Visualization of a Four-Dimensional Clock

In this paper, geometry and graphics are used to visualize a novel idea for a 4-dimensional analogue clock. This clock is constructed as an expansion of an everyday 3-dimensional circular analogue clock. It has 3 hands rotating inside a 3-dimensional spherical dial. The apex of each hand describes a winding curve on the 2-dimensional spherical surface as a representation of the passage of time in 4-dimensional space. Time, therefore, is expanded to 2 dimensions, which allows for the mutual permutation of space and time. The same reasoning may be extended to any higher-dimensional analogue clock as well. Such space and time keeping devices, sitting as they do on the frontier between the real and the imaginary, are best created as virtual devices inside the computer and may prove useful for future scientific investigations, of which several examples are provided in the conclusion.