Fractal MMORTS

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While looking at fractal animations, such as zooming in on the Mandelbrot set or this video fly-through of a so-called "mandelbox," one thing I've thought would be fun is to "build a city there."

I think the most fun way this could be done would be a kind of MMO version of the Civilization series of games. The most obvious technical hurdle (to me) of an MMO Civilization game is an exhaustion of in-game physical space and resources. A fractal-structured terrain would solve that problem by making resource usage exponential. This means a new player uses a very small amount of resources and thus needs little space. As technology advances, the player's civilization begins to need exponentially more energy/food/Vespene gas/whatever, and it subsequently develops exponentially larger "structures" and grows its boundaries exponentially larger.

The technical hurdles of this structure are a bit more subtle. First, well-known fractals have a "zoom-in" structure. The entire fractal, like the Mandelbrot set that started it all, can be viewed in finite space. One then magnifies one's view on a small subset of the set's boundary to find increasing detail, and thus increasing "real estate." This kind of game requires a fractal with a "zoom-out" structure. Since players begin at a small, finite level and work exponentially and, theoretically, infinitely outward, one needs a fractal that "works in reverse." This might be solved by performing a simple mathematical operation on a traditional fractal, such as z |-> 1/z for the Mandelbrot set.

The second technical hurdle is that we live in a 3D world, and thus build on 2D surface-boundaries of 3D objects (like mountains, or the Earth itself). The most well known fractals are 1D line-boundaries of sets in a 2D space. After a brief search, however, it may be possible to find sets in 3-space that satisfy the requirements (such as this attempt to find a "3D version of the Mandelbrot set," which they call the "Mandelbulb"). As pointed out in the previous paragraph, this would still require some mathematical operation to produce the desired "zoom-out" structure. And actually such an operation may be harder to find (and may not even exist!) in 3 dimensions.

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