Me: what is a "limit set of a group of Möbius transformations"? It draws some fancy fractals (also an appolonian gasket)? And is this connected to the "Kleinian group fractals"?
Claude Opus: A "limit set of a group of Möbius transformations" is a mathematical concept related to complex analysis and hyperbolic geometry. Möbius transformations are a special type of conformal mapping on the complex plane or the Riemann sphere. When you have a group of Möbius transformations, the limit set is the set of accumulation points of the orbits of points under the repeated application of the transformations in the group.
In simpler terms, if you take a point and apply the Möbius transformations in the group to it repeatedly, the limit set is the set of points that the transformed points approach arbitrarily closely.
The limit set of a group of Möbius transformations can indeed generate intricate fractal patterns. The appearance of the fractal depends on the specific group of transformations being used. Some well-known examples include:
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Apollonian gasket: This fractal is generated by a group of Möbius transformations that map circles to circles. It consists of a series of ever-smaller circles that are tangent to each other and fill up the space between three larger circles.
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Kleinian group fractals: Kleinian groups are a special class of discrete subgroups of the group of Möbius transformations. The limit sets of Kleinian groups often exhibit fractal properties and can generate stunning visual patterns. Some famous examples of Kleinian group fractals include the Schottky dance, the Maskit slice, and the Riley slice.
The study of limit sets and the fractals they generate is an active area of research in complex analysis, hyperbolic geometry, and related fields. These concepts have applications in various areas, including the study of dynamical systems, chaos theory, and computer graphics.
In summary, the limit set of a group of Möbius transformations is a mathematical object that arises from repeatedly applying the transformations to points in the complex plane. It is indeed connected to Kleinian group fractals, as Kleinian groups are a special type of Möbius transformation groups whose limit sets often display fractal properties.
Me: How exactly does this related to the hyperbolic geometry?
Claude Opus: The connection between the limit set of a group of Möbius transformations and hyperbolic geometry lies in the fact that Möbius transformations are isometries (distance-preserving transformations) of the hyperbolic plane.
In hyperbolic geometry, the hyperbolic plane is a two-dimensional surface with constant negative curvature. There are several models of the hyperbolic plane, but the most commonly used ones in the context of Möbius transformations are the Poincaré disk model and the Poincaré half-plane model.
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Poincaré disk model: In this model, the hyperbolic plane is represented by the interior of a unit disk in the complex plane. The boundary of the disk, called the circle at infinity, represents points at infinity in the hyperbolic plane. Möbius transformations that map the unit disk to itself are isometries of the hyperbolic plane in this model.
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Poincaré half-plane model: In this model, the hyperbolic plane is represented by the upper half of the complex plane. The boundary of the half-plane, which is the real axis, represents points at infinity in the hyperbolic plane. Möbius transformations that map the upper half-plane to itself are isometries of the hyperbolic plane in this model.
When you have a group of Möbius transformations acting on the hyperbolic plane, the limit set of the group lies on the boundary of the hyperbolic plane (the circle at infinity in the Poincaré disk model or the real axis in the Poincaré half-plane model). The dynamics of the group action on the hyperbolic plane are intimately related to the structure of the limit set.
For discrete groups of Möbius transformations (such as Kleinian groups), the limit set can be classified into different types based on its topological and dynamical properties. The limit set can be:
- Empty: In this case, the group action is called "elementary."
- A finite set of points: The group action is still considered "elementary."
- A perfect, nowhere dense set: This is the case for "non-elementary" groups, and the limit set is often a fractal.
The study of the limit sets of Möbius transformation groups acting on the hyperbolic plane has led to significant advances in our understanding of hyperbolic geometry, 3-manifolds, and related areas of mathematics.
In summary, the limit set of a group of Möbius transformations is closely related to hyperbolic geometry because Möbius transformations are isometries of the hyperbolic plane. The limit set lies on the boundary of the hyperbolic plane and encodes important information about the dynamics of the group action.
