About

The Mandelbrot set

The Mandelbrot set consists of points, c, in the complex plane that obey the following rule

1. Start with the complex number z = a + ib where a and b are real numbers.
2. Generate a new complex number, z', by multiplying z by itself and adding the result to c. z' = z² + c.
3. Repeat steps 1 and 2. If the complex number z goes toward infinity, then the starting point, c, is not a member of the Mandelbrot set. All numbers that remain bounded are members of the set.

It can be shown that if the magnitude of z is greater than 2 , then z will approach infinity. The code assumes that the number c is in the set if |z| > 2 after 256 iterations. In order to show how rapidly a number fails the test, we color the pixel corresponding to the number of iterations.

Mandelbrot set with variable initial value

The members of the Mandelbrot set follow the rule

z_n+1= z_n²+ c ; initial value z₀ = 0

c is a complex number, as is z

We look for points c in the complex plane, for which the series does not diverge. They form the Mandelbrot set. In the simulation they are colored red, lying within a range with the fractal boundary, the famous "apple man". Diverging points are colored green to blue, with the color shading indicating the speed of divergence.

Drawing a rectangle with the mouse defines a smaller range of calculation with correspondingly increasing resolution. Reset leads back to the initial condition.

The series has the members 0, c, c²+ c, c⁴ + 2c³ + c² + c,...

For deeper understanding this simulation modifies the series in such a way, that the initial z_o value is variable:

z₀ = real(z₀) + i * im(z₀) = k + i m

This series has the members z₀ , z₀²+c, ( z₀²+c)²+c, , ...

z₀ = 0 (k = 0 , m = 0) delivers the common Mandelbrot set, which is the default case for the simulation.

In the drawing z₀ corresponds to the white point, which can be drawn with the mouse. The fractal changes according to the change of the initial value. The values of the real and the imaginary part of z₀ are shown in two editable number fields. There one can input exact values (press the ENTER key after every change).

The fractal structure is caused by the nonlinearity of the series rule. It is observed for other nonlinear series as well, with different structure of the fractal.

mandelbrot.xlm was originally developed by Francisco Esquembre and Wolfgang Christian The variations were developed by Dieter Roess in October 2008

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

 

Translations

Code	Language		Translator	Run

Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee; Francisco Esquembre; Wolfgang Christian; Felix J. Garcia Clemente

Overview:

This document provides a briefing on the JavaScript simulation applet for exploring the Mandelbrot set with a variable initial value (z0). The applet, hosted by Open Educational Resources / Open Source Physics @ Singapore, allows users to interactively investigate the behavior of complex numbers under a specific iterative process and visualize the resulting fractal structure. This briefing will outline the core concepts of the Mandelbrot set, the extension implemented in this simulation, the functionalities of the applet, and its pedagogical implications.

Main Themes and Important Ideas/Facts:

1. The Standard Mandelbrot Set:

• The Mandelbrot set is defined as the set of complex numbers, denoted as 'c', for which the iterative sequence zn+1 = zn2 + c does not diverge to infinity, starting with an initial value of z0 = 0.
• The document clearly states the rule: "The Mandelbrot set consists of points, c, in the complex plane that obey the following rule: 1. Start with the complex number z = a + ib where a and b are real numbers. 2. Generate a new complex number, z', by multiplying z by itself and adding the result to c. z' = z2 + c. 3. Repeat steps 1 and 2. If the complex number z goes toward infinity, then the starting point, c, is not a member of the Mandelbrot set. All numbers that remain bounded are members of the set."
• A practical criterion for divergence is given: "It can be shown that if the magnitude of z is greater than 2, then z will approach infinity. The code assumes that the number c is in the set if |z| > 2 after 256 iterations."
• Points within the Mandelbrot set in the simulation are colored red, forming the characteristic "apple man" shape. Points that diverge are colored based on the speed of divergence (green to blue).

1. Extension: Variable Initial Value (z0):

• This simulation extends the standard definition by allowing the initial value of the sequence, z0, to be variable instead of being fixed at 0.
• The document explains: "For deeper understanding this simulation modifies the series in such a way, that the initial zo value is variable: *z0 = real(z0) + i * im(z0 ) = k + i m This series has the members z0 , z02+c, ( z02+c)2 +c, , ... z0 = 0 (k = 0 , m = 0) delivers the common Mandelbrot set, which is the default case for the simulation."
• In the interactive visualization, the variable z0 is represented by a white point that the user can manipulate with the mouse. "In the drawing z0corresponds to the white point, which can be drawn with the mouse."
• The fractal image dynamically changes as the value of z0 is altered, demonstrating the dependence of the bounded/divergent behavior on the starting point. "The fractal changes according to the change of the initial value."
• The real and imaginary parts of z0 can also be precisely controlled through editable number fields. "The values of the real and the imaginary part of z0 are shown in two editable number fields. There one can input exact values (press the ENTER key after every change)."

