Non-Euclidean geometry

Non-Euclidean geometry is a part of mathematics that looks at shapes and spaces in new ways. It is different from Euclidean geometry, which many people learn in school. Euclidean geometry has certain rules, but non-Euclidean geometry changes one of these rules. It changes the idea that parallel lines never meet, allowing lines to act in new ways.

There are two main types of non-Euclidean geometry: hyperbolic geometry and elliptic geometry. In hyperbolic geometry, lines that look parallel can meet at a far distance, and many lines can pass through a point without meeting a given line. In elliptic geometry, there are no parallel lines — any two lines will meet eventually.

These geometries help us understand curved surfaces and spaces, such as the Earth or the universe. They are useful in areas like physics and astronomy, where flat space does not always apply. By learning non-Euclidean geometry, mathematicians and scientists can describe the world more accurately and interestingly.

Principles

The big idea in geometry is how lines behave when they are next to each other. In regular geometry, called 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 rule was made by a thinker named Euclid a long time ago.

But in other types of geometry, things are different. In hyperbolic geometry, there are many lines from that point that never touch the first line. In 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 way to describe space.

Euclid wrote a book called Elements where he started with a few simple ideas and used them to prove many others. One of these ideas, called the parallel postulate, was harder to understand. For over a thousand years, many tried to prove this idea but could not.

Development of non-Euclidean geometry

In the 1800s, some mathematicians began to explore ideas 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 has rules called axioms. One rule is the parallel postulate. It says that if you have a point not on a line, there is exactly one line through that point that never meets the first line.

If we change this rule, we get different kinds of geometry.

One way is to say there are more than one line through the point that never meets the first line. This makes hyperbolic geometry.

Another way is to say there are no lines through the point that never meet the first line. This creates elliptic geometry, first studied by Riemann. Both types 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 special shapes that are not flat like the ones we usually see. In these shapes, the rules about lines and points are different. 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 thought Euclidean geometry was the only way to understand space. This idea was important because people believed it was 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 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 be linked to special numbers z = x + y ε where ε² can be –1, 0, or 1.

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

zz* = (x + yε)(x − yε) = x² + y²

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

Non-Euclidean geometry shows up 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} forms the unit hyperbola.

When ε² = 0, z is a dual number.

This way helps us understand angles in non-Euclidean geometry. The measures of slope in the dual number plane and hyperbolic angle in the split-complex plane are like angles in Euclidean geometry. They both come from the polar decomposition of a complex number z.

Kinematic geometries

Hyperbolic geometry helps us understand motion and the universe. In 1908, a scientist named Hermann Minkowski shared important ideas about "worldline" and "proper time". He saw that some events could be shown as a three-dimensional hyperbolic space.

Scientists use special numbers to describe how things move in space and time. These numbers connect geometry with the physics of motion.

Fiction

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

• In 1895, H. G. Wells wrote a 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 non-Euclidean geometry.
• Some writers, like H. P. Lovecraft, use this geometry to create strange and unusual settings.
• 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 curved surface.
• Robert Heinlein's The Number of the Beast uses non-Euclidean geometry to explain how characters can travel instantly through space and time.
• Zeno Rogue's game HyperRogue is set on a special curved space called the hyperbolic plane. The game's rules are designed around this geometry.
• In the Renegade Legion science fiction world, characters can travel faster than light using a special geometry.
• In Ian Stewart's Flatterland, the main character explores many different non-Euclidean worlds.