Board game geometry
Posted by Aapo Rantalainen on August 2, 2015
There are several board games played with square grid and TOKEN CAN MOVE DIAGONALLY. I’m not saying this is wrong or even hard to play with.
I’m describing what this mean mathematically.
This is true in this game-universe and with these game-rules. If we then use our physical reality and measure distances with ruler.
I repeat this is not a bug in rules. It is not even paradox. But how does it look in our reality if we somehow set that “nearby points have distance of one”?
First we (mathematically) define our game board. Let say we have three types of points:
– Corner: It has three neighbourhood points (E.g. point A)
– Side: Five neighbourhood points (E.g. point B)
– Center: Eight neighbourhood points (E.g. point D)
(Where neighbourhood point = point with distance 1)
And then we have some little more complicated relation between points:
Distance(A,B) = 1
Distance(A,C) = 1
Distance(A,D) = 1
Distance(B,C) = 1
Distance(B,D) = 1
Distance(C,D) = 1
…For each point in board.
Let’s construct this:
* Starting with one corner (A) (in 0,0,0)
* Helper points X Y Z
*Lines from A to each helper point X and Y and Z. *Ball with center A and radius 1. It intersects three helper lines on points B and C and D
(Note: it is not circle but ball, therefor it looks points B and C and D are not intersecting ball)
*This construction is now equivalent what we wanted.
And here is a physical construction of 20 points (marked by four tokens) from corner of the board.