1. Interactive Features of the Simulation:

• Zooming: Users can zoom into specific regions of the complex plane to observe the intricate details of the fractal boundary by drawing a rectangle with the mouse. "Drawing a rectangle with the mouse defines a smaller range of calculation with correspondingly increasing resolution."
• Reset: A "Reset" button returns the simulation to its initial conditions, likely with z0 = 0, displaying the standard Mandelbrot set. "Reset leads back to the initial condition."
• Variable z0 Manipulation: Users can change the initial value z0 by either clicking and dragging the white point on the display or by entering numerical values into the provided fields.

1. Educational Value:

• The simulation is presented as an open educational resource intended for learning and teaching mathematics, specifically related to sequences and series and complex numbers. It is part of a larger collection, "Learning and Teaching Mathematics using Simulations – Plus 2000 Examples from Physics."
• For teachers, the document highlights the key difference between this simulation and Julia Set simulations: the variability of the initial z-value. "There is one main difference between this Mandelbrot Set and the Julia Set simulation - that the initial z-value can be varied. The standard Mandelbrot Set follows this algorithm: z(n+1) = z(n)^2 + c, where z(0) is fixed at z = 0. This simulation, however, allows the user to vary the value of z(0) by either moving the white dot on the panel or by using the field provided."
• The simulation allows for a deeper understanding of the Mandelbrot set by illustrating how the choice of the initial condition affects the behavior of the iterative sequence and the resulting fractal.

1. Fractal Nature and Nonlinearity:

• The document briefly touches upon the underlying mathematical principles: "The fractal structure is caused by the nonlinearity of the series rule. It is observed for other nonlinear series as well, with different structure of the fractal." This emphasizes that the complex and self-similar patterns arise from the quadratic nature of the iterative formula.

Authors and Credits:

• The original Mandelbrot.xlm was developed by Francisco Esquembre and Wolfgang Christian.
• The variations, including the variable z0 implementation, were developed by Dieter Roess in October 2008.
• Other contributors are credited, including Dieter Roess (WEH- Foundation), Tan Wei Chiong, Loo Kang Wee, Francisco Esquembre, Wolfgang Christian, and Felix J. Garcia Clemente.

Technical Information:

• The simulation is a JavaScript HTML5 applet, making it embeddable in web pages using an iframe.
• It is part of the Open Educational Resources / Open Source Physics @ Singapore project.

Related Resources:

• The page includes links to other related simulations and resources, showcasing the breadth of interactive learning tools available on the platform. These include simulations related to vectors, electromagnetism, calculus, mechanics, waves, optics, and more.

License Information:

• The contents are licensed under the Creative Commons Attribution-Share Alike 4.0 Singapore License.
• Commercial use of the EasyJavaScriptSimulations Library requires a separate license from the University of Murcia (um.es).

Conclusion:

The "Mandelbrot Set (Variable z0) JavaScript Simulation Applet HTML5" provides a valuable interactive tool for understanding the Mandelbrot set and the impact of the initial condition on the iterative process that defines it. By allowing users to manipulate z0, the simulation offers a deeper insight into the dynamics of complex number iteration and the origins of fractal structures. Its accessibility as a web-based applet and its inclusion within a broader collection of educational simulations enhance its utility for both students and educators in mathematics and related fields.

he Mandelbrot Set: A Study Guide

Quiz

1. What is the fundamental rule that determines if a point c belongs to the Mandelbrot set?
2. Describe the iterative process used to test if a complex number c is part of the Mandelbrot set. What condition indicates that c is NOT a member?
3. Explain why the magnitude of z being greater than 2 is significant in the Mandelbrot set calculation.
4. In the standard Mandelbrot set algorithm, what is the fixed initial value of the complex number z (denoted as z0)?
5. How does the "Mandelbrot set with variable initial value" simulation differ from the standard Mandelbrot set calculation? What parameter is changed?
6. What does the white point in the variable z0 simulation represent, and how can it be manipulated within the simulation?
7. According to the text, what causes the intricate fractal structure observed in the Mandelbrot set and similar nonlinear series?
8. What visual cues in the simulation indicate whether a point c is a member of the Mandelbrot set or how quickly it diverges?
9. What action within the JavaScript simulation allows users to explore the Mandelbrot set in greater detail?
10. According to the "For Teachers" section, what is a key distinction between the Mandelbrot Set simulation described and a Julia Set simulation (mentioned but not detailed in the provided text)?

