PROCEDURAL GENERATION IN FOUR-DIMENSIONAL SPACE: FROM 4D GRADIENT NOISE TO VISUALIZATION OF QUATERNION FRACTALS - Novokhatko, I. O.; Fedotova, N. A. (Systems and Technologies, 2025)
The article offers a comprehensive review of procedural content generation (PCG) methods in four-dimensional (4D) space, which is a relevant direction in modern computer graphics. The theoretical and practical aspects of using the fourth dimension as a tool for creating complex animations and seamless textures are considered. The Euclidean 4D space and its mathematical foundations are formally defined. Unlike 3D, where rotation occurs around an axis, in 4D space it occurs around a plane, which allows for the unique phenomenon of double rotations4. The geometry of 4-polytopes, such as the tesseract, whose boundaries consist of 3D polyhedra, is also studied. Gradient noise algorithms are analyzed in detail, in particular, a comparison of classical Perlin noise with more productive and high-quality simplex noise is carried out. It is shown that the hypercubic Perlin noise mesh leads to undesirable visual artifacts and has a high computational complexity of O(n4), while the simplex noise (O(n2)) uses a more isotropic simplicial mesh. Methods for generating 4D fractal structures, such as the Mandelbrot and Julia sets, using quaternion algebra are investigated. A key property of quaternion algebra is the noncommutativity of multiplication, which allows the generation of complex and asymmetric 4D structures that are not possible in lower dimensions. The generation algorithm is based on an iteration formula, where the Mandelbrot quaternion set is defined as the set of constants C for which the iteration Z Z C n n 1 2 remains bounded. Since direct perception of 4D objects is impossible, the paper systematizes key visualization techniques: 3D slicing (slices), orthographic and perspective projections, and volumetric rendering using ray marching. Perspective projection creates the well-known "cube-in-cube" effect for the tesseract, giving a sense of depth along the fourth axis. Volumetric rendering using ray marching has been shown to be particularly effective for implicitly specified surfaces, as it uses sign distance functions (SDFs) for photorealistic visualization. The conclusions summarize the current state of the art and outline promising research directions related to hardware acceleration on GPUs and the use of generative models based on AI.