Hyperbolic geometry

Most people think of math using flat surfaces like a sheet of paper. But there is another way to look at space called hyperbolic geometry.

This kind of geometry happens on a plane with negative curvature. You can imagine this shape by looking at a saddle.

In this space, lines behave very differently than they do on a flat desk. The most important difference is how parallel lines work. In a flat world, there is only one line through a point that never touches another line. In hyperbolic geometry, you can draw many different lines through that same point. None of these lines will ever meet the first line.

Shapes like triangles and circles change their rules in this space. If you draw a triangle on a hyperbolic plane, its angles are small. The three corners will always add up to less than a straight line.

Mathematicians call this missing amount the defect. The area of the triangle actually depends on this defect. Circles are also strange in this world. The distance around the edge of a circle is much larger than you would expect. This happens because the surface curves away from itself. Even the way we measure distance and angles is linked together here.

People have studied these strange rules for a long time. One famous person was a Russian geometer named Nikolai Lobachevsky. Because of his work, some people call this Lobachevskian geometry.

Other people use the name Bolyai–Lobachevskian geometry. A mathematician named Felix Klein later gave it the name we use today. He wanted to group it with other types of geometry. He put it in a list with elliptic, parabolic, and hyperbolic geometry. This helped everyone understand how these different spaces relate to each other.

There are many ways to visualize or model these shapes. One way is called the Poincaré disk model.

You can also use a model called a hyperboloid. This model is used in special relativity to show events in time.

Some shapes in this geometry are very special. You can find polygons with an infinite number of sides. These are called apeirogons or pseudogons.

They can have sides of any length and still look like polygons. You can even use them to create beautiful patterns called tilings.

Hyperbolic geometry connects to many things you might see in real life. Some people use crochet to make models of these planes. These look like the wavy edges of a coral reef.

You can also see these patterns in complex tilings. These tilings cover the whole plane with repeating shapes.

Even though the rules feel strange, they are very consistent. They follow their own set of logical steps. This math helps us understand how curved space can work. It shows us that there are many ways to measure the world.