Mats wrote:I'm sorry but I'm still having problems understanding your argument. After substitution you get: "the glider's travel is always less than or equal to the c/4 ortogonal spaceship's travel time." In L_1 the glider is faster than a c/4 ortogonal spaceship.quintopia wrote:Substitute "a c/4 orthogonal spaceship" for every occurrence of LWSS and it should make more sense.
Yes. Whereas in L_inf, they are the same speed. This is exactly the argument I was trying to make. That we want a metric that considers the glider to be faster than the c/4 orthogonal space ship since the glider never takes any longer than the c/4 orthogonal space ship to reach any particular location, and frequently gets there much faster.
Mats wrote:Why would you need a normed space to sort spaceship speeds?quintopia wrote:makes absolutely no sense. L_p is a set of normed spaces. There is no norm that satisfies your lexicographical ordering.
*facepalm* You're the one that claimed your metric was based on a norm of some sort by calling it "L_'not quite infinity but very large'"
However, I will give you this: there is a norm that gives lexicographical ordering if you're willing to allow it to be a surreal-valued function.
In any case, which property of a norm is it that you think is not necessary to give a sensible measurement of distances travelled?
HartmutHolzwart wrote:it's the fastest velocity in empty space. Btw. a glider could not change its direction in empty space and would never get to (0,2x).
Yes of course we are only concerned with the behavior of spaceships in empty space, but I'm not sure why it matters that a glider could not travel to (0,x) in a vacuum: when discussing speeds of patterns, we can, without loss of generality, pause the simulation, rotate the pattern by 90 degrees, and restart the simulation. This takes zero generations, and, in a sensible metric, would not affect the total distance traveled, so the final speed calculation will still be reliable. In all the L_p-normed spaces, this property holds.