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### Re: Smallest slowest spaceship?

Posted: April 11th, 2011, 1:25 am
Mats wrote:
quintopia wrote:Substitute "a c/4 orthogonal spaceship" for every occurrence of LWSS and it should make more sense.
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.

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:
quintopia wrote:makes absolutely no sense. L_p is a set of normed spaces. There is no norm that satisfies your lexicographical ordering.
Why would you need a normed space to sort spaceship speeds?

*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.

### Re: Smallest slowest spaceship?

Posted: April 11th, 2011, 2:26 am
I seem to recall that this doesn't actually asymptotically approximate a perfect circle
In the very long term you must get c/2 common spaceship collisions in the diagonal corners seeding new small circles so the total pattern will eventually diverge from circular. As the circle radius grows at just under c/4, potentially colliding spaceships can only ever form within not much more than 15° of the diagonals with consequent new centres correspondingly close to the growing circle. More dramatically, it is also relatively easy to engineer patterns ("engines") at the growing edge which move at c/2 and thus appear to drag out a triangle bounded by two tangents to the circle. Aside from those kinds or localised irregularities, I don't see any reason to assume divergence from at worst a near regular n-gon with very high n. Having seen a lot of similar near circular growth phases in other rules, I have formed an impression that their circularity is like that of the cauliflower, a statistical averaging of bumps.

### Re: Smallest slowest spaceship?

Posted: April 12th, 2011, 8:41 am
137ben: I can't see why my sorting is more awkward or more arbitrary than a sorting based on Manhattan metric in a Moore neighbourhood.

All three metrics mentioned in this thread assign a definite speed to each spaceship. Your system assigns a speed, but then uses a "tie-breaker" method to arbitrarily say that two space ships moving at the same speed should be ranked in a specific order.