Realisation possibilities for polygons

[LagrangeEuler]

http://books.google.rs/books?id=vrc...on possibilities for polygons Nolting&f=false

[tex]\rho(m,n)\leq 3^{2m+2n}[/tex]

And this argument stands because
''At any lattice site j there are three independent posibilities to put next wall unit (''backward'' is not allowed)''

I don't understand this. Can you give me some more detail explanation?

 

[eljose79]

I would be happy to provide more detail about the argument presented in the forum post. The statement \rho(m,n)\leq 3^{2m+2n} is referring to the maximum number of possible realizations for a polygon with m horizontal and n vertical wall units. This means that for any given polygon with m horizontal and n vertical wall units, there are at most 3^{2m+2n} different ways to arrange the wall units in a valid configuration.

The reason for this is explained in the following sentence, which states that at any lattice site (or point on the grid), there are three independent possibilities for placing the next wall unit. This means that for each lattice site, there are three different ways to continue building the polygon. However, it is important to note that the constraint of not allowing the wall unit to be placed ''backward'' further limits the number of possible arrangements.

To understand this better, let's consider a simple example. Imagine we have a polygon with 2 horizontal and 2 vertical wall units. The first wall unit can be placed in any of the three positions at the bottom of the grid. Once this is done, there are three possible positions for the next wall unit, but one of them will be ''backward'' and not allowed. This leaves us with two valid options. Similarly, for the third wall unit, there are two possible positions, and for the fourth wall unit, there is only one possible position.

This pattern continues for larger polygons, where the number of possible arrangements decreases as more wall units are added. This is why the maximum number of realizations can be expressed as \rho(m,n)\leq 3^{2m+2n}.

I hope this explanation helps clarify the argument presented in the forum post. If you have any further questions, please don't hesitate to ask.