Answer Key

1. A point c belongs to the Mandelbrot set if, when used in the iterative formula z' = z² + c starting with z = 0, the magnitude of z remains bounded and does not approach infinity.
2. The process starts with z = 0, and in each iteration, a new complex number z' is calculated by squaring the previous z and adding c. If the magnitude of z exceeds 2 at any point, it is assumed that z will go to infinity, and thus c is not a member of the set.
3. It has been mathematically proven that if the magnitude of z becomes greater than 2 during the iteration process, then the sequence will inevitably tend towards infinity. This provides a practical escape condition for determining if a point is outside the Mandelbrot set.
4. In the standard Mandelbrot set algorithm, the initial value of the complex number z (z₀) is fixed at 0 (0 + 0i).
5. The "Mandelbrot set with variable initial value" simulation allows the user to change the starting value of z (denoted as z0), whereas in the standard Mandelbrot set, z0 is always 0. This modification explores how different initial conditions affect the resulting fractal.
6. The white point in the variable z0 simulation corresponds to the initial value of z (k + im). It can be manipulated by clicking and dragging the mouse on the panel or by entering specific real and imaginary values in the editable number fields.
7. The fractal structure is attributed to the nonlinearity of the series rule (z² + c). This type of complex behavior, generating intricate patterns upon iteration, is a characteristic of nonlinear systems.
8. Points c that are members of the Mandelbrot set (do not diverge) are colored red in the simulation. Diverging points are colored green to blue, with the specific shade indicating how quickly the sequence's magnitude exceeds the bound.
9. Users can zoom into specific regions of the Mandelbrot set within the simulation by clicking and holding the mouse, dragging diagonally to define a rectangle, and then releasing. This recalculates the fractal at a higher resolution within the selected area.
10. The key difference highlighted is that in the Mandelbrot Set simulation described, the initial z-value (z0) can be varied, while the text implies that in a Julia Set simulation, a different parameter might be variable, with the standard Mandelbrot having a fixed z0 of 0.

Essay Format Questions

1. Discuss the significance of iteration and the concept of boundedness in the definition and visual representation of the Mandelbrot set. How do these mathematical ideas translate into the fractal's characteristic features?
2. Explain the difference between the standard Mandelbrot set algorithm and the variation presented that allows for a variable initial value (z0). What insights into the behavior of complex number sequences can be gained by exploring this variation through the simulation?
3. Analyze the role of the complex plane in understanding the Mandelbrot set. How are points in the complex plane tested for membership, and how does their location relate to the visual structure of the fractal?
4. Describe how the provided JavaScript simulation facilitates a deeper understanding of the Mandelbrot set. Discuss the interactive features and how they allow users to explore the mathematical concepts involved.
5. The text briefly mentions that fractal structures are observed for other nonlinear series as well. Based on the information provided about the Mandelbrot set, discuss the potential connection between nonlinearity and the generation of complex, self-similar patterns in mathematical and potentially other systems.

Glossary of Key Terms

• Mandelbrot Set: A set of complex numbers c for which the sequence defined by z₀ = 0 and z_(n+1) = z_n² + c does not diverge to infinity.
• Complex Number: A number of the form a + ib, where a is the real part, b is the imaginary part, and i is the imaginary unit (√-1).
• Complex Plane: A geometric representation of complex numbers where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
• Iteration: The process of repeatedly applying a function or operation, using the output of one application as the input for the next.
• Bounded: A sequence is bounded if the absolute value (or magnitude in the case of complex numbers) of its terms does not grow indefinitely.
• Divergence: A sequence diverges if its terms grow without limit or do not approach a finite value.
• Magnitude (of a complex number): The distance of a complex number z = a + ib from the origin in the complex plane, calculated as |z| = √(a² + b²).
• Fractal: A geometric shape that exhibits self-similarity on different scales, meaning that parts of the shape resemble the whole.
• Nonlinearity: A property of a system or equation where the output is not directly proportional to the input, often leading to complex and unpredictable behavior.
• Initial Value (z0): The starting value of a sequence in an iterative process. In the standard Mandelbrot set, z0 is fixed at 0, but the simulation allows this value to vary.

Sample Learning Goals

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For Teachers

This is one of the multiple variations of the Mandelbrot Set simulation. There is one main difference between this Mandelbrot Set and the Julia Set simulation - that the initial z-value can be varied.

The standard Mandelbrot Set follows this algorithm: z(n+1) = z(n)^2 + c,

where z(0) is fixed at z = 0.

This simulation, however, allows the user to vary the value of z(0) by either moving the white dot on the panel or by using the field provided.

