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Pi = 4. Discuss.

91 posts in this topic

Quote:
Original post by JoeCooper
We do have preconceived notions, assumptions and specifications, which is perfectly OK and in fact mandatory. If we can't agree on what a circle is and what Pi is supposed to do, than we might as well skip the diagrams, make up numbers and call it a day.

This approach is only being questioned because it looks like the reduction approach. Since the perimeter is not reduced, it is not the reduction approach, and is therefore one of an infinite number of totally random activities that also don't do anything relevant or useful, like watching Stargate. Your arguments are equally valid in support of me watching Stargate, counting the minutes, adding a break to refill my coffee, dividing it by 9 and calling it Pi; I can't tell that this is the wrong approach unless I have some idea of what I'm looking for.


Right. Pi is indeed a well defined concept, and we can determine that some procedures should converge to it, and some should not. Its just that you have so far failed to enumerate what the characteristics of such a procedure should be, and the characteristic 'its not pi for any of the iterations I have inspected' is not very helpful in general. It is infact sufficient an observation for fixed iterations like this, but is rather useless for all other conceivable algorithms, correct or incorrect, none of which produce pi for any n.

In other words; 'its not pi' is not much of an answer to the question 'why isnt this pi?'
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Can you answer why my Stargate method for producing Pi doesn't yield Pi without falling back on the fact that it's not designed to find Pi?

Can you at least show why the perimeter-of-a-square method should work?

Quote:
In other words; 'its not pi' is not much of an answer to the question 'why isnt this pi?'


That's not my answer. My answer is that I can't use it for what Pi is used for. I can't take Pi=4 and get the object's area without, at some point, winding up with the 3.14etc. figure somewhere else in the equation, at which point we're only renaming things.

Quote:
failed to enumerate what the characteristics of such a procedure should be


It's a useless activity. You earlier enumerated the characteristics and only succeeded in suggesting that an intentionally broken procedure should work. The whole joke is designed to play on the idea that you'll look at it in those terms, in order to mock you for doing so. It's recognizable as looking like a correct procedure in the same sense that the lunar maria are recognizable as a face.

[Edited by - JoeCooper on December 2, 2010 11:04:57 AM]
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When you switch from a finite number of iterations to an infinite number of iterations, the square with cut corners becomes a perfect circle and the perimeter changes from 4 to pi without warning.

From http://news.ycombinator.net/item?id=1927096:
"The question is about the sequence of approximations to that line: why the length of the approximations doesn't converge to the length of the limit. And the answer is that it doesn't have to, because even though the approximations are very similar to the limit line in one respect (geometric closeness), they are all very different from it in another respect (directions and angles of travel). If we had a sequence of approximations whose direction of travel converged correctly, the length would converge correctly too."
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Quote:
Original post by JoeCooper
Can you answer why my Stargate method for producing Pi doesn't yield Pi without falling back on the fact that it's not designed to find Pi?

I gave a stab at a general rule, by saying it had to do with the lack of convergence of its tangent direction. Fractals can have any length.

Quote:
Can you at least show why the perimeter-of-a-square method should work?
Why would I?

Quote:
Quote:
In other words; 'its not pi' is not much of an answer to the question 'why isnt this pi?'


That's not my answer. My answer is that I can't use it for what Pi is used for. I can't take Pi=4 and get the object's area without, at some point, winding up with the 3.14etc. figure somewhere else in the equation, at which point we're only renaming things.

Indeed you are renaming things, because again you give that exact same answer: 'its not pi'.

Quote:
It's a useless activity. You earlier enumerated the characteristics and only succeeded in suggesting that an intentionally broken procedure should work.


This starts to smell like trolling. I never said it should work, I said you failed to distinguish what sets it apart from working methods. (that is, aside from the trivial answer; it not working)
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Original post by JoeCooper
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the construction in the OP still seems to fulfill the usual definition of a circle: all the points in a plane that are a given distance away from a given point


Maybe that's not the case. Again, if you just zoom in, it's stair-steps, it's not a circle. The shape on your screen isn't really a circle either, given that it's also painted onto such a grid.


Sorry, I wasn't precise enough in what I was saying. It is a circle if we interpret "repeat to infinity" as "take the limit as this sequence is iterated to infinity" or, if we number the steps and consider the nth step, "take the limit as n approaches infinity".

Quote:

We do have preconceived notions, assumptions and specifications, which is perfectly OK and in fact mandatory. If we can't agree on what a circle is and what Pi is supposed to do, than we might as well skip the diagrams, make up numbers and call it a day.


This isn't a very good way to proceed. You still haven't explained why the one is correct and the other isn't. That is, why the procedure that yields 3.141... is correct and the one that yields 4 isn't. The only explanation you have is that you knew beforehand that 3.141... is the correct answer, but by what procedure did you come up with that number? How did you decide that that procedure yielded the correct answer?

