I try to learn a little about finite geometry and I have now encountered the following exercise:
Exercise:
Construct the affine plane $\mathrm{AP}(\mathbb{Z}_3)$. Determine it's parallel classes and the corresponding Latin squares for the classes of finite nonzero slope.
I understand how you construct the lines
(i) slope of 0: $y=0$, $y=1$, $y=2$
(ii) Slope 1: $y=x$, $y=x+1$, $y=x+2$
(ii) Slope 2: $y=2x$, $y=2x+1$, $y=2x+2$
But then, when I look at the solution, they have drawn a pictures but I can't see how they relate to each other and why. The picture looks like this (now drawn in paint ;) )
Hope someone can explain this to me. I read another example from my text-book but i never understood it then. So i thought, perhaps if I do an exercise I will grasp it. But no luck with that either. I'm not really sure I even understand what a parallell class is. why have they drawn the picture like this? Why do the left-top get the number 3 and so? Would be happy if someone could answer this. Thanks
These three lines form the parallel class of lines having slope 2. I think they are shown as kind of weird in this drawing, but it is arbitrary what order they are in. For example, consider $y = 2x$, this line is satisfied by $(0,0)$, $(1,2)$, $(2,1)$. These three points make up the blue line in the picture.
None of these three lines intersect, and they partition the points of the plane; that is what makes them a parallel class.