a journey into the fourth dimension
This paper shows that by a simple (one way) mapping from quaternions to complex numbers, the problem of generating a 4D 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 4D 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 also of fast raytracing such a fractal using the table so generated are discussed. Further speed up in raytracing can be achieved by estimates of a distance of a point from the Mandelbrot set. Animation is a key factor in visualizing 4D objects. Three types of animation are attempted: (a) translation in 4D, (b) rotation in 4D and (c) fly-through in 3D.
paper map complexity iter functional values order fractal distance point animation factor visualization object types translation rotate 3d