Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a type of non-Euclidean geometry. Unlike the geometry you might learn in school, hyperbolic geometry changes one important rule. Instead of having just one line that can go parallel to another line, there are at least two lines that will never meet that other line.

The hyperbolic plane is a special kind of flat space where every point looks like a saddle. This idea connects to shapes called pseudospherical surfaces, which curve in a way that keeps their curvature the same but negative. These surfaces help us understand how the hyperbolic plane behaves.

One way to picture hyperbolic geometry is through the hyperboloid model. This model shows how points in space relate to each other, much like ideas in special relativity. It helps scientists think about how events happen over time. Different thinkers studied this geometry, and Felix Klein gave it the name we use today. In the former Soviet Union, it was often called Lobachevskian geometry, after the Russian mathematician Nikolai Lobachevsky.

Properties

Relation to Euclidean geometry

Hyperbolic geometry is closely related to Euclidean geometry. The main difference is the parallel postulate. In Euclidean geometry, parallel lines never meet, but in hyperbolic geometry, there can be many lines through a point that don’t meet a given line. This change leads to many new ideas and properties that are different from Euclidean geometry.

Lines

In hyperbolic geometry, lines behave somewhat like they do in Euclidean geometry. Two points always define a unique line, and lines can keep going forever. When two lines cross, they form equal opposite angles, just like in Euclidean geometry. But when you add a third line, things can look different. For example, there can be many lines that don’t touch either of two crossing lines.

Non-intersecting / parallel lines

Non-intersecting lines in hyperbolic geometry are different from those in Euclidean geometry. For any line and any point not on that line, there are at least two lines through the point that never meet the first line. Some of these lines get closer and closer to the first line but never touch it, while others stay a certain distance apart.

Circles and disks

In hyperbolic geometry, the size of a circle is different from in Euclidean geometry. The distance around a circle is always more than what you’d expect from its radius. This is because of the special curved shape of the hyperbolic plane.

Hypercycles and horocycles

In hyperbolic geometry, there are special curves called hypercycles and horocycles. A hypercycle is a curve where every point is the same distance from a line, but not in a circle shape. A horocycle is another special curve where the lines reaching out from it all run in the same direction forever.

Triangles

Triangles in hyperbolic geometry are different from those in Euclidean geometry. The angles in a triangle always add up to less than 180 degrees. This difference is called the "defect" and it tells us about the area of the triangle.

Regular apeirogon and pseudogon

Special shapes called regular apeirogons and pseudogons exist in hyperbolic geometry. These are shapes with an infinite number of sides that can still look like real polygons, unlike in Euclidean geometry.

Tessellations

Just like in Euclidean geometry, we can cover the hyperbolic plane with regular shapes in repeating patterns. These patterns are called tessellations and they use shapes like triangles and squares, but they look different because of the curved space.

Coordinate systems for the hyperbolic plane

Giving directions or locations is trickier in hyperbolic geometry than in flat geometry. There are many ways to set up a coordinate system, but they all need careful choices to work well.

Distance

We can measure distances in hyperbolic geometry using special formulas. These formulas help us find the distance between two points by using coordinates, similar to how we do it in flat geometry but with extra steps to account for the curved space.

History

See also: Non-euclidean Geometry

For thousands of years, mathematicians have tried to prove a key rule in geometry called the parallel postulate. Many great thinkers worked hard to solve this puzzle, but they couldn't prove it. Their efforts, however, helped discover a new kind of geometry called hyperbolic geometry.

In the 1700s, a mathematician named Johann Heinrich Lambert used special math tools to study shapes in this new geometry. Later, in the 1800s, four amazing mathematicians—Nikolai Lobachevsky, János Bolyai, Carl Friedrich Gauss, and Franz Taurinus—realized they had found a whole new way to understand geometry. They shared their ideas with the world, and soon others proved that this new geometry was just as valid as the old one.

The discovery of hyperbolic geometry changed how people thought about math and philosophy. It showed that there could be more than one correct way to describe the shapes and spaces around us.

Physical realizations of the hyperbolic plane

There are special shapes in regular space called pseudospheres that have a constant negative curved shape and a limited area.

According to Hilbert's theorem, it's impossible to perfectly place a whole hyperbolic plane—a surface with constant negative Gaussian curvature—inside our 3D space without stretching or cutting it.

Other helpful models of hyperbolic geometry exist in regular space, though they don't keep the exact measurements the same. A famous paper design based on the pseudosphere was created by William Thurston.

Artists have used crochet to show hyperbolic planes, with the first example made by Daina Taimiņa.

In 2000, Keith Henderson showed a simple paper model called the "hyperbolic soccerball" (exactly, a truncated order-7 triangular tiling).

Instructions for making a hyperbolic quilt, designed by Helaman Ferguson, are available from Jeff Weeks.

