I’ve discovered a technique, using macros in Excel (which you do NOT have to enable in order to see it), to create what I’m now calling a “factor field” in mere minutes. Since I don’t know of anyone else who has done graphs such as these (although I’m sure there probably is someone out there who has), I’ll just make up my own names for the stuff right now and then, if I find “official” names for them later on, I can switch to those. You can view the entirety of the Excel sheet by downloading it here (note: it is about 6.5 megs in size, and again if you are concerned about viruses you do not need to enable the macros to see the graphs).
As you can see in the picture I uploaded in the previous post, the factor field is an extremely interesting place filled with lots of symmetry. But to begin exploring it, let us look at the beginning of the factor field, which is represented in this picture:
This picture shows line 1 to about line 510. The width is restricted to only 256, due to Excel’s limitations. I’ve given the slopes of the lines there for ease of reference. The slopes should be read in the following manner: 1/1 means the slope moves 1 right and 1 down, repeated. 1/2 means the slope moves 1 right and 2 down. 1/3 = 1 right and 3 down, etc.
Some interesting tidbits:
* The 1/1 slope is at a 45 degree angle to the straight line forming the left-hand side of the graph.
* If you draw a base-line from any value on the left side of the graph to connect it to the 1/1 slope, the resultant figure is a right triangle with equal length arms.
* Furthermore, the 1/2 slope will hit the line (if it’s at an even number) at the midpoint of the line. The 1/3 slope will hit at a point 1/3 of the distance from the left, etc.
More could be said, but first let’s look at some interesting landmarks in the factor field. First is what I call the “Starburst” image. (The previous post is a picture of one.) This is formed by the spikes that occur on the 6th interval, when a line with a 1/1 slope departs both upward and downward from the feature. Thus, we have the first Starburst at 6:
Another feature of the Starburst is that the arms that branch out mimic the straight line in terms of which point shows up. Thus, if there is a gap in the straight line, there will be a gap in the vertical line at the same point. The only exception (which isn’t much of one) is that due to the 1/1 sloping line, the line that goes upward is not as long as the line that goes downward. While I need to do more testing to find out if this is a universal thing, it appears to only be the last point that does not lie on the upper line, but this would correspond to the value of 0 anyway.
A second feature that occurs is the parabola, some of which are shown here:
Thus, the factor field seems to be far more complex than first imagined! As usual, more will follow at a later date :-)
