对偶原理(principle of duality)反映了布尔代数中逻辑恒等式之间存在一种普遍的对偶关系。它指出,如已有一个逻辑恒等式,则把该式中0和1,与运算和或运算相互替换,则新的关系式依然成立。例如,由0+x=x可得到1·x=x等等。我们已知布...对偶原理(principle of duality)反映了布尔代数中逻辑恒等式之间存在一种普遍的对偶关系。它指出,如已有一个逻辑恒等式,则把该式中0和1,与运算和或运算相互替换,则新的关系式依然成立。例如,由0+x=x可得到1·x=x等等。我们已知布尔代数仅为格代数中变量只取二值的简单情况,因此,在讨论变量取多值的格代数中对偶原理是否继续存在便应予考察与讨论,本文将以三值情况为例进行讨论。展开更多
In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analy...In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.展开更多
文摘对偶原理(principle of duality)反映了布尔代数中逻辑恒等式之间存在一种普遍的对偶关系。它指出,如已有一个逻辑恒等式,则把该式中0和1,与运算和或运算相互替换,则新的关系式依然成立。例如,由0+x=x可得到1·x=x等等。我们已知布尔代数仅为格代数中变量只取二值的简单情况,因此,在讨论变量取多值的格代数中对偶原理是否继续存在便应予考察与讨论,本文将以三值情况为例进行讨论。
基金Project supported by the National Basic Research Program (973) of China (Nos. 2004CB719402 and 2002CB312106), the National Natural Science Foundation of China (Nos. 60375020 and 50305033), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20020335112)
文摘In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.