Models of the hyperbolic plane

There are special shapes called pseudospheres that can show hyperbolic geometry. One well-known example is the tractoid, which acts like a model for this geometry, similar to how a cone or cylinder can model flat geometry. However, the whole hyperbolic plane cannot be shown this way, so other models are used.

Four main models help us understand hyperbolic geometry: the Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz or hyperboloid model. These models all describe the same ideas but in different ways. They can also be used in more than two dimensions.

The Beltrami–Klein model

Main article: Beltrami–Klein model

This model uses the inside of a unit circle to represent the whole hyperbolic plane. Lines in this model are the chords of the circle. For more dimensions, it uses the inside of a unit ball, with chords as lines.

• Lines are straight in this model, but angles look wrong, and circles don’t look like circles.
• Distance is measured using a special math rule involving the cross-ratio.

The Poincaré disk model

Main article: Poincaré disk model

This model also uses the inside of a unit circle, but lines are shown as curves that touch the edge of the circle, plus lines through the center.

• Angles look correct in this model.
• Circles inside stay circles, but their centers look closer to the middle than they really are.
• Special circles called horocycles touch the edge of the disk.
• Other curves called hypercycles end at the edge at special angles.

The Poincaré half-plane model

Main article: Poincaré half-plane model

This model uses the top half of a flat plane, with a line at the bottom that isn’t part of the model.

• Lines are either curves that touch the bottom line or straight lines going up from it.
• Distances are measured using a special log rule.
• Like the disk model, angles look right.
• This model is what you get when you stretch the disk model very wide.

The hyperboloid model

Main article: hyperboloid model

This model uses a special curved shape in three dimensions to show the hyperbolic plane. It connects to ideas about space and time.

• It shows how points in space can be related to movement over time.
• The distance between points shows how fast things move past each other.
• This model can also be used in three dimensions.

The hemisphere model

The hemisphere model is less common but helps show how the other models connect.

It uses the top half of a unit sphere. Lines are curves that touch the edge of this half-sphere.

The hemisphere model links to other models using special views:

• One view matches the Poincaré disk model.
• Another matches the hyperboloid model.
• A third matches the Poincaré half-plane model.
• One view matches the Beltrami–Klein model.
• Another view matches the Gans Model.

Connection between the models

All these models describe the same basic idea. They just use different ways to draw it. The key feature of the hyperbolic plane is that it curves in a special way, no matter how you draw it. Lines called geodesics act the same in all models.

Other models of hyperbolic geometry

The Gans model

In 1966, David Gans suggested a flat way to show the hyperboloid model. It shows the whole flat plane but uses curves for lines.

• Unlike other models, this one uses the whole Euclidean plane.
• Lines look like parts of hyperbolas.

The conformal square model

This model turns the Poincaré disk into a square. It’s good for art because it keeps angles right.

The band model

Main article: Band model

This model uses a strip between two parallel lines. It keeps distances right along the middle line.

Different ways to move shapes in hyperbolic geometry include turning, sliding, and flipping. Every movement can be made with a few flips. These ideas work in flat and curved spaces too, but they look different.

In art

M. C. Escher's famous prints Circle Limit III and Circle Limit IV show the conformal disc model (Poincaré disk model). The white lines in III are close to geodesics, called hypercycles. We can see the negative curvature of the hyperbolic plane through how it changes the angles in triangles and squares.

For example, in Circle Limit III, every vertex is part of three triangles and three squares. In the Euclidean plane, their angles would add up to 450°, but in the hyperbolic plane, the angles of a triangle must be smaller than 180°. Another property we can see is exponential growth. In Circle Limit III, the number of fish within a distance of n from the center grows exponentially. Since the fish have the same hyperbolic area, the area of a ball of radius n must also grow exponentially.

The craft of crochet has been used to show hyperbolic planes, with the first one made by Daina Taimiņa. Her book Crocheting Adventures with Hyperbolic Planes even won a prize in 2009. There is also a game called _HyperRogue, a roguelike set on different tilings of the hyperbolic plane.

Higher dimensions

Main article: Hyperbolic space

Hyperbolic geometry isn’t just for flat surfaces; it works in any number of dimensions, not just two. Whether you’re thinking about a flat plane or something more complex, hyperbolic geometry can describe it all.

Homogeneous structure

Hyperbolic space of dimension n is a special kind of space that has special symmetries. It can be described using certain mathematical groups and transformations.

The group O(1, n) acts on a special kind of space called Minkowski space R^1,n, and this action helps create models of hyperbolic n-space. In smaller dimensions, there are special connections between different mathematical groups that give extra ways to understand the symmetries of hyperbolic spaces. For example, in two dimensions, certain groups allow us to see the hyperbolic plane in different models, and in three dimensions, we can study the shapes of hyperbolic space by looking at special complex matrices.