A funny thing happened at my most recent gaming session. I forgot about the movement rules in D&D 3.5 - and when one of the other players tried to explain it, I just sat there unable to "get it." Sad. I knew it was a basic geometry principal at work, but I couldn't remember which it was.
First, let me remind you of the movement rules. When you are making diagonal movement the first diagonal step counts as 1. The second counts as 2. The third counts as 1 again. The pattern repeats.
But at first glance, the distance between the diagonal of the square shape compared to the vertical or horizontal size didn't seem different enough to count for anything, and my gut instinct was to count all movement as 1 point. Then, I pulled out a pencil and measured the straight distance across 5 grid squares. And then I measured the distance across 5 grid squares diagonally, and could see the diagonal distance was greater.
Here in this illustration, we see Howard the Kobold who wants to run up and shatter the big jade throne before the golem guards can activate. Howard moves 1 space forward, then 1 diagonal, then 1 straight across, then 1 diagonal. The movement costs are 1 + 1 + 1 + 2 because the first diagonal costs 1, and the second costs 2.
Of course the geometric principal that was floating around helplessly in my brain was the
Pythagorean theorem. This rule states that relationships of the sides of a triangle are
a 2 + b2 = c2. More to the point, it means that the relationship between the length of the sides of the square and the diagonal of the square is that diagonal length is going to be equal to the
square root of 2.
We can solve for
c using the formula
squareroot(a2 + b2 ) = c(I couldn't figure out how to make the square-root symbol cover the a & b... Sorry!)
But it always ends up being the square root of 2, which is an irrational number.
Still, if you use a calculator to solve the value, you get something like 1.4142136 - which is slightly less than
1.5.
Since there is no half-square movement in D&D, the rule of diagonal movement costing first 1, then 2, then 1 again is a rough mathematical equivalent to having it cost 1.5 movement to move diagonal. If you end on a 1.5 it costs 1. But if you go two diagonal spaces, it costs (1.5 + 1.5) a total of 3.
Oh Geometry, why did I forget so much of you? And why don't schools use gaming to teach math? Doesn't troop and unit movement seem more fun than just solving problems?
I still remember physics from Nuclear A-School in the Navy - we always dealt with the terrible fates of metaphorical cats. Though we never really shot a cat out of a cannon into a brick wall, we certainly knew how to calculate the theoretical velocity of the cat at the moment of impact. (assuming a vacuum through which to fire said feline, a device to measure the speed of the cat, and somebody to keep the animal rights people from stopping the important experiment!)
