Non-Euclidean geometry

Non-Euclidean geometry is a branch of mathematics that explores shapes and spaces in ways that are different from the familiar Euclidean geometry. While Euclidean geometry, which many people learn in school, is based on certain rules or axioms, non-Euclidean geometry changes one of these key rules. Specifically, it replaces the idea that parallel lines never meet with new ideas that allow lines to behave differently.

There are two main types of non-Euclidean geometry: hyperbolic geometry and elliptic geometry. In hyperbolic geometry, lines that seem to be parallel can actually meet at infinity, and there are many different lines that can be drawn through a point that will never meet a given line. In elliptic geometry, there are no parallel lines at all — any two lines will eventually meet.

These different geometries help us understand curved surfaces and spaces, like the surface of the Earth or the shape of the universe. They are important in fields like physics and astronomy, where the usual flat space of Euclidean geometry doesn’t always apply. By studying non-Euclidean geometry, mathematicians and scientists can describe the world in more accurate and interesting ways.

Principles

The big idea in geometry is how lines behave when they are next to each other. In regular geometry, which we call Euclidean geometry, if you draw a line and then draw another line from a point not on the first line, there will be only one line that never touches the first one. This is a rule made by a thinker named Euclid a long time ago.

But in other types of geometry, things change. In one kind called hyperbolic geometry, there are many lines from that point that never touch the first line. In another kind called elliptic geometry, every line from that point will eventually touch the first line. So, the way lines move apart or come together can be very different!

History

See also: Euclidean geometry § History, History of geometry, and Hyperbolic geometry § History

Background

Euclidean geometry, named after the Greek mathematician Euclid, is some of the oldest math we know. For a long time, people thought it was the only correct way to describe space.

Euclid wrote a book called Elements where he started with a few simple ideas and used them to prove many other ideas. One of these ideas, called the parallel postulate, was harder to understand than the others. For over a thousand years, smart people tried to prove this idea using the other ideas, but they couldn’t.

Development of non-Euclidean geometry

In the 1800s, some mathematicians began to explore ideas that were different from Euclid’s. They changed the parallel postulate and found new ways to describe space. This led to the creation of non-Euclidean geometry.

Two mathematicians, Nikolai Ivanovich Lobachevsky and János Bolyai, published books about a new kind of geometry called hyperbolic geometry. Around the same time, Bernhard Riemann talked about another type of geometry called elliptic geometry.

Terminology

The term "non-Euclidean geometry" was first used by Carl Friedrich Gauss. Today, this term usually means either hyperbolic or elliptic geometry.

Axiomatic basis of non-Euclidean geometry

Euclidean geometry is based on a set of rules, or axioms. One important rule is the parallel postulate, which says that through a point not on a line, there is exactly one line parallel to the given line. If we change this rule, we get different kinds of geometry.

There are two ways to change the parallel postulate. The first way is to say that through a point not on a line, there are more than one line parallel to the given line. This leads to what is called hyperbolic geometry. The second way is to say that through a point not on a line, there are no lines parallel to the given line. This creates elliptic geometry, a type of geometry first studied by Riemann. Both of these geometries follow the other rules of Euclidean geometry, but with this one rule changed.

Main article: Hilbert's system
Main articles: Undefined terms
Main article: Birkhoff
Main article: Absolute geometry
Main article: Negation
Main article: Playfair's axiom
Main article: Riemann
Main article: Elliptic geometry

Models

Models of non-Euclidean geometry are mathematical models of geometries that are not like the flat shapes we are used to. In these shapes, rules about lines and points change. In one type, called hyperbolic geometry, there are infinitely many lines that can pass through a point without meeting another line. In another type, called elliptic geometry, lines never stay parallel — they always meet up somewhere.

Euclidean geometry, which is the math we use for flat surfaces, can be thought of like a flat piece of paper. A simple way to imagine elliptic geometry is by using a sphere, where lines are like big circles (such as the equator or meridians on a globe). For hyperbolic geometry, a surface called the pseudosphere works well because of its special curvature.

Main article: Elliptic geometry

Main article: Hyperbolic geometry

In three dimensions, there are many different models of geometry. Besides the simple ones like Euclidean, elliptic, and hyperbolic, there are mixed types and even one special geometry where every direction acts differently.

