Secant Animation 1 ("Wessel")
The second animation ever created based on the secant method.
This is a grand tour of a fractal with static parameters. The fractal formula is based on using the secant method to find the roots of the cosine function. Additional discussion of the secant method in general is in the technical section. Some examples of this technique applied to solving other equations can be found in the still images section.
|Fast download||17.6 MB 416x234 390 Kbps Fast Start|
|Medium Quality||85.1 MB 800x450 2 Mbps Fast Start|
|High Quality||163 MB 800x450 4 Mbps Fast Start|
|Fast download||12.7 MB 416x236 400Kbps|
The MP4 files have been encoded in a way that supports progressive download. That means clicking on an MP4 file should cause your browser to open QuickTime (if you have it installed) and begin playing right away (after some buffering period). If you are having problem with this, please read this page in the technical section for some possible reasons.
|Date Generated:||22-27 Mar 09|
|Resolution:||800 x 450 (16:9)|
|Video Length:||4:48 of fractal, 5:09 total|
|Rendering Time:||118 hours|
|Fractal Type:||Secant method with cosine|
Caspar Wessel (1745-1818) was a Danish-Norwegian mathematician who in 1799 was the first to recognize the representation of complex numbers as points or vectors in the two-dimensional plane. This geometrical interpretation of complex numbers is now deeply ingrained in our understanding of how to visualize these entities and how to do arithmetic with them. This same result was independently rediscovered by Gauss and Argand a few years later, and played a major role in the widespread acceptance of these numbers, which at the time were considered somewhat dubious.
This animation is a tour with static parameters, meaning the numerical values that define the shape of the fractal do not change as the viewpoint moves over the fractal and zooms into it. It was inspired by the video "Muhtoombah" from the Biocursion DVD.
Animations of this type that move around and zoom in and out rely heavily on a system for moving the video viewpoint and magnification smoothly. It is a particularly challenging mathematical problem to move and zoom simultaneously in a way that is pleasing to the eye, and it has taken several revisions of the math that controls this to get it right.
SecantAnimation1 is the first video to really showcase this new motion system (SA2 also used it for its parameter transitions but had no viewpoint motion). Eighteen keyframes were used to control the movement of the video viewpoint. Each keyframe specifies a location and magnification, and has associated with it parameters that are converted to velocity and acceleration data that control how the viewpoint moves. This has to be fine-tuned manually, and many painstaking revisions were made to get the final version just right.
This is the first fractal animation I've done in 16:9 format. I felt that the wider 16:9 format showcased some of the views of this fractal better than the more square 4:3 format. The 800x450 resolution was chosen to be a nice round number with approximately the same number of pixels as a 640x480 image has. This is not a high-definition format, just widescreen.
The coloring method is an improvement of the dynamic mapping technique that was used, for example, in Canyon1 and Canyon2; more details of the issues can be found on those pages. The improved version utilized here removes the low-pass filtering of the minimum and maximum count ranges and instead allows precision manual control of the envelope of the count span. By manually adjusting the exact shape of the envelope, I have been able to achieve results superior to what even very good digital filters were able to produce, although am still a little unsatisfied with the zoom-in at time index 4:10 (around frame 7500).
The mathematical operation that generates this fractal is the use of the secant method to solve cos(x)=0, with the second data point generated from the logarithm of the first data point.