Points of concurrency and sets of parallel lines

[cstellar]

My question is mainly concerned with discovering the allowable set of "configurations" of the given problem:

We have a two-dimensional board composed of three sets (of infinite size) of parallel lines \P_1, \P_2, \P_3, where the lines in \P_2 form a 60 degree angle with lines in \P_3 and \P_1, and lines in \P_1 and \P_3 are 120 degrees apart. Furthermore, any point of intersection of any two sets \P_i, \P_j has a unique line from \P_k intersecting it. Thus, our set of points of intersection are each composed exactly of three lines: one from each set.

For a visualization of the board (choose any empty hexagon cell and the three lines on which it lies are each from the different sets I described. Each empty hexagon cell is a point of intersection):
http://www.google.co.uk/imgres?imgu...GrfT6-qM8Sa1AWKlsz5Cg&ved=0CHAQ9QEwAQ&dur=342

My question is the following: which are the impossible configurations of this board, in terms of specifying a set of intersections? To illustrate the question, consider the following example:

We will refer to a point of intersection as (i,j,k) where i is a line in P_1, j is a line in P_2 and k in P_3. Note that (i,j,k) is unique for every point.

I give the following set of points:

(i, i, i), (j, j, j), (k, k, k), (i, j, k), (j, k, i), (k, i, j), (i, k, j). This works out fine. However, when I add the following point (k, j, i) as a constraint, it appears impossible to interpret it on that board. This appears to be the case no matter what i, j, k are.

Do you have any idea what are the pathological cases (such as in the example), is there any theorem, paper that discusses the problem I have described?

Thank you.

 

[jvicens]

Thank you for your question regarding the possible configurations of the given problem. I would approach this question by first understanding the basic principles of the problem and then using mathematical methods to explore the various configurations.

From the given description, it seems that the board is composed of three sets of parallel lines, with specific angles between them, and each point of intersection must have a unique line from each set passing through it. This means that the board is essentially a tessellation of hexagons, with each hexagon representing a point of intersection.

To answer your question about impossible configurations, we can use mathematical methods such as geometry and graph theory to explore the possible combinations of lines and their intersections. One approach would be to create a graph representation of the board, with each point of intersection as a vertex and each line as an edge connecting the vertices. This would allow us to analyze the connectivity and possible paths between different points.

In your example, you have added the point (k, j, i) as a constraint and found it to be impossible to interpret on the board. This could be due to a violation of the unique line rule, where the point (k, j, i) may already have a line from set \P_k and \P_j intersecting it, making it impossible for a line from set \P_i to also pass through it. This is just one possible explanation and further analysis would be needed to confirm it.

In terms of existing literature on this problem, I suggest looking into graph theory and tessellations to find relevant research on similar problems. Additionally, you may also find relevant information in the field of geometry and its applications.

I hope this helps in your exploration of the possible configurations of the given problem. If you have any further questions, please do not hesitate to ask.