Main article: Thurston geometry

Uncommon properties

Euclidean and non-Euclidean geometries share many properties, especially those not related to how lines run alongside each other. These shared features are studied in absolute geometry, also called neutral geometry.

Some special shapes show how these geometries differ:

• A Lambert quadrilateral has three corners that are right angles. Its fourth angle is smaller than a right angle in hyperbolic geometry, exactly a right angle in Euclidean geometry, and larger than a right angle in elliptic geometry. This means rectangles only exist in Euclidean geometry.
• A Saccheri quadrilateral has two sides of equal length that meet a base at right angles. Its other two angles, called summit angles, are smaller than right angles in hyperbolic geometry, exactly right angles in Euclidean geometry, and larger than right angles in elliptic geometry.
• The angles of any triangle add up to less than 180° in hyperbolic geometry, exactly 180° in Euclidean geometry, and more than 180° in elliptic geometry.

Importance

Before new ideas about shapes and space were shared by Beltrami, Klein, and Poincaré, people believed that Euclidean geometry was the only way to understand space. This idea was very important because it was thought to be a basic truth about how our minds work.

When non-Euclidean geometry was discovered, it changed many areas beyond math and science. It even affected ideas about how we know things, especially in philosophy. This discovery was like a big shift in how people thought about the world.

Non-Euclidean geometry showed that there are different ways to think about space, which was a major change in science. Some called Lobachevsky the "Copernicus of Geometry" because his work was so important. This new way of thinking also influenced how geometry was taught in schools, especially in Victorian England. Even famous writers like Lewis Carroll wrote about these changes in geometry learning.

Planar algebras

In analytic geometry, a plane can be described using Cartesian coordinates:

C = { (x, y) : x, y ∈ R }

The points can sometimes be identified with special numbers z = x + y ε where ε² can be –1, 0, or 1.

The usual flat plane, called Euclidean geometry, happens when ε² = −1, which is linked to an imaginary unit. Here, the distance of z from the starting point is given by:

zz* = (x + yε)(x − yε) = x² + y²

For example, {z | zz* = 1} forms the unit circle.

Non-Euclidean geometry appears in the other cases. When ε² = +1, a hyperbolic unit is used. Then z becomes a split-complex number, and we use j instead of ε. Here:

zz* = (x + yj)(x − yj) = x² − y²

and {z | zz* = 1} creates the unit hyperbola.

When ε² = 0, z is a dual number.

This method helps explain angles in non-Euclidean geometry. The measures of slope in the dual number plane and hyperbolic angle in the split-complex plane match angles in Euclidean geometry. They both come from the polar decomposition of a complex number z.

Kinematic geometries

Hyperbolic geometry has been used in the study of motion and the universe, thanks to ideas from Hermann Minkowski in 1908. He introduced important ideas like "worldline" and "proper time" into physics. Minkowski noticed that certain sets of events could be seen as a three-dimensional hyperbolic space.

Scientists also use special numbers to describe motion and changes in viewpoints. These numbers help show how things move in space and time, connecting geometry with the physics of motion.

Fiction

Non-Euclidean geometry often appears in science fiction and fantasy stories.

• In 1895, H. G. Wells wrote a short story called "The Remarkable Case of Davidson's Eyes." To understand this story, you need to know about special points on a sphere, which are part of a type of non-Euclidean geometry.
• Some writers, like H. P. Lovecraft, use non-Euclidean geometry to create strange and unusual settings. In his stories, places might follow their own unique rules that seem very wrong or unsettling to those who see them.
• In Robert Pirsig's book Zen and the Art of Motorcycle Maintenance, the main character talks about a special kind of geometry called Riemannian geometry.
• In The Brothers Karamazov, a character named Ivan talks about non-Euclidean geometry.
• Christopher Priest's novel Inverted World is about life on a planet shaped like a spinning curved surface.
• Robert Heinlein's The Number of the Beast uses non-Euclidean geometry to explain how characters can travel instantly through space, time, and even between different universes.
• Zeno Rogue's game HyperRogue is set on a special curved space called the hyperbolic plane. The game's rules and places are designed around the ideas of this geometry.
• In the Renegade Legion science fiction world, characters can travel faster than light and send messages using a special geometry created in the 22nd century.
• In Ian Stewart's Flatterland, the main character, Victoria Line, explores many different non-Euclidean worlds.