Me: So, I have two mobius transformations. Can you please briefly explain an algorithm to calculate limit set of a Kleinian group of those transformations. For example, to draw an Appolonian gasket?
Claude Opus: Certainly! Here's a brief explanation of an algorithm to calculate the limit set of a Kleinian group generated by two Möbius transformations and draw an Apollonian gasket:
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Define the Möbius transformations:
- Let's call the two Möbius transformations f and g.
- Each transformation can be represented by a 2x2 complex matrix.
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Choose a starting point:
- Select a starting point z in the complex plane.
- This point can be chosen arbitrarily, but it should not be a fixed point of either transformation.
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Iterate the transformations:
- Create a set called limit_set to store the points in the limit set.
- Initialize an empty list called orbit to store the orbit of the starting point.
- Append the starting point z to the orbit list.
- Repeat the following steps for a large number of iterations (e.g., 10,000 or more):
- Randomly choose either transformation f or g.
- Apply the chosen transformation to the last point in the orbit list.
- Append the resulting point to the orbit list.
- If the new point is close enough to any point in the limit_set (within a small tolerance), skip to the next iteration.
- Otherwise, add the new point to the limit_set.
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Plot the limit set:
- Create a plot of the complex plane.
- Plot the points in the limit_set on the complex plane.
- Customize the plot as desired (e.g., set the aspect ratio to equal, adjust the limits, add labels).
To draw an Apollonian gasket, you can use specific Möbius transformations that map circles to circles. One common choice is to use transformations that map the unit circle to three mutually tangent circles. Here's an example:
- Transformation f:
(z - i) / (z + i) - Transformation g:
(z - exp(i*pi/3)) / (z + exp(i*pi/3))
These transformations, along with their inverses and compositions, will generate an Apollonian gasket when used in the algorithm described above.
Keep in mind that the algorithm presented here is a simplified version, and there are more efficient and sophisticated methods to compute limit sets, such as the "Schottky algorithm" or the "Klein-Maskit combination theorem." These methods take advantage of the specific structure of the Möbius transformations and can generate limit sets more quickly and accurately.
Implementing this algorithm in a programming language like Python, along with a suitable plotting library (e.g., Matplotlib), will allow you to visualize the limit set and create stunning fractal images of Apollonian gaskets and other Kleinian group fractals.
Me: What is limit set or Kleinian group even are? Why does this algorithm computes this fractal?
Claude Opus: A Kleinian group is a discrete subgroup of the group of Möbius transformations acting on the extended complex plane (also known as the Riemann sphere). In other words, it is a group consisting of certain Möbius transformations that, when applied repeatedly, yield a self-similar fractal pattern.
The limit set of a Kleinian group is the set of accumulation points of the orbits of points under the action of the group. Intuitively, if you take a point and keep applying the transformations from the Kleinian group to it, the limit set is the set of points that the transformed points approach arbitrarily closely.
The limit set of a Kleinian group has the following properties:
- It is a closed subset of the extended complex plane.
- It is invariant under the action of the Kleinian group, meaning that applying any transformation from the group to a point in the limit set results in another point in the limit set.
- It is either the entire extended complex plane, a circle, or a fractal.
The algorithm I described earlier computes an approximation of the limit set by iteratively applying the Möbius transformations to a starting point and keeping track of the resulting points. The idea behind this algorithm is that as you keep applying the transformations, the points will eventually converge to the limit set.
The reason this algorithm generates a fractal is that the limit set of a Kleinian group often exhibits self-similarity at different scales. Self-similarity means that smaller portions of the fractal resemble the larger structure. This self-similarity arises from the repeated application of the Möbius transformations, which are conformal mappings that preserve angles and shapes locally.
In the case of the Apollonian gasket, the specific Möbius transformations used in the algorithm map circles to circles in a way that creates a pattern of tangent circles. As the algorithm iterates, it generates smaller and smaller circles that fill the gaps between the larger circles, resulting in the characteristic fractal pattern of the Apollonian gasket.
