Fun with Fractals in Visual Studio .NET
- Aug 11, 2006
- Debugging 101 for Visual Studio
- Fun with Fractals in Visual Studio .NET
- Exploring Excel Functions, Part 9: Analyzing Frequency Distributions
- File Management in .NET
- Exploring Excel Functions 8: Predicting the Future
- Exploring Excel's Functions, Part 7: Rounding with Excel
- Binary File Access in the .NET Framework
- Text File Access in the .NET Framework
- Using Geometric Transforms for Text Effects in .NET
- Understanding MDI Applications in .NET
- A .NET Framework Text and Font Primer
- Exploring Excel's Functions, Part 6: TTEST() Function
- Troubleshooting Excel PivotTables
- Troubleshooting Word Tables
- Exploring Excel's Functions, Part 5: The Power of Choice
- Seven Things I Hate About Word Printing
- Exploring Excel's Functions, Part 4: Database Functions
- Exploring Excel's Functions Part 3: CELL() Shocked
- Exploring Excel's Functions Part 2: ADDRESS() and INDIRECT()
- Exploring Excel's Functions: IF() Only!
- Advanced Find-and-Replace Tools in Word
- Going Beyond Basic Spaces and Hyphens in Word
- Get Organized with Word's Outline Tools
- Managing Your Money in Microsoft Excel: Basic Financial Calculations
- Structured Exception Handling in Visual Studio .NET
- Multithreading with the .NET Framework
- .NET Tools for Working with XML
- Storing Information: Variables and Constants in C
- Introducing Web Programming with .NET
- Using Web Forms
- Introducing Web Services
- XML and the .NET Framework
- ASP.NET Programming: Using Web Forms
- Understanding the Common Features of Web Controls
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You may have heard of fractal images, and you’ve likely seen them more than once because they’re frequently used to adorn calendars, T-shirts, and the like. But did you know that you can generate fractal images on your computer? This article shows you how to create what is perhaps the most famous fractal—the Mandelbrot set—using Visual Studio .NET. First, however, a little necessary background material.
Complex Numbers
The name complex numbers is sort of misleading. Complex numbers are not called that because they’re complicated, but rather because they have two parts. Any complex number is represented as follows:
a + bi
The components a and b are "regular" numbers such as 2, −1.09, or 306. These are called real numbers. The "i" component is what makes things interesting—it stands for the square root of −1. You probably remember from your high school math class that negative numbers can’t have square roots, but mathematicians have found out that you can do all sorts of interesting calculations, including fractals, if we pretend that they can. Such numbers are called imaginary numbers. So a complex number has a "real" part and an "imaginary" part.
When working with complex numbers, the complex plane is often used. This is very much like the graphs you did in high school math, where the x value was plotted on the horizontal axis and the y value on the vertical axis. In the complex plane, however, the real part of a complex number is plotted on the horizontal axis and the imaginary part on the vertical axis. Figure 1 shows the complex plane with a few points plotted on it.
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Figure 1 The complex plane.
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