Abstract
Abstract The Hopf fibration mapping circles on a 3-sphere to points on a 2-sphere is well known to topologists. While the 2-sphere is embedded in 3-space, four-dimensional Euclidean space is needed to visualize the 3-sphere. Visualizing objects in 4-space using computer graphics based on their analytical representations has become popular in recent decades. For purely synthetic constructions, we apply the recently introduced method of visualization of 4-space by its double orthogonal projection onto two mutually perpendicular 3-spaces to investigate the Hopf fibration as a four-dimensional relation without analogy in lower dimensions. In this paper, the method of double orthogonal projection is used for a direct synthetic construction of the fibers of a 3-sphere from the corresponding points on a 2-sphere. The fibers of great circles on the 2-sphere create nested tori visualized in a stereographic projection onto the modeling 3-space. The step-by-step construction is supplemented by dynamic three-dimensional models showing simultaneously the 3-sphere, 2-sphere, and stereographic images of the fibers and mutual interrelations. Each step of the synthetic construction is supported by its analytical representation to highlight connections between the two interpretations.
Highlights
Mathematical visualization is an important instrument for understanding mathematical concepts
Our constructions or models being technically impossible to create by hand without computer graphics, we use interactive 3D geometry software in which the images are placed in a virtual three-dimensional modeling space, and the observer is able to reach any point of this space and rotate views by manipulating the viewpoint in the graphical interface
Two groups might benefit from our paper: computer scientists interested in computer graphics, for whom we present a method of constructive visualization applied to the example of the Hopf fibration; and mathematicians, or students of mathematics, for whom we give a synthetic construction of the Hopf fibration and interactive graphical models to help better understand and intuitively explore it
Summary
Mathematical visualization is an important instrument for understanding mathematical concepts. Using the modeling tools of computer graphics, we are able to construct image representations of four-dimensional objects to enhance their broader understanding. Our constructions or models being technically impossible to create by hand without computer graphics, we use interactive (or dynamic) 3D geometry software in which the images are placed in a virtual three-dimensional modeling space, and the observer is able to reach any point of this space and rotate views by manipulating the viewpoint in the graphical interface. Two (not necessarily disjoint) groups might benefit from our paper: computer scientists interested in computer graphics, for whom we present a method of constructive visualization applied to the example of the Hopf fibration; and mathematicians, or students of mathematics, for whom we give a synthetic construction of the Hopf fibration and interactive graphical models to help better understand and intuitively explore it
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