Unveiling Infinity: Plotting the Mandelbrot Set by Hand

The Mandelbrot set, a captivating fractal, holds a unique allure: its intricate beauty arises from a remarkably simple equation. While computers effortlessly generate stunning visualizations, understanding the set’s foundation requires a deeper dive. This article guides you through the fascinating process of plotting the Mandelbrot set *by hand*, revealing the mathematical elegance that underlies this iconic image. Be prepared for a rewarding journey into the heart of complex numbers and iterative processes!

What is the Mandelbrot Set?

Before diving into the plotting process, let’s define what we’re dealing with. The Mandelbrot set is a collection of complex numbers, *c*, that exhibit a specific behavior when plugged into a particular equation. A complex number takes the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1). The core equation is:

z_n+1 = z_n² + c

Where:

• z₀ is initially set to 0 (0 + 0i).
• c is the complex number we’re testing for membership in the Mandelbrot set.
• z_n+1 is the next value in the sequence.
• The equation is iterated repeatedly.

A complex number *c* belongs to the Mandelbrot set if the sequence of *z* values remains bounded, meaning they don’t escape to infinity. If the sequence diverges (grows infinitely large), then *c* is *not* part of the set. The “escape” is typically defined as the magnitude of *z* exceeding a certain threshold, usually 2.

Materials You’ll Need

• Graph Paper: Essential for plotting complex numbers. Choose a grid size that allows you to represent the real and imaginary axes with sufficient detail. Larger paper is generally better.
• Pencil: For making accurate marks on the graph paper.
• Eraser: For correcting mistakes (and you will likely make some!).
• Calculator: A calculator capable of handling complex number arithmetic is extremely helpful. While you can do it manually, it significantly speeds up the process. Many online calculators can perform complex number calculations.
• Compass (Optional): Useful for visualizing the magnitude of complex numbers.
• Colored Pencils or Pens (Optional): To visually represent the escape rate of points outside the Mandelbrot set.

Setting Up Your Graph

1. Draw the Axes: On your graph paper, draw the real axis (horizontal) and the imaginary axis (vertical). The point where they intersect is the origin (0 + 0i). Carefully label the axes with numerical values. A good starting range is typically between -2 and 1 on the real axis, and -1 to 1 on the imaginary axis. The scale you choose is crucial for detail. For instance, each grid square could represent 0.1, 0.05, or even smaller increments for higher resolution.

2. Choose Your Scale: Decide how many grid squares will represent a unit of 1 on both the real and imaginary axes. Consistency is key. A finer scale will allow you to plot more points and see more detail in the Mandelbrot set, but it will also require more calculations.

3. Prepare for Complex Numbers: Remember that each point on your graph represents a complex number in the form a + bi. The *a* value corresponds to the point’s position on the real axis, and the *b* value corresponds to its position on the imaginary axis.

The Plotting Process: Step-by-Step

This is where the real work begins. We’ll systematically test complex numbers to determine if they belong to the Mandelbrot set.

1. Choose a Complex Number (c): Start with a relatively simple complex number. For example, let’s begin with c = 0 + 0i (the origin). This is the easiest point to calculate.

2. Initialize z: Set z₀ = 0 + 0i.

3. Iterate the Equation: Repeatedly apply the equation z_n+1 = z_n² + c.

* Iteration 1: z₁ = z₀² + c = (0 + 0i)² + (0 + 0i) = 0 + 0i

* Iteration 2: z₂ = z₁² + c = (0 + 0i)² + (0 + 0i) = 0 + 0i

* Iteration 3: z₃ = z₂² + c = (0 + 0i)² + (0 + 0i) = 0 + 0i

* And so on…

In this case, the sequence remains at 0 + 0i. It’s clearly bounded.

4. Check for Boundedness: After each iteration, calculate the magnitude of *z_n+1*. The magnitude of a complex number a + bi is calculated as √(a² + b²). If the magnitude of *z_n+1* ever exceeds 2, we can conclude that the sequence will diverge (escape to infinity), and the complex number *c* is *not* in the Mandelbrot set. We also need to define a maximum number of iterations. Usually 255 or 100. If the magnitude of z is still <= 2 after this limit we consider the number as being part of the set.

5. Plot the Point:

* If the sequence remains bounded (magnitude of *z* stays below 2) after a reasonable number of iterations (e.g., 20-30, or even more for greater accuracy), then plot the complex number *c* on your graph paper. You might use a solid color to represent points within the Mandelbrot set.

* If the sequence diverges (magnitude of *z* exceeds 2), then the complex number *c* is *not* in the Mandelbrot set. You can leave it blank, or use a different color to indicate how quickly it escaped (more on this later).

6. Repeat: Choose another complex number *c* and repeat steps 2-5. Systematically test different points on your graph. A methodical approach is best, such as working across rows and columns of your grid.

Example: c = 1 + 0i

Let’s try another example to illustrate the process. This time, we’ll use c = 1 + 0i.

