Two concepts are helpful at this juncture.
First, all the information needed to specify a sliver may be reduced to a single number. Since there is an infinite number of numbers between zero and one, we can place every number between them with the aid of the decimal point. Simply by placing the point after the first digit (putting into standard form) (by using decimals) it should be possible to reduce all of the information needed to reproduce a particular sliver into a single number.
Let's see how that work for the following example:
Take the above tic-tac-toe 'sliver'. If X is a '2', O is a '1', and an empty space represented by zero, then, reading from left to right, top to bottom, the information needed to define the sliver would be 102221100. A single number, therefore, is enough to specify the whole diagram. In
Now, if we place a decimal point in front of it, we may 'squeeze' this number into something manageable: into a number greater than zero and less than one - in this case 1.02221100 And we can do that for every possible sliver - noughts and crosses, chessboard, digital photograph and 3-D hologram.
That is the first of this post's concepts: the information needed to specify a sliver may be reduced to a single piece of information.
The second concept is that we may represent that number by a point on a Cartesian plane.
Let's start with any sliver. And then let's specify another by changing one of its pixels. We may think of them as 'neighboring' slivers. Placed on our our Cartesian plane, they would be neighbors.
Let's start with any sliver. And then let's specify another by changing one of its pixels. We may think of them as 'neighboring' slivers. Placed on our our Cartesian plane, they would be neighbors.
If you change any pixel in the universe, then the set of slivers that you'd create would all fit, as points, about the original point (because there is an infinite number of points adjoining it, more than enough to be able to plot every one-pixel-different sliver).
In turn, each of those slivers may be surrounded by slivers one-pixel-different-from-it. And so on ad infinitum.
That's how to visualize the Sliver Catalog on paper.
