Open thread, August 14 - August 20, 2017

1) If only positions should match, then the answer is yes. Just roll two identical balls at each other.

2) If both positions and velocities should match, then the answer is no. Here's a sketch of a proof:

Assume without loss of generality that some balls have nonzero velocity along the X axis. Let's define the following function of time: take all balls having nonzero X velocity, and take the lowest X coordinate of their centers. A case analysis shows that the function can change from increasing to decreasing, but never the other way. Therefore it cannot be periodic. But since it's determined from the configuration, that means the configuration can't be periodic, QED.

The full answer is: they cannot return if there are only finitely many balls, but they can if there are infinitely many.

Let's first assume that there are finitely many balls. As Thomas pointed out, we can assume that the center of mass is fixed. Let's consider R defined to be the distance from the center of mass to the furthest ball and call that furthest ball B (which ball that is might change over time). R might be decreasing at the start - we might start with B going towards the center of mass. But if R decreased forever then we would know that they never return to their starting location (since R would be different)! So at some point it must become at least as large as it was at the start. At that point either the derivative of R is 0 or it is positive. In either case, R must increase forever onwards - which again shows it can't return to its original starting point. Why is it always increasing from that point onwards? Well, the only way for the ball B to turn around and start heading back towards the center is if there is another ball further away than it to collide with it. But that can't be, since B is the furthest out ball! (Edit: I see now that this is essentially equivalent to cousin_it's argument.)

For infinitely many balls, you can construct a situation where they return to their original position! We're going to put a bunch of balls on a line (you don't even need the whole plane). In the interval [0,1], there'll be two balls with initial velocity heading in towards each other at unit speed, with one ball at the left edge of the interval and one ball at the right. Then do the same thing for each interval [k,k+1]. When you let them go, each pair in each interval will collide and then be heading outwards with unit interval. Then they'll collide at the boundary with the next interval with the ball from the next interval. That sets them back at the starting position. I.e. all balls collide first with their neighbor on one side, then their neighbor on the other side, setting them back to their starting position.

Possibly not a rational answer (so possilbly not living up to the less wrong philosophy!) but given the assumption of an infinite plane I would think the probability is vanishingly small of returning to the original position and velocity.

Something would need to constrain the vectors taken to prevent any ball from taking off in some direction that could be described as "away from the group". Perhaps that could be understood be be on a path for which the the path of no other ball can possible intersect. At that point this ball can will never change it's current velosity and never return to it's oiginal position.

I cannot offer a proof that such a condition must eventually occur in your experimnt but my intuition is that it will. If so that vanishing small probablity that everyting return to some orginal state goes to zero.

Two balls which are orbiting around its common mass center, they do return to a previous position. If there is no gravity, a finite bunch of rolling balls will never again return to the present state. Never again.

I know several reasonable algorithms for stable sorting in O(n log n) time and O(sqrt(n)) extra space, like Mrrl's SqrtSort. That's good enough for all practical purposes, because anyone who wants to sort a billion elements can afford an extra array of 30000 elements. And all known algorithms using less extra space essentially emulate O(sqrt(n)) bits of storage by doing swaps inside the array, which is clearly a hack.

Radix sort has its own rabbit hole. If you're sorting strings that are likely to have common prefixes, comparison sorting isn't the best way, because it needs to look at the initial characters over and over. There are many faster algorithms for string sorting based on ideas from radix sort: American flag sort, three-way radix quicksort, etc. The Haskell package Data.Discrimination generalizes the idea from strings to arbitrary algebraic datatypes, allowing you to sort them in almost linear time (in terms of total size, not number of elements).

The situation changes if you regularly travel, though.

Yeah... that might be interesting for Tinder. "Users who fucked this one also fucked this, this, and that one" X-D

(It is, of course, unimportant, but I think I know what I was thinking about - continued fractions, and what do people even do when the fraction has functions, not numbers, for terms.)

Thank you. We definitely had to learn about attractors in college, this is familiar...

"according to basically any model of the world, fooling yourself does not result in more paperclips."

Paul Almond at one time proposed that every interpretation of a real thing is a real thing. According to that theory, fooling yourself that there are more paperclips does result in more paperclips (although not fooling yourself also has that result.)

But what does the code for that look like. It looks like maximize(# of paperclips in world), but how does it determine (# of paperclips in world)? You just said it has a model. But how can it distinguish between real input that leads to the perception of paperclips and fake input that leads to the perception of paperclips?

As I said though, how do you program it to want REAL paperclips as opposed to just perceiving that it is getting paperclips.

I have parts of my brain that are like that, and I suspect that most people do. But if "meditation only causes you to pay attention to those thoughts" then you are probably doing it wrong.