Let's discuss Mathematics

[Mise]

Well personally I ignored movement speed because I couldn't be bothered to work it out.

 

[civac]

Well personally I ignored movement speed because I couldn't be bothered to work it out.

Alright, but it's the key to the problem. Movement cost will overwhelm the building costs of settlers very quickly. In fact as LulThyme pointed out you can simplify the problem by ignoring building costs.

EDIT: I've just reread your post, it's not clear, but you seem to be implying that you can have exponential growth in the number of cities (not just population) in Civ 2. I am really puzzled and really can't see how to get around the movement problem.

I will be very interested to hear your answer.

Yes, it's possible to attain exponential growth in number of cities. It seems I wasn't entirely clear when stating the problem. Unlike Civ4, in Civ2 or Smac you can get around the quadratic growth barrier. You may use any unit or building or technology existing in those games. Its unfortunate that you are not familiar with these games. I'd assumed that most people here know Civ2 at least. I will wait a while longer before giving the solution.

 

[Mise]

Yeah, I see that now very interesting problem.

 

[sanabas]

You seem to be mixing up very different things. The fact that the ratio of consecutive Fibonnaci number converges is well-known (and yes, this implies that the sequence grows exponentially). What this has to do with our problem at hand, I don't know. I think too many of you were focused on the production of settlers (comparing to the rabbit problem), while the problem is essentially one of movement speed.

You can place your cities in a spiral if you want, that doesn't change the fact that, after n turns from the start, everything you own will be contained in a 2n+1x2n+1 square centered at your starting point, simply because of movement speed (again, just think of starting with an infinite number of settlers). In particular, the number of cities is bounded by a quadratic function.

I think an experiment might convince you:
Try to really achieve your Fibonnaci sequence for the number of cities.
That is, forget about city growth, just try to get an exponential number of cities.
I'll even allow you to build cities right next to each other, for simplicity.
And cities build settlers in 1 turn. In fact, I'll be even nicer, cities can create as many settlers as you want in a single turn (but the turn after the city is founded)!
BUT settlers can only move one square per turn (let's say they can move+found in one turn).
Now, try to match the Fibonnaci sequence with the number of your cities. That is, try to have F_n cities after n turns.
Even with all my generosity, you will still run into problems at some point.



Well, the gritty details depend on how exactly you set up the cities, but there are handwavy arguments that should be convincing.

Most "natural" shapes your empire could or would take have an area that is proportional to the square of the "boundary" and hence also proportional to the square of the area that is within a fixed distance of the boundary. This implies that the ratio between this latter area and the whole thing goes to 0. If the city densities are more or less bounded in both regions, that will lead to the city ratio to also go to 0.

To take an explicit example : suppose we start with an infinite supply of settlers at the origin, and try to build them so they are at least 3 squares apart (2 squares in between them) as fast as possible. After n turns, we will have very close to 4n^2/9 cities (4 for the 4 quadrants, 1/9 for the density). For n much larger than x, the number of those which will be within x of a border city will be something like 8nx/9 (the empire is basically a big square with side 2n, so perimeter 8n, you could make more precise but this is the highest order term).

For fixed x, 8nx/9 over 4n^2/9 goes to 0.

I think the bit I was missing/problem with what I was thinking is that there is a fixed minimum distance between cities. And so while looking at city production akin to rabbit production is good in theory, it reaches a point that the cities need to be closer together than that minimum distance for it to work, which means it breaks down.

And yeah, I'm now happy that the ratio of exterior cities to total cities will approach zero, for the same reason.

Yes, it's possible to attain exponential growth in number of cities. It seems I wasn't entirely clear when stating the problem. Unlike Civ4, in Civ2 or Smac you can get around the quadratic growth barrier. You may use any unit or building or technology existing in those games. Its unfortunate that you are not familiar with these games. I'd assumed that most people here know Civ2 at least. I will wait a while longer before giving the solution.

I know both games (sort of), it's just that it's probably been 10 years since I played either of them. I assume there's infinite movement thanks to either air drops or railway.

 

[LulThyme]

I know both games (sort of), it's just that it's probably been 10 years since I played either of them. I assume there's infinite movement thanks to either air drops or railway.

