When we consider the number of steps involved in moving from any n-dimensional object to its respective n+1 or n-1 dimensional representation, we can apply the rule of expanding or contracting each point or set of points (line segments) in a direction which is perpendicular to the corresponding point or set of points. For instance, a point can be made into a line by drawing a series of points, all of which are perpendicular to the original point. A line can be made into a square by moving the line perpendicular to itself. This principle holds for all n-dimensional objects. The beauty of this process is that we need not know what the resulting n+1 or n-1 dimensional representation will look like, we need only to follow the rules.
We are, of course, limited to three-dimensional projections of all objects with dimensionality greater than three, but we can't be faulted for living in a three dimensional universe. Now, suppose we consider that we had the starting n-dimensional object and the resulting n+1 or n-1 dimensional projection. We say projection because we don't actually have the n>3 representation, only its projection into the 3-D plane. Now that we have the start and end objects, suppose we wanted to view the arbitrary k steps which bring us from the start to the end object. Suppose that I also want the added convenience of specifying k to be any chosen number >= 0. Our previous technique now seems to be flawed, but can still be interpreted in a variety of different ways.
Suppose we say that we can move each n-dimensional constituent of the starting object, perpendicular to itself for k points until we reach or destination object. Assuming that out n+1 dimensional object is a square with a perimeter measurement of 16 points. This implies that our one dimensional object is a line segment of length 4 points. Now, we can say that k=4 to move from the line segment to the square, but we do not satisfy the added criteria that we have the freedom to vary k. In this example k is automatically pre-chosen. If k greater then 4 or k less than 4, we end up with an object which is not a square. Lets consider another approach.
The element which is missing, relies on our perception of time intervals. If we consider k in units of points and the time interval required to render that point, then we have the freedom to vary k to any arbitrary number, and thus satisfy our previous criteria. This will then allow us to reach our destination object of any specified proportions. There is a technique in computer programming which demonstrates this particular method very well. The technique is called morphing
Morphing begins with two images, one specified as the start and the other the end image. Now, the specific algorithms for morphing may vary from least-work to filtered, but the concept underlying each technique is the same. Perimital contours are generated for each object, and are then broken up into vertices. These vertices are then given a ratio correspondence between the objects. Each correspondence is then fitted with a line of interpolation. Now, if you consider a single point which moves from the start image to the end image, through this line of interpolation, you can envision the result. As the point moves from the source image, it is influenced less by the colors of the source image and more by that of the end image. This process is done for each corresponding vertex, and the result is an image which seems to morph into something else. Its a simple, but elegant algorithm.
The contour samples are taken in k intervals. This specifies the number of steps involved in the morph. If we specify the number of steps in k, then the samples are easily defined, because we simply scale the samples until we reach k samples. Its now easy to see how this algorithm could be applied to the interim pictures found when morphing between objects of different dimensionalites. This response was put together by Envall, Kathy, and Caroline.