This is important because in math and science you can often get two different values for an unknown quantity and you need to determine which procedure is correct.

As for why the original construction "should" produce pi, intuitively, if curve A converges to curve B, then the length of curve A should converge to the length of curve B. So, if we know the what length curve A converges to and curve B is a circle of known diameter, then we should be able to calculate pi from its primary definition: the ratio of a circle's perimeter to its diameter. In the original construction, B is a circle with a diameter of 1 and A is a curve that converges to a circle and whose length converges to 4.

The problem is that the "intuitive" part there is wrong.

Quote:
Original post by BlueSalamander
"If we had a sequence of approximations whose direction of travel converged correctly, the length would converge correctly too."


So, an obvious necessary condition is that the curve itself converges, but that's apparently not sufficient. Given that the curve converges, is a converging gradient sufficient?
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Original post by BlueSalamander
When you switch from a finite number of iterations to an infinite number of iterations, the square with cut corners becomes a perfect circle and the perimeter changes from 4 to pi without warning.


No it doesn't. It only appears to be a perfect circle if your concept of a 'perfect' circle is that of a 2 dimensional structure bent around a point of a 2d plane so you can't actually look close enough to see the fine details. A true perfect circle has no width on the actual 'line', it has space outside the circle, space inside the circle, and values that fall exactly between the two, which get truer and truer as you zoom in.

Cutting the corners will ALWAYS produce a jagged edge, and will always have a perimeter of 4. 1/10^10^10^10 of an inch still counts, and they don't magically go away.

The model presented in the OP that Pi = 4 is flawed because they aren't producing a circle, or even a circle like structure as they approach infinity.
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Original post by Way WalkerThat is, why the procedure that yields 3.141... is correct and the one that yields 4 isn't. The only explanation you have is that you knew beforehand that 3.141... is the correct answer, but by what procedure did you come up with that number? How did you decide that that procedure yielded the correct answer?


I don't. I just expect Pi to relate the area, radius and circumference, and two approximations of Pi (which is all we're talking about) can be compared to each other by how well they accomplish that.

If I try to use Pi to get the area from the radius, than figures of 3.14, 3.1 or even 3 all give closer answers than 4.

While you'd be correct pointing out that under my model there are situations where Pi=4 would be "good enough", it is still easy to show that the "Bible says Pi is 3" approach gives a quantifiably superior result. 3 is a closer approximation to Pi than 4, and we know that by trying to use it in any application that demands Pi.
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Hmm, somehow I don't doubt that this one would confound Archimedes significantly. It's a disconcerting result.

BlueSalamander's link is the only thing in this thread that really comes close to addressing the paradox. As Eelco has been insisting, the only truly acceptable answer is one born of thorough mathematical rigour: a principled and formal approach which pins down this vague notion of circumference in perfect logical detail, and leaves absolutely no room for conflicting interpretation - lest one be ostracised and cast into the depraved wilderness of pseodumathematics.

JoeCooper: you've touched on some significant notions in parts, but you speak it like a philosophy student, and (seem to) get frustrated when others remain unconvinced of the ideas you attempt to convey (kind-of like a philosophy student >_> ). You've given no good reason why a valid "iterative" approximation of a quantity should necessarily converge (you said "reduce"?) indefinitely to some limiting value, as opposed, to say, starting and remaining at the correct value through all steps; and you haven't explained why the relationship between a circle's circumference and it's area must hold, independent of the results called into question by the paradox.


Anyway, I'll try to formalize this once I've had some sleep, but I expect to say something about metric, Euclidean distance vs the way they do things in Manhattan, and/or something along those lines. For now, I'm not really satisfied that this paradox has been addressed.

Quote:
Original post by Hodgman
Is 1/infinity * infinity equal to 1?

God no.

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I don't get the argument. It seems fairly obvious to me that as you "remove corners" to inifity it's the area that is converging to be the same area as that of the circle, not the diameter. No matter how small you make the steps, they're still steps.

You can easily draw any sort of shape you want with the same area as that of another shape, but they both can have very different perimeters. A square or any other shape other than a circle doesn't have a "diameter" from which to calculate the area, so trying to calculate its perimeter using a non-existant construct (what's the diameter of a square?) obviously won't work. "Removing corners" doesn't turn a shape with straight edges into a circle with a diameter, no matter how many times you do it.

*Edited to straighten out my own thoughts.
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Quote:
Original post by Fenrisulvur
Quote:
Original post by Hodgman
Is 1/infinity * infinity equal to 1?

God no.