Same here. I've played thousand of hours of Civ 2 but it's been a long time.

I thought of railroads already but then convinced myself it wouldn't work, because you still need go get to the new square. But now that I think about it, as the number of engineers grows, you should be able to get the railroad to expand arbitrarily fast each turn (engineers had two moves right? and they could build same turn as move?). I'll try to think if the details work out.

Airdrops I had completely forgotten about.. I seem to recall there being a maximum number of times per turn each airport could be used. That might still not be a problem though the fact that you need to wait one turn for the airport might.

 

[LulThyme]

Yes, it seems airdrops are not enough, even if you forget about the unit limit per turn (which I'm not sure exists). Just the delay to build them means that you can't expand the outer border of your empire faster than by foot.

On the other hand, railroads work!
You can produce a number of engineers proportional to your number of cities in constant time and hence you can produce railroad at speed proportional to the number of cities and thus expand at an exponential rate.

Here is an explicit solution :
We will build all our cities in a straight line, with say 3 spaces in between.
Starting with one city, at each step :
-each city builds enough engineers to railroad 3 squares in one turn (constant time)
-each city builds one settler (constant time)
-by induction, all cities existing cities are already connected, we use the engineers and settlers to double the number of cities in one turn (constant time).
Voila! Double the number of city in constant time.

Great problem civac!

It's interesting that it depends not only on unlimited railroad movement, but also on engineers having 2 movement!

EDIT : it seems this requires "nice" terrain (not too many mountains and such). Is this the solution you had in mind?

 

[civac]

Indeed, your solution is flawless as usual.

In Smac, you can build magtubes which work just like railroads. The other possibility is to build colony pods with air drop capability. Once a certain late game technology has been researched air drops can be made into any square of the map no matter how far away. Obviously, this allows exponential growth. However, this is a rather boring solution, the railroad/magtube one is far more interesting.

 

[Souron]

So in Civ 2, engineers are ungraded settlers. You can never build both. . .

 

[Atticus]

Wednesday is called the Little Friday here, so it's time for a little bump:

Can an irrational power of a rational number be rational? I just ate, so I'm too lazy to think, and some student is probably just happy to give it a go.

 

[Mise]

Irrational powers of 1 or 0 are rational

 

[Atticus]

You're right!

Are they the only ones?

 

[suiraclaw]

a^(ln(b)/ln(a))=b

So yes, an irrational power (ln(b)/ln(a)) of an rational number (a) can be rational (b).

 

[Atticus]

Ok.

The motivation behind this question were symbolic calculation programs like Mathematica. I was wondering, how they handle with numbers, and thought they could think numbers as "multidimensional": having "rational dimension" and "irrational dimensions".

I mean, suppose you sum numbers like 4+2sqrt(5) and 3 +4\pi. The machine would sum rationals separately and irrationals separately, so that the number would be (7,2,4), where first digit is "rational dimension", the second one "sqrt(5) dimension" and the third one "\pi dimension". Otherwise the machine would have to operate with rounded numbers, and I suppose that won't do in symbolic maths program.

This was just something I though upon when walking to school at morning.

 

[civ-addicted]

Does anybody know how programs like Matlab, Maple etc. calculate Convex Programming? For my thesis, i need to code a little something-something that does that, otherwise my Professor will be pretty disappointed of my approach

 

[peter grimes]

How can we describe the motion of a point on the rim of a bike tire as the wheel rotates?

I was thinking about this riding home yesterday after I saw a guy with LED blinkers in his spokes.

I *think* I know what the plot would look like, but I don't know how to express it mathly.

I think it would be a series of upside-down U's, slanted in the direction of motion of the bike.

I arrived at this by thinking what it would look like at night with just a single LED on. How close am I?

 

[IdiotsOpposite]

How can we describe the motion of a point on the rim of a bike tire as the wheel rotates?

I was thinking about this riding home yesterday after I saw a guy with LED blinkers in his spokes.

I *think* I know what the plot would look like, but I don't know how to express it mathly.

I think it would be a series of upside-down U's, slanted in the direction of motion of the bike.

I arrived at this by thinking what it would look like at night with just a single LED on. How close am I?