As with the other Mandelbrot Set simulations, zooming is enabled. Simply click and hold the mouse until the coordinates appear at the bottom left of the screen, and drag diagonally southeast. Give it some time to recalculate, and you get a zoomed-in version of the Mandelbrot in its full detail.

 

Research

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Video

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 Version:

1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

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Frequently Asked Questions: Mandelbrot Set Simulation

• What is the Mandelbrot set? The Mandelbrot set is a collection of complex numbers, denoted as c, that exhibit a specific behavior when subjected to an iterative mathematical process. This process begins with a complex number z, and in the standard definition of the Mandelbrot set, the initial value of z (denoted as z₀) is 0. A new complex number z' is generated by squaring z and adding c: z' = z² + c. This step is repeated. If the magnitude (or absolute value) of z remains bounded (does not approach infinity) after numerous iterations, then the starting complex number c is considered a member of the Mandelbrot set.
• How is membership in the Mandelbrot set determined in the simulation? The simulation determines if a complex number c belongs to the Mandelbrot set by iteratively applying the formula z_(n+1) = z_n² + c. For the standard Mandelbrot set, the initial value z₀ is 0. The simulation sets a limit on the number of iterations (256 in this case). If, after these iterations, the magnitude of z remains less than or equal to 2 (i.e., |z| ≤ 2), the point c is considered to be within the Mandelbrot set. If the magnitude exceeds 2 at any point during the iterations, it is assumed that z will approach infinity, and thus c is not part of the Mandelbrot set.
• What is unique about this particular Mandelbrot set simulation? The key difference in this simulation compared to the standard Mandelbrot set visualization is that it allows the user to vary the initial value of z (denoted as z₀). In the standard Mandelbrot set, z₀ is fixed at 0. This simulation introduces a variable z₀, which can be a complex number k + im, where k is the real part and m is the imaginary part. By changing z₀, the fractal structure that is generated is modified, offering a deeper understanding of how the initial conditions influence the behavior of the iterative process.
• How can the initial value z₀ be changed in the simulation? The initial value z₀ can be altered in two ways within the simulation. Firstly, the user can interact with the graphical display by clicking and dragging a white point that represents z₀. As this point is moved, the fractal image dynamically updates to reflect the change in the initial value. Secondly, the simulation provides two editable number fields where the user can directly input the exact real (k) and imaginary (m) components of z₀. After entering a value in either field, pressing the ENTER key applies the change and updates the fractal.
• How are points inside and outside the Mandelbrot set visualized in the simulation? In the simulation, points c that are determined to be within the Mandelbrot set (i.e., the iterative sequence remains bounded) are colored red. These points collectively form the recognizable "apple man" shape with its intricate fractal boundary. Points c that are found to diverge (i.e., the magnitude of z exceeds 2 during the iterations) are colored with shades ranging from green to blue. The specific shade of green or blue assigned to a diverging point indicates the speed at which the sequence diverges; different colors represent different numbers of iterations before the divergence condition is met.
• What does zooming in on the fractal reveal? Zooming in on the Mandelbrot set fractal, which is enabled in this simulation by drawing a rectangle with the mouse, reveals increasingly intricate and self-similar structures at smaller scales. This is a fundamental characteristic of fractals – their property of exhibiting similar patterns at different magnifications. As the user zooms in, the simulation recalculates the points within the selected region with higher resolution, uncovering the infinite detail and complexity of the Mandelbrot set's boundary.
• Why does the Mandelbrot set exhibit a fractal structure? The fractal structure of the Mandelbrot set arises from the nonlinearity of the defining iterative rule, z_(n+1) = z_n² + c. This simple quadratic equation, when applied repeatedly to complex numbers, leads to incredibly complex and self-similar patterns. The boundary of the Mandelbrot set is where infinitesimal changes in the complex number c can drastically alter the behavior of the sequence (either remaining bounded or diverging), resulting in the intricate and infinitely detailed fractal boundary. This type of behavior is also observed in other nonlinear series, although the specific structure of the resulting fractal may differ.
• What is the educational value of being able to vary the initial value z₀ in the Mandelbrot set? Allowing the variation of the initial value z₀ enhances the educational value of the Mandelbrot set simulation in several ways. It provides a deeper understanding of how the initial conditions of a nonlinear system can significantly impact its long-term behavior. By observing how the fractal structure changes as z₀ is altered, learners can gain insight into the sensitivity to initial conditions, a key concept in chaos theory and dynamical systems. Furthermore, it allows for exploration beyond the standard definition, encouraging a more nuanced understanding of the iterative process and the factors that contribute to the formation of the Mandelbrot set's complex geometry.