1. Choose c: c = 1 + 0i

2. Initialize z: z₀ = 0 + 0i

3. Iterate:

* Iteration 1: z₁ = (0 + 0i)² + (1 + 0i) = 1 + 0i

* Iteration 2: z₂ = (1 + 0i)² + (1 + 0i) = (1 + 0i) + (1 + 0i) = 2 + 0i

* Iteration 3: z₃ = (2 + 0i)² + (1 + 0i) = (4 + 0i) + (1 + 0i) = 5 + 0i

4. Check for Boundedness: The magnitude of z₃ is √(5² + 0²) = 5, which is greater than 2. Therefore, c = 1 + 0i is *not* in the Mandelbrot set.

5. Plot: Since it’s not in the set, leave this point blank or use a specific color to indicate a fast escape.

Example: c = -1 + 0i

Let’s test c = -1 + 0i

1. Choose c: c = -1 + 0i

2. Initialize z: z₀ = 0 + 0i

3. Iterate:

* Iteration 1: z₁ = (0 + 0i)² + (-1 + 0i) = -1 + 0i

* Iteration 2: z₂ = (-1 + 0i)² + (-1 + 0i) = (1 + 0i) + (-1 + 0i) = 0 + 0i

* Iteration 3: z₃ = (0 + 0i)² + (-1 + 0i) = -1 + 0i

* Iteration 4: z₄ = (-1 + 0i)² + (-1 + 0i) = (1 + 0i) + (-1 + 0i) = 0 + 0i

4. Check for Boundedness: This sequence alternates between -1 and 0. It is bounded, since the magnitude is never larger than 1

5. Plot: Plot this point as part of the set.

Calculating z²

A critical part of the iteration process is calculating the square of a complex number. Here’s how to do it:

If z = a + bi, then z² = (a + bi)² = (a + bi)(a + bi) = a² + 2abi + (bi)² = a² + 2abi – b² = (a² – b²) + (2ab)i

So, z² = (a² – b²) + (2ab)i. This means:

• The real part of z² is (a² – b²).
• The imaginary part of z² is (2ab).

Using this formula, you can efficiently calculate z² for each iteration.

Visualizing Escape Rate (Optional)

If you want to add another layer of information to your plot, you can use different colors to represent how quickly points outside the Mandelbrot set escape. Here’s how:

1. Track Iterations: Keep track of the number of iterations it takes for the magnitude of *z* to exceed 2.

2. Assign Colors: Assign a color scheme based on the number of iterations. For example:

* Red: Escapes within 5 iterations.

* Orange: Escapes within 10 iterations.

* Yellow: Escapes within 15 iterations.

* (And so on…)

3. Plot with Colors: Plot the points outside the Mandelbrot set using the color corresponding to their escape rate. This will create a visually stunning representation of the fractal’s surroundings.

Tips for Success

• Be Organized: Keep your calculations neat and organized. Label each iteration clearly.
• Double-Check: Complex number arithmetic can be tricky. Double-check your calculations to avoid errors.
• Use a Calculator: A calculator capable of handling complex numbers will save you a significant amount of time and reduce the risk of errors.
• Be Patient: Plotting the Mandelbrot set by hand is a time-consuming process. Be patient and persistent.
• Start Simple: Begin with simple complex numbers (e.g., those with a real or imaginary part of 0) to get a feel for the process.
• Zoom In: Once you have a basic outline, zoom in on interesting areas of the set to reveal more detail. Change the scale and recalculate.
• Embrace the Imperfection: Your hand-drawn Mandelbrot set will not be perfectly accurate. That’s part of its charm. It’s a testament to the effort you put into understanding this fascinating fractal.
• Use a Spreadsheet: If you want to get more accurate with a smaller amount of effort use a spreadsheet (e.g. Google Sheets, Excel) to perform the calculations. The spreadsheet makes it easier to change the parameters and rerun the calculations.

Challenges and Considerations

• Computational Intensity: Plotting the Mandelbrot set by hand is computationally intensive. It requires numerous calculations for each point you test.
• Accuracy Limitations: Due to the manual nature of the process, your results will have limited accuracy. The finer the detail you want, the more calculations you need.
• Time Commitment: Be prepared to invest a significant amount of time in this project.

Why Bother Plotting by Hand?

With computers readily available to generate stunning Mandelbrot set images, why bother plotting it by hand? The answer lies in the deeper understanding you gain by manually performing the calculations. This hands-on experience provides a tangible connection to the mathematics underlying the fractal. You’ll develop an intuitive grasp of how the iterative process works and how the simple equation creates such intricate and beautiful structures. It’s a unique and rewarding way to appreciate the elegance of mathematics.

Exploring Beyond the Basics

Once you’ve mastered the basic plotting technique, consider these extensions:

• Julia Sets: Explore Julia sets, which are closely related to the Mandelbrot set. Julia sets are generated using the same equation (z_n+1 = z_n² + c), but instead of varying *c*, you keep *c* constant and vary the initial value of *z₀*.
• Different Equations: Experiment with different equations to generate other types of fractals.
• 3D Visualization: Research how the Mandelbrot set can be extended to three dimensions.

Conclusion

Plotting the Mandelbrot set by hand is a challenging but incredibly rewarding experience. It’s a journey into the heart of complex numbers, iterative processes, and the fascinating world of fractals. While computers can generate beautiful images, the act of manually plotting the set provides a deeper understanding and appreciation for the mathematical elegance that underlies this iconic image. So, grab your graph paper, pencil, and calculator, and embark on your own exploration of infinity!