Complex Numbers Continued
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Lesson Summary:
In this lesson on Complex Numbers Continued, the instructor discusses complex transformations and fractal geometry. The video demonstrates various transformations such as homo-thetti, multiplication by i, and multiplication by 1+i. The instructor also explains the concept of Julia sets and Mandelbrot sets, which are the filled-in sets of certain transformations. The lesson concludes with a zoomed-in view of the beautiful Mandelbrot set and how it represents the essence of chaos in modern science.
Lesson Description:
More discussion about complex numbers and the complex plane, including complex transformations and fractal geometry.
Questions answered by this video:
- What are complex numbers?
- How can you transform images in the complex plane?
- What does a transformation look like when you multiply by 1 + i?
- How do you transform an image using the transformation z -> z^2, z -> -1/z, z -> z + k/z, or z -> e^z?
- What happens when you perform the transformation z -> z^2 repeatedly for points inside the unit circle?
- What is the Julia set?
- How are fractals created using complex numbers?
- What is the rabbit in complex fractals?
- What is the Mandelbrot set and how does it relate to the Julia set?
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Staff Review
This lesson picks up from where the previous video left off. Transformations of figures are discussed by multiplying coordinates of images by complex numbers. The resulting geometry is very interesting. Fractals in complex geometry are also shown and explained in great detail, including Julia and Mandelbrot sets.
