Physically Interacting with Four Dimensions

Abstract

ALL RIGHTS RESERVED ivTo my wife, my daughter Jasmine and our loving parents. The research presented here creates physically realistic models for simple interactions with objects and materials in a virtual 4D world. We provide methods for the construction, multimodal exploration, and interactive manipulation of a wide variety of 4D objects. One basic achievement of this research is to exploit the free motion of a computer-based haptic probe to support a continuous motion that follows the local continuity of a 4D surface, allowing collision-free exploration in the 3D projection. In 3D, this interactive probe follows the full local continuity of the surface as though we were in fact physically touching the actual static 4D object.Our next contribution is to support dynamic 4D objects that can move, deform, and collide with other objects as well as with themselves. By combining graphics, haptics, and collision-sensing physical modeling, we can thus enhance our 4D visualization experience. Since we cannot actually vii viii place interaction devices in 4D, we develop fluid methods for interacting with a 4D object in its 3D shadow image using adapted reduced-dimension 3D tools for manipulating objects embedded in 4D. By physically modeling the correct properties of 4D surfaces, their bending forces, and their collisions in the 3D interactive or haptic controller interface, we can support full-featured physical exploration of 4D mathematical objects in a manner that is otherwise far beyond the real-world experience accessible to human beings.

Relaxing topological surfaces in four dimensions

“…The real difference is that now these masses are embedded in dimensions above three and the vector operations are performed in dimensions above three. For the most part, vector operations in high dimensions are simple extensions of their 3D counterparts [38]. For example, computing the addition of two four-vectors is a matter of forming a resultant vector whose components are the sum of the pairwise coordinates of the two operand vectors.…”

Computer interfaces transform mathematical studies

“…We can use the same methods of this touch-based multimodal paradigm to help us understand and manipulate the much more complicated case of shapes with self-intersecting surfaces, [1][2][3] which arise naturally when projecting objects from 4D down to 3D. If sound cues are added to describe the passage of the computer hand across a visual obstruction, one can explore a knot or a surface without necessarily having to use vision at all.…”