RGB Color Cube

Consider the a path along the surface, i.e., faces of the RGB color cube. That is, on every point on the path at least one component is either 0 or 255.

Let a path be a function of position with respect to time. Define a path on the surface of a RGB color cube, i.e., saturated colors, whose brightness, defined as a certain linear combination of the color components, increases constantly as a function of time. Let the path start at the black corner, spiral around the surface of the cube, and end at the white corner.