I think someone forgot to use their parentheses. I think Hodgman was asking if (1 / infinity) * infinity was equal to 1, not 1 / (infinity * infinity).
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Original post by Mantear
I think someone forgot to use their parentheses. I think Hodgman was asking if (1 / infinity) * infinity was equal to 1, not 1 / (infinity * infinity).

Eh, I maintain that the inline division operator is an obscure and ambiguous notation.

Anyway, that's indeterminate, and it's not clear just skimming Hodgman's post where he derived the form from.
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Quote:
Original post by Fenrisulvur
Quote:
Original post by Hodgman
Is 1/infinity * infinity equal to 1?

God no.



I think he meant the (1/infinity)*infinity.

crap I was late...
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you speak it like a philosophy student, and (seem to) get frustrated when others remain unconvinced of the ideas you attempt to convey (kind-of like a philosophy student >_> )


Sorry. Incidentally, I'm surrounded by arts & humanities folk and I've never gotten along well with other programmers.

Go figure.

I'm not trying to troll anyone, I'm just having a hard time communicating here.

[Edited by - JoeCooper on December 2, 2010 4:08:34 PM]
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Original post by Fenrisulvur
Quote:
Original post by Mantear
I think someone forgot to use their parentheses. I think Hodgman was asking if (1 / infinity) * infinity was equal to 1, not 1 / (infinity * infinity).

Eh, I maintain that the inline division operator is an obscure and ambiguous notation.

Anyway, that's indeterminate, and it's not clear just skimming Hodgman's post where he derived the form from.
I'm assuming that "1/inf" is the smallest value that is still greater than zero.
The "squiggly circle", or "reduced square circle" has an infinite number of bumps on it's surface, each of which adds an infinitely small (but greater than zero) length to the circumference.
length of bump = 1/inf
number of bumps = inf
extra length added by bumps = (1/inf)*inf
perimeter of this "circle" = Pi*D + (1/inf)*inf

If we let D be 1, we know Pi to be 3.1415...
and we know that the perimeter of this bumpy circle to be 4, then we can assume that (1/inf)*inf is equal to 4 - 3.1415..., or 0.8584073464102067615373566167205... which is also known as the trollface constant.

I'm glad that's resolved.
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This is silly. It's like looking at a parabola and stating that since it has an asymptote of x = 0, it must at some point into infinity have x = 0 actually lying on the parabola's curve.
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Can we assume that x^2+y^2=r^2 is a valid test for whether or not a vertex lies on a circle? Is that a given?

If so, in a regular polygon (the method this spoofs that - I think - we're trying to differentiate it from), every specified vertex lies on a circle.

But you draw a line from that vertex to the next and the midpoint of that line must deviate from a circle; x^2+y^2 will be less than r^2.

Clearly, it's not a circle, many points on it will do not satisfy the equation of a circle. But if I do something like this...



This shows the sum total of the mid-point's hypotenuse divergence from r in a regular polygon. As we add vertices, the divergence from a circle approaches zero; the difference between our circle approximation and a hypothetical true circle is reduced.

The excavation method does not have this characteristic.

Both the square excavation method and regular polygon contain vertices which satisfy the points, alternating with ones that do not. But as we iterate the square excavation method, the sum total of the vertice's deviation increases.

I'm not really sure how to make a function for this one - I could try if necessary - but for now I'll but I'll run some numbers here manually.

In the first iteration, showing the square, we have four specified vertices for which their hypotenuse is sqrt(2) (1.41etc) each, so we can say they deviate from r^2 (1) by 0.41etc, or 1.65ish total. (There are also four points that do fit on the circle, at the mid-points.)

On our second iteration, the deviant vertices' average hypotenuse is sqrt(cos(45)^2+1) or 1.22ish, so the deviation is that, minus one; 0.22ish. There are now 8 of these, so the total sum of the midpoint vertices' deviation from r is now increased to 1.76.

This contrasts with the regular polygon which reduces deviation from the definition of a circle by adding vertices.

Thus we can quantify that squares are an inferior approximation of a circle compared to a regular polygon.

Or something.

Quote:
You've given no good reason why a valid "iterative" approximation of a quantity should necessarily converge (you said "reduce"?) indefinitely to some limiting value


In the above pitch, a correct procedure should converge onto zero when measured this way because a non-zero value shows failure to satisfy the equation of a circle. It should be more like a circle because the goal is to approximate a circle's properties by creating a shape to approximate a circle, then measuring its properties.

Is any of this relevant or am I barking up the wrong tree?

[Edited by - JoeCooper on December 3, 2010 8:37:55 AM]
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Quote:
Original post by Hodgman
I'm assuming that "1/inf" is the smallest value that is still greater than zero.

