Implementing 4D Visualization
Implementing 4D Visualization
Marc Vicuna
Carleton University, Ottawa, Canada
In this project, I implemented Mathematica visualisations of 4-dimensional data (4D polytopes, 4D points clouds and 4D cellular automata using various 4D visualisation techniques, extending existing Mathematica graphics functions. I focused on intuitiveness rather than fidelity to ease understanding of 4D geometry and data analysis for formal research. The goal was to allow users to understand fourth (spacial) dimensional data and geometry. I implemented 4D points clouds, many 4D regular and irregular polytopes through general 4D geometry graphing and experimented with visualisation of 4D cellular automata. Intersection, projection, simple reduction and composite approaches to 4D visualization were attempted, but only projection and simple reduction were considered both intuitive and informative. Most if not all current 4D visualization techniques are adopted for space 4D data, thus, 4D cellular automata being dense require new methods. Further optimization of code and a more expensive array of primitives and options will be needed to make 4D visualization better understood.
Key ideas before reading
Key ideas before reading
A polytope is a figure generalizing the notion of a polygon in plane or a polyhedron in solid geometry to spaces of any number of dimensions.
A projection is the distortion of scale to lower dimensionality. We can intuitively think about it as the shadow cast by an object. The shadow of a line is a point, the shadow of a polygon is a line, the shadow of a polyhedron is a polygon, and the shadow of a 4-polytope is a polyhedron. As the light source (observer) gets closer from n-dimensional objects, the greater the difference of shadow size between objects. Thus, projection graphs far objects smaller then closer objects.
A projection is the distortion of scale to lower dimensionality. We can intuitively think about it as the shadow cast by an object. The shadow of a line is a point, the shadow of a polygon is a line, the shadow of a polyhedron is a polygon, and the shadow of a 4-polytope is a polyhedron. As the light source (observer) gets closer from n-dimensional objects, the greater the difference of shadow size between objects. Thus, projection graphs far objects smaller then closer objects.
Graphics4D
Graphics4D
ListPointPlot4D
ListPointPlot4D
4D Cellular Automata
4D Cellular Automata
Concluding remarks
Concluding remarks
Keywords
Keywords
Acknowledgment
Acknowledgment
References
References