Hyperbolic triangle

A triangle has three sides and three corners.

Most triangles are flat. These special ones are not. They look thin and curvy. They live on a bumpy surface.

These triangles are very different. In a flat triangle, the corners add up to a certain amount. In these triangles, the corners add up to less.

Some of these triangles are very large. They can even have corners that reach far away.

These are called ideal triangles. They are the biggest ones you can make.

They can also fit together in pretty ways. You can see them in patterns.

These triangles are truly amazing.

A hyperbolic triangle has three sides and three corners.

Most triangles we see are flat. These are not flat. They live on a curved surface. They often look thin.

These triangles act differently than flat ones. In a flat triangle, the three corners always add up to 180 degrees. In a hyperbolic triangle, the corners add up to less than 180 degrees. This difference is called the defect. The size of the triangle depends on this defect. A larger defect means a larger area.

Some triangles are very special. They can have corners that reach far away. We call these ideal points. A triangle with three ideal points is an ideal triangle.

These are the largest triangles possible in this space.

These shapes can also make beautiful patterns.

You can fit many of them together to cover a surface. This is called a tiling. Mathematicians use special models to study them. One model is called the Poincaré disk. This helps us see how these curvy shapes work.

A hyperbolic triangle is a special shape made of three sides and three corners.

In the flat world we see every day, triangles follow very strict rules. But these triangles live in a curved space called the hyperbolic plane. They might look thin or stretched out. This happens because the surface they sit on is not flat. You can imagine them living on a surface shaped like a saddle. Even in a huge, high-dimensional space, three points will always form a flat plane. This means we can study these shapes as simple, flat drawings.

These triangles act in ways that surprise us. In a flat triangle, the three angles always add up to exactly 180 degrees. However, in hyperbolic geometry, the sum of the angles is always less than 180 degrees. This missing amount is called the defect. The area of the triangle is actually tied to this defect. If the defect is large, the triangle has a large area. There is even a maximum possible area for these triangles.

This is very different from flat math where triangles can be any size. Every hyperbolic triangle also has a circle tucked inside it. This is called an inscribed circle.

History shows us that people have studied these ideas for a long time. One person named Ferdinand Karl Schweikart described a special type of triangle in 1818. He looked at a triangle with two corners that reach far away. These far-away corners are called ideal points. These points sit on the very edge of the hyperbolic space. A triangle with all three corners at the edge is an ideal triangle. This is the largest triangle you can possibly make in this geometry. Because the corners are at the edge, the angles are all zero.

Mathematicians use special tools to draw these shapes on paper. One famous tool is called the Poincaré disk model. In this model, the hyperbolic plane sits inside a large circle. The edges of the triangle look like curved lines. These lines are actually parts of circles that hit the edge at right angles.

Another way to see them is the half-plane model. This model uses a flat plane where the bottom edge is the boundary. These models help us see how the shapes stay the same even when moved. They help us calculate distances and angles without getting lost in the curves.

Learning about these triangles helps us understand how space can work. They are related to other types of geometry, like the kind used on a sphere. If you swap certain math rules, you can move between spherical and hyperbolic math. This connection shows that math is a giant, connected web. We can use these shapes to create beautiful, repeating patterns called tilings. You can see these patterns in many different ways across the hyperbolic plane. They show us that even simple shapes can have amazing secrets.

A hyperbolic triangle is a fundamental shape within the hyperbolic plane. It is defined by three non-collinear points, known as vertices, and the three segments connecting them, known as sides or edges.

While these triangles exist in a curved space, any three points in a hyperbolic space of any dimension will always lie on the same plane. This allows mathematicians to study planar hyperbolic triangles even when working in much higher dimensions. These shapes are essential for understanding non-Euclidean geometry, where the rules of flat space no longer apply.

To understand how these triangles function, one must look at their unique geometric properties. Every hyperbolic triangle possesses an inscribed circle, which is a circle that fits perfectly inside the triangle. However, not every hyperbolic triangle has a circumscribed circle that passes through all three vertices. This occurs if at least one vertex is an ideal point, or if the vertices lie on a horocycle or a one-sided hypercycle. Furthermore, hyperbolic triangles are characterized as being "thin." This means there is a maximum distance, denoted as delta, from any point on an edge to one of the other two edges. This specific characteristic led to the mathematical concept of delta-hyperbolic space.

Hyperbolic triangles exhibit several distinct types based on their vertices. A special category includes triangles with ideal vertices, where the corners sit on the ideal boundary of the plane. If two sides are limiting parallel, they approach each other but never intersect, forming an angle of zero at an omega point. A triangle with exactly one ideal vertex is called an omega triangle. Another specific type is the triangle of parallelism, which has one ideal vertex and one right angle. Ferdinand Karl Schweikart described the Schweikart triangle in 1818, which features two ideal vertices and one right angle. Finally, an ideal triangle has all three vertices at the ideal boundary. Because its angles all sum to zero, it is the largest possible triangle in hyperbolic geometry.

The history of these shapes is tied to the discovery of how curvature affects geometry. Johann Heinrich Lambert first proved a vital theorem relating a triangle's area to its angle sum. In Euclidean geometry, the angles of a triangle always sum to 180 degrees. In hyperbolic geometry, the sum of the angles A, B, and C is always strictly less than 180 degrees. This difference is known as the defect of the triangle. The area of a hyperbolic triangle is directly proportional to this defect. Specifically, the area equals the defect multiplied by the square of the Gaussian curvature constant. This relationship is mathematically similar to Girard's theorem used in spherical geometry.

Mathematicians use various planar models to visualize these complex shapes on a flat surface. One common method is the Poincaré disk model, where the hyperbolic plane is contained within a unit circle. In this model, hyperbolic lines appear as the interior parts of circles that intersect the unit circle at right angles.

Another method is the Poincaré half-plane model, which uses the upper half of the complex plane. In this model, the real axis acts as the boundary at infinity, and hyperbolic lines are represented by circles or lines that intersect the real axis at right angles. These models allow for the use of Möbius transformations to move shapes without changing their fundamental properties.

Trigonometry in the hyperbolic plane relies on different functions than those used in flat geometry. Instead of standard sine and cosine, hyperbolic trigonometry uses hyperbolic functions such as sinh, cosh, and tanh. These formulas depend on measuring sides in absolute length, which is a unit that sets the Gaussian curvature to negative one. For a right triangle with angle C as the right angle, the sine of angle A is the hyperbolic sine of the opposite side divided by the hyperbolic sine of the hypotenuse. There are also complex laws, such as the hyperbolic law of cosines and the hyperbolic law of sines, which allow for the calculation of sides and angles in any hyperbolic triangle.

The study of hyperbolic triangles reveals deep connections to other mathematical fields. There is a striking analogy between hyperbolic and spherical trigonometry. If one replaces the sides in spherical formulas with imaginary values, the formulas transform into hyperbolic ones. This connection shows that these geometries are part of a larger, interconnected system. Hyperbolic geometry also allows for the creation of complex, repeating patterns known as tilings.

These tilings, such as an order-7 triangular tiling, demonstrate how equilateral triangles can fill a hyperbolic plane in ways that are impossible in flat space.