It's important to note that the algorithm I provided is a simplified version and may not be the most efficient or accurate way to compute the limit set. More advanced techniques, such as the Schottky algorithm or the Klein-Maskit combination theorem, take advantage of the algebraic structure of the Kleinian group to generate the limit set more efficiently.
Nonetheless, the basic algorithm I described captures the essence of how the limit set of a Kleinian group can be approximated and visualized as a fractal. The repeated application of the Möbius transformations and the accumulation of points in the limit set give rise to the intricate and self-similar patterns characteristic of Kleinian group fractals.
Me: What does word Kleinian mean? Why you can't just say "apply all those transformations randomly and find their limit points"? Why there is a need to this and what is a "discrete subgroup"?
Claude Opus: The word "Kleinian" refers to the mathematician Felix Klein, who made significant contributions to the study of group actions on complex spaces and the geometry of surfaces. Kleinian groups are named after him because they are a particular type of group that he studied extensively.
Now, let's address your question about why we can't just say "apply all those transformations randomly and find their limit points." The reason is that the limit set of a Kleinian group has a specific mathematical definition and structure that goes beyond just randomly applying transformations.
A discrete subgroup of a group is a subgroup that consists only of isolated points, as opposed to a continuous range of points. In the context of Kleinian groups, discreteness means that the group elements are sufficiently separated from each other, and there are no infinite sequences of group elements that converge to another group element.
The discreteness property is crucial because it ensures that the group action on the extended complex plane is well-behaved and that the limit set has a fractal structure. If the group were not discrete, the limit set could be the entire extended complex plane, and the fractal structure would be lost.
Furthermore, the limit set of a Kleinian group is not just the set of limit points obtained by randomly applying transformations. It has additional properties that make it mathematically interesting and relevant:
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The limit set is invariant under the action of the Kleinian group, meaning that applying any transformation from the group to a point in the limit set results in another point in the limit set.
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The limit set is a closed subset of the extended complex plane, which means it contains all its accumulation points.
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The limit set has a specific topological structure. It can be classified as either the entire extended complex plane, a circle, or a fractal, depending on the properties of the Kleinian group.
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The dynamics of the group action on the extended complex plane are closely related to the structure of the limit set. The limit set encodes information about the orbits of points under the group action and the geometry of the corresponding hyperbolic 3-manifolds.
Studying Kleinian groups and their limit sets is important because they provide a rich interplay between group theory, hyperbolic geometry, and complex analysis. They have applications in various areas of mathematics, such as the classification of 3-manifolds, the study of chaotic dynamical systems, and the theory of conformal mappings.
The algorithm I described earlier is a way to approximate the limit set numerically and visualize it as a fractal. While it captures the essence of the limit set, it is not a complete mathematical characterization of the limit set or the Kleinian group itself. The algorithm relies on the discreteness and the specific properties of the transformations to generate the fractal pattern.
In summary, the term "Kleinian" refers to the mathematician Felix Klein, and Kleinian groups are named after him. The limit set of a Kleinian group is not just a random collection of limit points but has specific mathematical properties that make it interesting and relevant. The discreteness of the group is essential for the fractal structure of the limit set and the well-behaved action on the extended complex plane.
Me: What is a extended complex plane?
Claude Opus: The extended complex plane, also known as the Riemann sphere, is an extension of the complex plane that includes a point at infinity. It is a way to visualize the complex plane as a sphere, where the point at infinity is represented as the "north pole" of the sphere.
To understand the extended complex plane, let's first recall the complex plane. The complex plane is a two-dimensional representation of complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Each point in the complex plane corresponds to a unique complex number of the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i^2 = -1.
Now, imagine taking a sphere and placing it tangent to the complex plane at the origin (0, 0). The south pole of the sphere touches the origin of the complex plane. We can then establish a correspondence between the points on the complex plane and the points on the surface of the sphere (excluding the north pole) using a stereographic projection.
