and other arthropods weren’t always so small. About 315 million years ago during the Carboniferous Period, they were not only abundant: they were enormous.

Fossil fuels are made from the remains of extinct organisms that have been exposed to millions of years of heat and pressure. But in the case of coal, these organisms consisted largely of some downright bizarre plants that once covered the Earth, from Colorado to China.

Part of why we’re so fascinated with extinct dinosaurs it’s just hard for us to believe that animals that huge actually existed. And yet, they existed! From the Jurassic to the Cretaceous Periods, creatures as tall as a five-story building were shaking the Earth.

Besides the blue of the oceans, the dominant color of our planet, as we know it, is green. But imagine a time when the Earth looked a little … purple.

Crocodiles, horseshoe crabs and tuatara are animals that have persisted for millions of years, said to have gone unchanged since the days of the dinosaurs. But even the most ancient-looking organisms show us that evolution is always at work.

We know whales as graceful giants bound to the sea. But what if we told you there was actually a time when whales could walk.