Weekly Reflections 1

A General Documentation of our Conjectures:

1. Fixing a,b,c,d as the spans of lines (i.e., m#(a, _; b, _; c, _; d, _)) there are only two ways to fill in the blanks so as to have a coherent/realizable/valid configuration: m#(a, c; b, d; c, a; d, b ) and m#(a, d; b, a; c, b; d, c). The former we will call Pattern 1; the latter, Pattern 2.
2. These two patterns have different properties:
   
3. Pattern 1 configurations seem to always have two extra orbits of 4-fold intersections, (abbreviated here as 2-Pair-intersection, or 2PIs) where 2 of the lines are from one symmetry class of lines, and the other 2 are from another. Furthermore, these 2PIs are different from the ones constructed by the symbol, resulting in every pair existing: 4 symmetry classes of lines, 2 classes through each of these intersections; 4 choose 2 = 6 possible combinations.
   
4. Pattern 2 configurations do not always have two extra 2PIs. In some cases, there is only 1 extra; other cases, there are none. There are also 3-fold intersections, usually of three different symmetry classes. There is a possibility that the properties have connections to the parity of m; however, there are too many possibilities to make any conjectures at this point. Further analysis is needed.
5. You can always check algorithmically to see if a 2PI exists. There are only floor[m/2] distinct possibilities to check. Furthermore, if all the intersections of one color occur inside the innermost intersection of another color, you know certainly that a 2PI of those colors does not exist.
6. Pseudo Conjecture: Based on Branko Grunbaum's work, there are 6 (isomorphically) distinct configurations given a,b,c,d,m, 3 of Pattern 1 and 3 of Pattern 2 (they are polars of each other).
7. Conjecture: The 3 configurations of Pattern 1 given a,b,c,d,m all have the same line structure. What this means is that you could place the configurations right on top of each other and you would get a [6,4] configuration.
8. Conjecture: The 2PIs that are missing from each of the Pattern 1 configurations are always the same: the innermost (orbit 1) and outermost (orbit 6) symmetry classes, orbits 2 & 5, or orbits 3&4.