The stereographic projection works as follows:
- Draw a straight line from the north pole of the sphere to any point in the complex plane.
- The point where this line intersects the surface of the sphere (excluding the north pole) is the corresponding point on the Riemann sphere.
Using this projection, every point in the complex plane is mapped to a unique point on the surface of the sphere, and vice versa (excluding the north pole). The north pole of the sphere is not associated with any point in the complex plane and is considered the point at infinity.
The extended complex plane has some important properties:
- It is a compact space, meaning that it includes the point at infinity and has no boundary.
- Möbius transformations, which are the transformations used in the study of Kleinian groups, are well-defined on the extended complex plane. They map circles and lines to circles and lines on the Riemann sphere.
- It provides a natural setting for studying the behavior of complex functions, particularly their behavior near infinity.
The concept of the extended complex plane is crucial in the study of Kleinian groups because the limit sets of Kleinian groups are subsets of the extended complex plane. By considering the point at infinity, we can study the global behavior of the group action and the structure of the limit set in a more complete and unified way.
In summary, the extended complex plane, or the Riemann sphere, is an extension of the complex plane that includes a point at infinity. It is obtained by stereographically projecting the complex plane onto a sphere and considering the north pole as the point at infinity. The extended complex plane is a compact space where Möbius transformations are well-defined, and it provides a natural setting for studying the limit sets of Kleinian groups and the global behavior of complex functions.
Me: What is the standard way to present this point at infinity in code as a data structure? For example, to apply a mobius transformation.
Claude Opus: When working with the extended complex plane in code, it's common to use a special representation for the point at infinity. One standard way to represent the point at infinity is to use a separate flag or a special value that indicates infinity.
In many programming languages, floating-point numbers have a special value called "Infinity" or "Inf" that can be used to represent the point at infinity. However, when dealing with complex numbers, we need to consider both the real and imaginary parts.
Here's an example of how you can represent the point at infinity in Python using a tuple:
import math
def is_infinity(point):
return math.isinf(point[0]) or math.isinf(point[1])
def apply_mobius(z, a, b, c, d):
if is_infinity(z):
if c == 0:
return (a / c, b / d) # Infinity maps to a/c
else:
return (float('inf'), float('inf')) # Infinity maps to infinity
else:
numerator = (a * z[0] + b, a * z[1])
denominator = (c * z[0] + d, c * z[1])
if denominator == (0, 0):
return (float('inf'), float('inf')) # z maps to infinity
else:
real_part = (numerator[0] * denominator[0] + numerator[1] * denominator[1]) / (denominator[0]2 + denominator[1]2)
imag_part = (numerator[1] * denominator[0] - numerator[0] * denominator[1]) / (denominator[0]2 + denominator[1]2)
return (real_part, imag_part)In this code:
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We define a function is_infinity that checks if a given point (represented as a tuple) is the point at infinity. It returns True if either the real or imaginary part is equal to the special value math.inf (infinity).
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The apply_mobius function takes a point z (represented as a tuple) and the parameters a, b, c, d of the Möbius transformation.
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If the input point z is the point at infinity, we check if c is equal to zero. If c is zero, the point at infinity maps to a/c under the Möbius transformation. Otherwise, it maps to infinity.
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If the input point z is not the point at infinity, we apply the Möbius transformation using the formula (az + b) / (cz + d). We compute the numerator and denominator separately for the real and imaginary parts.
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If the denominator of the transformed point is zero, it means that z maps to the point at infinity under the Möbius transformation.
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Otherwise, we compute the real and imaginary parts of the transformed point using the standard formula for complex division.
This code demonstrates one way to handle the point at infinity when applying Möbius transformations in Python. The point at infinity is represented using the special value math.inf, and the code checks for this value to handle the special cases appropriately.
Note that the specific representation of the point at infinity may vary depending on the programming language and the library you are using for complex numbers. Some libraries may have built-in support for the extended complex plane and provide their own representation for the point at infinity.