Such a value does not exist. Whenever you believe you have found such a value x, then x/2 is smaller, positive and not zero either (this works regardless of whether we are working with real or rational numbers).
In actual textbook math (not school-math, university-level math) there are preciously few points were an "infinity" can be. There are some more specialized branches of mathematics which allow actually using infinity as a more regular symbol, but even there you have to be extremely careful what you are allowed to do with it. There is a Wikipedia article about that.
The rest of the post kinda dies with that.


Maybe I should not have skimmed the thread like that but so far it seems arguments are on the "drawing sketches, waving hands"-level largely.

The only proper definition for the convergence of a sequence S(n) to the limit C I know is

and I don't think I have seen anyone trying to apply this to the problem. That would of course first require us to put the happy sketch into an actual formula you can work with but I feel a bit lazy today.
My personal guess is the result will be that the happy sketch idea simply does not converge because for every finite value the circumference is 4 while the limit would have to be pi. So even for it's not possible to find an as required.
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My personal guess is the result will be that the happy sketch idea simply does not converge because for every finite value the circumference is 4 while the limit would have to be pi.


Isn't that just saying that it doesn't work because the result isn't Pi? Or, it should converge on Pi and it doesn't?

I just got my rear handed to me on a platter for that.
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Well, as I said that is my personal guess. Actually proofing that would require a bit of work I currently do not have the time (and honestly, also not the motivation) for.

I'd be happy though if the rest of the post steers the topic toward the slightly more mathematical. A lot of the things I read while I glanced over the thread made me wince really, really badly even though my last 'real' math is seven, eight years behind me now. It's a bit frustrating, like watching people who are unwilling to believe that but do not have the mathematical background to actually argue their point or even understand the proves people who do know what they are talking about are supplying. :(
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In my first post I asserted that the limit curve is not a circle. I was wrong. It does converge to a circle, and this is clear just from the definitions.

The root issue here is that limits and derivatives do not generally commute, and they fail to do so here for a very clear geometric reason.

Let c_n(t) : [0,1] --> R^2 designate the curve after the nth iteration with c_0 being the square, and let c(t) designate the unit circle.

The length of the limit curve is the integral over [0,1] of |(limn-->infty c_n)'(t)|. The limit of the lengths of the curves is limn-->infty of the integral over [0,1] of |c_n'(t)|. The limits don't commute with the integrals. The reason in this case is that c'(t) is a vector that could point in any direction depending on t whereas c_n'(t) is always either horizontal or vertical. Observe that this does *not* happen when one uses the traditional circumscribed n-gons that we're all familiar with.

[Edited by - nilkn on December 3, 2010 12:27:13 PM]
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I think the trick also comes from fractal nature of the problem - unlike typical shapes, the perimeter and area of a fractal are not necessarily correlated.

Koch snowflake has infinite perimeter, but finite area.

I'd say that the problem here comes from implication that value of Pi follows directly from perimeter, whereas even when using the curve it must be derived from area.
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The limit curve is 'infinitely' squiggly/pointy (Non-differentiable everywhere).

The area is being monotonically reduced each time with the limit converging to PI*r^2 as any area outside of the circle eventually gets cut out. The perimeter is not getting reduced each time, and has a limit of 4. We created a shape with the same area as a circle, but not a circle.

The limit need not equal the value of the function. For example the piece wise function { 1 if x = 0; 0 otherwise } the limit as x->0 exists and equals 0. The value at 0 is 1.
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Quote:
Original post by Platinum314
We created a shape with the same area as a circle, but not a circle.


But, you didn't do that. You always have a saw tooth edge that comes near the circle, but must always include an area that exists outside the boundary of the circle. It therefore must have an area Larger than the circle itself, not equal to.
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Quote:
Original post by Platinum314
The limit curve is 'infinitely' squiggly/pointy (Non-differentiable everywhere).

The area is being monotonically reduced each time with the limit converging to PI*r^2 as any area outside of the circle eventually gets cut out. The perimeter is not getting reduced each time, and has a limit of 4. We created a shape with the same area as a circle, but not a circle.

The limit need not equal the value of the function. For example the piece wise function { 1 if x = 0; 0 otherwise } the limit as x->0 exists and equals 0. The value at 0 is 1.


In this case however the curves not only converge to the circle but they do so uniformly. Let c_n(t) be the curve after the nth iteration. For any epsilon > 0 draw an annulus a_epsilon of width epsilon whose central circle is the unit circle. For sufficiently large n the curve c_n(t) is contained entirely within this annulus a_epsilon. This is the condition for uniform convergence to the limit curve (the circle in this case).

The issue in this case is that two functions can be close yet their derivatives can differ by a lot. More precisely, the derivative of a limit of a sequence of functions need not coincide with the limit of the derivatives. I actually think you can say something stronger in this case case: despite the fact that the limit curve is everywhere differentiable the sequence of derivatives c_n'(t_0) for a fixed t_0 in [0,1] does not have a limit.
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