Started by Dec 01 2010 05:07 AM

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92 replies to this topic

Posted 01 December 2010 - 10:30 PM

So far, I havnt really seen any 'explanations' that go beyond 'but the polygon method confirms to our expectations but this one does not!'

How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.

I think the fundamental difference is that the one arbitrarily closely approximates the tangent of the circle while the other does not. A sufficiently fractal curve could have any length while satisfying the above properties, while one with constraints on its tangent direction could not.

Still not a proof, but I think its a correct intuition.

How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.

I think the fundamental difference is that the one arbitrarily closely approximates the tangent of the circle while the other does not. A sufficiently fractal curve could have any length while satisfying the above properties, while one with constraints on its tangent direction could not.

Still not a proof, but I think its a correct intuition.

Posted 01 December 2010 - 10:47 PM

Quote:

Original post by Eelco

How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.

They're different because it

It's the

Posted 01 December 2010 - 10:51 PM

Quote:

Original post by JoeCooperQuote:

Original post by Eelco

How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.

They're different because itdoesn'treduce the perimeter, and the perimeter is what we're measuring. No reducing is done. The figure's perimeter isn't changed at all. You might as well not change the figure.

Theareais, and if you measurethat, you'll find it most certainly does give us Pi.

Is that a valid distinction though? We can conceive of perimeters that do change, yet do not converge to the correct value; say, a grid outline on one side and a polyhedron on the other, or a more subtle instance of such a curve. Just because it changes, doesnt make it right, nor does the fact that it doesnt change tell you anything, without a preconceived notion of the quantity you are trying to figure out.

Posted 01 December 2010 - 11:20 PM

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*.

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

Posted 01 December 2010 - 11:37 PM

We know the circle has `Pi*diameter` circumference and `Pi*(0.5*diameter)^2` area.

The reduced square will always have an area greater than the circle's, plus it will always have a circumference of 4*diameter (also greater than the circle's).

If we take the points of the reduce square which are touching the circle's edge, and join those points into a polygon, then that polygon will always have an area and circumference less than the circle's.

One gives us maximum values, the other minimum values. Pi is somewhere between the two.

[edit]Another way to realise that this reduce square thingy is obviously going to have a large circumference is to take the idea to the extreme... Imagine a circle, then pick a point on it and travel around clockwise. You travel along in infinitely small steps, at each step, you move the edge inwards and outwards, creating a bump of an infinitely small distance. We just added an infinite number of bumps to the edge, which have a greater than zero length... What's the new circumference? Is`1/infinity * infinity` equal to 1? If so, the new circumference is... ~`Pi*Diameter+1`??

[Edited by - Hodgman on December 2, 2010 8:37:03 AM]

The reduced square will always have an area greater than the circle's, plus it will always have a circumference of 4*diameter (also greater than the circle's).

If we take the points of the reduce square which are touching the circle's edge, and join those points into a polygon, then that polygon will always have an area and circumference less than the circle's.

One gives us maximum values, the other minimum values. Pi is somewhere between the two.

[edit]Another way to realise that this reduce square thingy is obviously going to have a large circumference is to take the idea to the extreme... Imagine a circle, then pick a point on it and travel around clockwise. You travel along in infinitely small steps, at each step, you move the edge inwards and outwards, creating a bump of an infinitely small distance. We just added an infinite number of bumps to the edge, which have a greater than zero length... What's the new circumference? Is

[Edited by - Hodgman on December 2, 2010 8:37:03 AM]

Posted 02 December 2010 - 03:39 AM

Quote:

Original post by JoeCooper

Wedohave 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 todo, 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 thatalsodon'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 approachunless 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?'

Posted 02 December 2010 - 04:04 AM

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?

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.

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]

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

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

[Edited by - JoeCooper on December 2, 2010 11:04:57 AM]

Posted 02 December 2010 - 04:32 AM

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."

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."

Posted 02 December 2010 - 05:28 AM

Quote:

Original post by JoeCooper

Can you answer why my Stargate method for producing Pi doesn't yield Piwithout 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:Why would I?

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

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)

Posted 02 December 2010 - 05:36 AM

Quote:

Original post by JoeCooperQuote:

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 n

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?

Posted 02 December 2010 - 05:36 AM

Quote:

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 aperfect circleand 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.

Posted 02 December 2010 - 05:42 AM

HITLAR didn't forget about Pythagoras! He squared the circle and circled the square and invaded Cube Earth!

Posted 02 December 2010 - 06:18 AM

Quote:

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

Posted 02 December 2010 - 06:35 AM

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.

God no.

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

Is1/infinity * infinityequal to 1?

God no.

Posted 02 December 2010 - 06:44 AM

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.

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.

Posted 02 December 2010 - 06:48 AM

Quote:

Original post by FenrisulvurQuote:

Original post by Hodgman

Is1/infinity * infinityequal to 1?

God no.

I think someone forgot to use their parentheses. I think Hodgman was asking if

Posted 02 December 2010 - 06:55 AM

Quote:

Original post by Mantear

I think someone forgot to use their parentheses. I think Hodgman was asking if(1 / infinity) * infinitywas equal to 1, not1 / (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.

Posted 02 December 2010 - 08:05 AM

Quote:

Original post by FenrisulvurQuote:

Original post by Hodgman

Is1/infinity * infinityequal to 1?

God no.

I think he meant the

crap I was late...

Posted 02 December 2010 - 08:08 AM

Quote:

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]

Posted 02 December 2010 - 11:07 AM

Quote:I'm assuming that "1/inf" is the smallest value that is still greater than zero.

Original post by FenrisulvurQuote:

Original post by Mantear

I think someone forgot to use their parentheses. I think Hodgman was asking if(1 / infinity) * infinitywas equal to 1, not1 / (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.

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 =

number of bumps =

extra length added by bumps =

perimeter of this "circle" =

If we let D be 1, we know Pi to be 3.1415...

I'm glad that's resolved.