This page contains links to some mathematics that probably won't get you through your exams, (though if you have a project to do it might give you some ideas.) I take no responsibility if you spend too much time on some of these links!
Beginners (especially kids) may start out with simple cosmetic modifications to existing games, perhaps using the sprite editor to replace Pac-Man with a favorite Pokemon or a happy face. However, it is a small step from there to creating new objects, using a very intuitive graphic user interface that allows you to assign actions to events such as creation, collision, mouse input, etc. Animation technicalities are hidden in the language. At this level, a beginner could create a simple game with a ball bouncing off walls within an hour or so , or even less time using preexisting sprites. Even at this level, the package has an object-oriented, event-driven flavor.
The package also contains good basic resource editors. Resources are generally standard file types, so that the designer can work with other graphics editors, sound file editors, etc. when special functionality is needed. For instance, Windows Sound Recorder and a microphone can be used to create custom sound effects; or a more advanced program such as Audacity can be used instead. POV-Ray could be used for photorealistic sprites.
While the drag 'n' drop interface is actually powerful enough to create challenging and professional-looking arcade games, there will come a time when the user wants to do something not provided for. At this point GML, the Game Maker Language, comes into its own. This Java-like language allows practically anything to be done - and is enough like Java or C++ to give the GameMaker graduate a head start on more mainstream programming. It has many built-in capabilities, including file IO and linking to DLL's. (Nonprogrammers, please ignore the last half sentence.)
For those who want to use this to teach programming, especially at the grade school level, there is a collection of information for teachers and site licenses are available. The basic package can be downloaded for free. Alternatively, you can register the full version inexpensively. There is also a well-reviewed users's manual available separately.
This package also has promise as a practical way for teachers (or others) to create educational software. With so much of the technical stuff hidden away - but little loss of flexibility - one should be able to concentrate on the game logic, and cut the labor involved enormously. I think one might need GML to do this really well, though; there is a limit to what one can expect of the drag 'n' drop interface.
A fractal object is one that has "fine structure" at all size scales (or, at least, over a very wide range of scales). Zooming in doesn't make it simpler. Approximate examples from nature include trees, circulatory systems, snowflakes, and shorelines. Stock market charts are also approximately fractal, as well as being chaotic. However, a chaotic system may be smooth at small scales. Conversely, a system that changes in time in a fractal fashion (that is, with a fractal graph) need not be chaotic.
Self-similar objects are ones that look the same at different scales. The term can be used to mean that there exists an exact similarity, or more loosely to mean that there is a strong resemblance. For instance, a branch of a tree is not a perfect copy of the whole tree, but a small branch can be an effective model tree in a diorama.
Self-similar objects are usually fractal; if they have any defining features, they must have smaller and smaller copies of those features at different scales. However, an object like a straight line can be self-similar because of being "plain at every scale" too. It is more common for a fractal object not to be self-similar; the Mandelbrot set is a good example. Zooming in, it continues to have fine structure - in fact, it becomes more and more complicated. Systems that exhibit approximate self-similarity may be chaotic (stock markets); systems involving exact self-similarity are generally non-chaotic.
I wanted to include a link to an insanely great freeware fractal generator here, but so far I haven't found one that's obviously the right choice.
Unfortunately, the DOS version doesn't work well under Windows, because Windows desn't like software playing around directly with the video card. There have been various Windows ports - possibly the best is Paul de Leeuw's MANPWIN which does most of what the latest versions of FRACTINT do.
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