A journey into the fourth dimension [visualization]

It is shown that by a simple (one-way) mapping from quaternions to complex numbers, the problem of generating a four-dimensional Mandelbrot set by iteration of a quadratic function in quaternions can be reduced to iteration of the same function in the complex domain, and thus, the function values in 4-D can be obtained by a simple table lookup. The computations are cut down by an order. Simple ways of displaying the fractal without shading and ways of fast ray tracing such a fractal using the table so generated are discussed. Further speedup in ray tracing can be achieved by estimates of a distance of a point from the Mandelbrot set. Animation is a key factor in visualizing 4-D objects. Three types of animation are attempted: translation in 4-D, rotation in 4-D, and fly-through in 3-D