You are close. The equation itself is called a Cycloid, and does indeed look like a series of upside-down U's. Here's a nice gif that shows how it works:

Interestingly enough, if you took that and flipped it upside down, then hung a pendulum from one of the cusps, the motion of that pendulum would ALSO form a cycloid.

 

[peter grimes]

Ahh!

In my head i thought that the dot would have to move backwards relative to the ground just after passing through level with the center of the hub.

I also didn't see that at the top of the rotation the arc would flatten out.

Thanks!

 

[PlutonianEmpire]

Ok, here's an apparently complex system for you guys.

I have a planet and a moon, similar to Earth and its moon, but with differences.

The planet's mass is 0.605434 Earth masses, with a radius of 5,469.6462 km. The planet's moon is 0.027646 Earth masses, with a radius of 2,258.2968 km (for comparison, Luna is about 0.012267 Earth masses, AFAIK). The planet is reasonably Earthlike and habitable and all that jazz.

Considering the planet is less massive than Earth, and the planet's moon is about 2.25 times greater than Luna, the tidal effects are gonna be HUGE, as far as I can tell.

The planet's Moon forms about 200 million years after the planet's formation, in a collision similar to the one that produced ours. After the collision, the planet's rotation period is 7 hours. The planet's moon orbits about 88,000 km from the center of the planet, with an orbital period of about 90 hours. This is the starting point. (Or would the moon likely form further than 88,000 km?)

After 1.7 billion years, what is the planet's new rotation speed, and the moon's new orbital distance, considering the increased tidal effects in this system? And exactly how much different is this system's tidal effects from the Earth/Luna tidal effects? And how do I reach to your conclusions?

 

[peter grimes]

You are close. The equation itself is called a Cycloid, and does indeed look like a series of upside-down U's. Here's a nice gif that shows how it works:

Ok, so thinking about this a bit more I think it's close to the motion by described by a person standing on the equator or earth as it orbits the sun, but only in te special case of the axis being 90° to the orbital plane.

However, earth does not rotate on its axis the same way a bike tire meets the earth - it rotates faster.... so perhaps a cycloid more closely resembles the path described by a person standing on the equator of the moon?

..because the moon rotates on its axis in synch with its orbit around the earth?
Do I have that right?

 

[nc-1701]

Ok.

The motivation behind this question were symbolic calculation programs like Mathematica. I was wondering, how they handle with numbers, and thought they could think numbers as "multidimensional": having "rational dimension" and "irrational dimensions".

I mean, suppose you sum numbers like 4+2sqrt(5) and 3 +4\pi. The machine would sum rationals separately and irrationals separately, so that the number would be (7,2,4), where first digit is "rational dimension", the second one "sqrt(5) dimension" and the third one "\pi dimension". Otherwise the machine would have to operate with rounded numbers, and I suppose that won't do in symbolic maths program.

This was just something I though upon when walking to school at morning.

Maple keeps irrational numbers, and even rational ones to a lesser extent, in purely symbolic form. So pi just means pi unless you use a command to convert it to a floating point number. When a number is in symbolic form it is purely symbolic and has no meaning as a number. However Maple is smart enough to do symbolic algebra with it. Here are some examples of how it would handle things...

4/pi would produce 4/pi as an output if you want a usable number you need to say evalf(4/pi) which gives an approximation. However 4*pi/pi would return pi. Essentially it can treat it as a symbolic value exactly the same way you would calculating by hand in terms of cancellation etc. But you can't use these for things like boolean tests, so if you ask it if 4 > pi it will return a datatype error. If you use floating point arithmetic you get approximations you can use for more things, but obviously these are just approximations.

The same thing happens with rational numbers that don't easily simplify. It will keep them as symbolic values and do arithmetic with them, but it can't perform boolean tests on them or treat them as numeric values.

To my knowledge there is no other way to handle this, and it may even be impossible to improve on it at all.


PlutonianEmpire: I don't there there is really enough information to work out the tidal effects there, since it would depend on each objects rotational speeds and composition aka how much do the tides deform them? I'm sure depending on the answers to those questions just about any distance could be a stable system, once stable they should be tidally locked eg always face each other with the same side. At that point tidal effects should no longer significantly effect the orbits.
But given a bunch of arbitrary values I have no idea how to calculate how long that would take etc.