Roots of unity
The visualization shows, for a given period \(n\), the \(n\)-th complex roots of unity, and also $k$-th roots of unity where $k|n$ and $k < n$. The roots are evenly distributed on the unit circle in the complex plane. Each root is labeled with its corresponding \(n\)-value for easy identification.
The positions of the \(n\)-th roots of unity are given by the formula: \[ z_k = e^{i \frac{2k\pi}{n}}, \quad k = 0, 1, 2, \ldots, n-1. \] The transformation from the unit circle to the boundary of the cardioid, corresponding to the bifurcation points of the Mandelbrot set, is described by the following equations: \[ z_x(\theta) = \frac{1}{2} \cos(\theta) - \frac{1}{4} \cos(2\theta), \] \[ z_y(\theta) = \frac{1}{2} \sin(\theta) - \frac{1}{4} \sin(2\theta), \] where \(z_x\) and \(z_y\) are the real and imaginary components of the complex number \(z\), and \(\theta\) is the angle corresponding to the root.
These transformations illustrate how the \(n\)-th roots of unity map to the bifurcation points of the Mandelbrot set, revealing the relationship between the roots and the set’s hyperbolic components.
Description
- Period: Selects the corresponding \(n\) for the \(n\)-th roots of unity.
Bifurcation points
This visualization extends the previous one by adding a transformation of the unit circle into a smaller circle with a radius of $0.25$ and a center at $-1$, representing the period-2 component. The transformations used are:
For component with a period of two, the transformation is simpler, involving a shift of the center to \(-1\) and a change in the radius to \(\frac{1}{4}\). The transformation equations for this case are: \begin{align*} z_x(\theta) &= -1 + \frac{1}{4} \cos(\theta),\\ z_y(\theta) &= \frac{1}{4} \sin(\theta). \end{align*}
Only bifurcation points with the exact period $n$ are displayed in this visualization. Unlike the previous version, divisors of $n$(where $k | n$ and $k < n$) are no longer shown, ensuring that only the contact points of the chosen period are visible.
Description
- Period: Period of hyperbolic components for which the bifurcation points with the cardioid or the period-2 component are displayed.
Basins of attraction and Newton's method
This visualization is based on the following principle: for any point \( c \), located at the center of a hyperbolic component with period \( p \), the magnitude of the point of the critical orbit \( |z_p| \) will return to \( 0 \) after \( p \) iterations. For points in the complex plane, whether inside or outside the Mandelbrot set, that are sufficiently close to the hyperbolic component with period \( p \), \( |z_p| \) reaches a new minimum at the \( p \)-th iteration (except for the initial value \( z_0 = 0 \)). The closer \( |z_p| \) approaches zero after \( p \) iterations, the closer the point \( c \) is to the center of the hyperbolic component.
To calculate the precise center of the hyperbolic component corresponding to the basin of attraction, Newton's method is applied. Newton's method is a widely known iterative approach for finding the roots of a holomorphic function \( f(z) \). The iterative formula is defined as follows: \[ z_{n+1} = z_n - \frac{f(z_n)}{f'(z_n)}, \] where \( f'(z_n) \) is the derivative of the function \( f(z) \) at \( z_n \). This method is particularly effective due to its quadratic convergence rate, which allows it to rapidly refine the estimate of the root.
In this context, \( f(c) \) represents \( z_p(c) \), the \( p \)-th iterate of the critical orbit starting at \( z_0 = 0 \). By using an initial guess for \( c \), Newton's method iteratively improves this value until \( |f(c)| < \epsilon \), where \( \epsilon \) is a predefined tolerance. This ensures an accurate determination of the center of the hyperbolic component.
The visualization highlights basins of attraction, with each basin corresponding to a hyperbolic component of the Mandelbrot set. These basins are visualized using the described principle, and the center of the hyperbolic component corresponding to this basin of attraction is calculated with Newton's method.
Description
- Arrows: Use the arrow keys to navigate the canvas in all directions.
- Zoom In Button: Increases the zoom level for a closer view.
- Zoom Out Button: Decreases the zoom level for a wider view.
- Reset Button: Resets the view to the default position.
- Transparency: Adjusts the transparency of the Mandelbrot set to make overlapping features more visible.
- Rendering Time: Displays the time taken to render the visualization.
- Zoom: Indicates the zoom level.
- Center: Displays the result of the center calculation for the hyperbolic component.
Hyperbolic components
In the Mandelbrot set, hyperbolic components correspond to roots of polynomials \( P_n(c) = f^n(0) = 0 \), where the degree of \( P_n \) is \( 2^{n-1} \). This suggests that there are \( 2^{n-1} \) solutions, or roots, for each polynomial. However, the number of hyperbolic components does not match this directly due to the fact that components of period \( n \) are also components of periods \( 2n, 3n, 4n, \dots \). For example, a component with period 3 is also a component with period 6, 9, etc.
To account for this, we introduce the term \( U_n \), which represents the number of components with exactly period \( n \). The number of hyperbolic components of period \( n \) is given by: \[ U_n = 2^{n-1} - p, \] where \( p \) is the sum of hyperbolic components for all divisors \( k \) of \( n \), with \( k < n \): \[ p = \sum_{k|n,\, k < n} U_k. \]
This formula gives the correct number of hyperbolic components observed in the Mandelbrot set.
The key idea is that the polynomial \( P_n(c) \) is divisible by all polynomials \( P_k(c) \) where \( k \) divides \( n \). Therefore, to isolate the components of exactly period \( n \), we define a reduced polynomial \( Q_n(c) \), which eliminates solutions for divisors of \( n \): \[ Q_{n}(c) = \frac{P_{N}(c)}{\prod_{k|n,\, k < n} Q_{k}(c)} \quad \text{for} \quad Q_{1}(c) = P_{1}(c). \] Solving \( Q_n(c) = 0 \) gives the roots, or the centers, of the hyperbolic components for a given period. These roots will either be real or complex conjugates due to the symmetry of the Mandelbrot set along the real axis.
Description
- Rendering Time: Displays the time taken to render the visualization.
- Number of hyperbolic components for period: Number of hyperbolic components for period for given period.
- Period: Period of hyperbolic components.
- Reload Button: Re-renders the visualization after parameter values have been changed.