ôô

What Are Euclidean and Non-Euclidean Geometry?

How many degrees are in the angles of a triangle? What’s the shortest distance between two points? Do seemingly parallel lines ever cross? Think you know the answers? Think again! Keep on reading to find out why.

By
Jason Marshall, PhD
Episode #217

By the end of the last episode, we had proven that the interior angles of a triangle always add up to 180 degrees. Or so it seemed. At the very end, I challenged you to try your hand at a project with a balloon that hopefully forced you to question this conclusion.

Did you take me up on that challenge and try the project? If not, it’s not too late to give it a try. And you definitely should since it will force you to think about questions like "How many degrees are in a triangle?" and “Do parallel lines ever cross?” in a whole new way.

Why is that? Well, it turns out that the kind of geometry you learned in school isn’t the only kind of geometry. And, as a simple balloon can show you, the implications of this other kind of geometry are rather surprising.

.

Balloons, Triangles, and Angles

A few months ago, my daughter got her first balloon at her first birthday party. Ever since that day, balloons have become just about the most amazing thing in her world. After her party, she decided to call her balloon “ba,” and now pretty much everything that’s round has also been dubbed “ba.” A ball? A “ba.” The Moon? Yep, also a “ba."

Why did she decide that balloons—and every other round object—are so fascinating? I might be biased in this belief, but I’ve come to the conclusion that it’s because they’re so good at demonstrating some amazing properties of what’s called non-Euclidean geometry. OK, that’s almost certainly not true…but the math fan in me would like to believe it.

As I described last time, you can get a glimpse at one of these properties by performing a simple maths-and-crafts project. All you have to do is get an uninflated balloon, lay it on a flat surface, and draw as close to a perfect triangle on it as you can. If you have a protractor, this would be a good time to measure its angles and make sure they add up to approximately 1800.

Now, blow up the balloon and take a look at your once-perfect triangle. What happened to it? Do its angles still add up to 1800? To understand what you see, we need to talk about the differences between what’s called Euclidean and non-Euclidean geometry.

What Is Euclidean Geometry?

Since we’re talking about geometry, we’d first best establish what we mean by “geometry.” In broad terms, geometry is the realm of math in which we talk about things like points, lines, angles, triangles, circles, squares and other shapes, as well as the properties and relationships between the properties of all these things.

The type of geometry we typically learn in school is known as Euclidean geometry.

The type of geometry we typically learn in school—and the type of geometry we usually think of when we think of “geometry”—is known as Euclidean geometry. Why such a proper name? Euclidean geometry gets its name from the ancient Greek mathematician Euclid who wrote a book called The Elements over 2,000 years ago in which he outlined, derived, and summarized the geometric properties of objects that exist in a flat two-dimensional plane. This is why Euclidean geometry is also known as “plane geometry."

In plane geometry, the interior angles of triangles add up to 1800, two parallel lines never cross, and the shortest distance between two points is always a straight line.

What Is Non-Euclidean Geometry?

But it turns out that not everything lives in a two-dimensional flat world and therefore not everything is bound by the laws of plane Euclidean geometry. For example: you, me, and all of humanity live on the surface of the Earth, and the Earth is not flat. It is, in fact, an approximately spherical object. Which means that the rules of plane geometry do not rule our lives.

To understand what this means, let’s go back for a minute to balloons. An uninflated balloon is a flat object, and therefore lives within the realm of Euclidean geometry. In this world, nicely drawn triangles have 1800. But as soon as you inflate your balloon, its surface is no longer flat—it becomes spherical, and that brings it into the realm of what’s known as non-Euclidean geometry.

The term non-Euclidean sounds very fancy, but it really just means any type of geometry that's not Euclidean—i.e., that doesn’t exist in a flat world. A non-Euclidean geometry is a rethinking and redescription of the properties of things like points, lines, and other shapes in a non-flat world. Spherical geometry—which is sort of plane geometry warped onto the surface of a sphere—is one example of a non-Euclidean geometry.

Non-Euclidean Geometry in the Real World

In flat plane geometry, triangles have 1800. In spherical geometry, the interior angles of triangles always add up to more than 1800. You saw this with your inflated balloon, but you can also see it by thinking about the Earth.

In spherical geometry, the interior angles of triangles always add up to more than 1800.

Imagine you start at Earth’s North Pole and walk south until you reach the equator. You then walk directly east until you travel 1/4 of the way around the planet. Finally, you turn back north and return to the North Pole. If you think about it, you’ll see that the path you’ve traveled is a triangle on the spherical surface of the Earth. And the crazy thing is that all three angles of this "triangle" are right angles—so its interior angles add up to 90 x 3 = 2700.

Here’s another crazy thing: the pair of lines representing the two sides of the triangle marking the north-south legs of your journey are “parallel" to each other—in the sense that they both run in the north-south direction. But they intersect at the North Pole! And the South Pole! So even though they’re going in the same direction, they’re not parallel like the never-crossing parallel lines of plane geometry.

And just in case that’s not enough to get you thinking that non-Euclidean geometry is full of surprises, here’s another one. In a well-known case of “Huh? How is that possible?" it turns out that the shortest path to fly from Florida to the Philippines (which, mind you, is at a more southern latitude than Florida) is to fly over Alaska! How is that possible? I’m going to let you think about that…but be sure to check back next time for the answer.

Wrap Up

OK, that's all the math we have time for today.

Please be sure to check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Sphere image courtesy of Shutterstock.