Pi - No Patterns, because Pi is the pattern

[Quantum Quack]

Why would the circle not exist?

because the line will only be proved to be zero (< 1/infinity) in thickness...

The outside volume of the circle inside the box would be an infinite resolving number as will the volume inside the circle....the difference between the two when compared to the whole box, should be 1/infinity as far as I can work out... I'm making up a diagram to show the thought experiment [ double reduction absurdo - or what I interpret this to mean ]

 

[Quantum Quack]

Steps
1] Calculate the area of the Square.. = x
2] Calculate the area outside the circle[but inside the square] = y
3] Calculate the area of the circle = z


Then you will find if I am not mistaken logically the following:

x= y+z+(< 1/infinity)

The end result is a circle [line] that is both non-existent and existent simultaneously.
The circle line becomes an event horizon or vanishing point and nothing more

As shown in exaggerated form below:

Of course this is on the premise that Pi can not ever be resolved finitely

 

[Captain Kremmen]

The end result is a circle [line] that is both non-existent and existent simultaneously.

Yes, because it is theoretical.
There are an infinite number of circles, and any other shape, that can fit in the square,
and your calculation applies to all of them.
What does it prove?
Other than a line has no width?

@CBork
I imagine that Archimedes would have given everything he owned and lived in rags
to have thought of Taylor's Series. Simple, and beautiful.
Why does it work?

If you studied it hard enough,
I expect you would find that it has its basis in Geometry,
and is cookie cutting by another route.

 

[Quantum Quack]

Yes, because it is theoretical.
There are an infinite number of circles, and any other shape, that can fit in the square,
and your calculation applies to all of them.
What does that prove?
Other than a line has no width?

it proves a lot of things.....
one of which:
It proves when applied to 3 dimensional space the simultaneous existence and non-existence of zero.

My bet is that no matter how you try to resolve the calculations you will always end up with a value <1/infinity but a value all the same.
the line has a width of <1/infinity
Using the double reduction absurdo method mentioned earlier

 

[Quantum Quack]

The reason why this is exciting for me is graphically shown in this image of Baileys beads.

especially when you compare with the image posted earlier:

which relates back to what I posted earlier:

However Pi in Nature is not calculated as it is already existent. [in any curved surface I would imagine]
and it emulates something that is already existent and that makes it a very remarkable number. [concerning the fundamentals of this universe] IMO

 

[Captain Kremmen]

Just because you can't calculate the area of the circle absolutely precisely given a radius,
does not mean that the circle doesn't have an exact area.
What if you placed a circle with an area of 1 square centimetre inside a square of 4 square centimetres?

 

[Quantum Quack]

Just because you can't calculate the area of the circle absolutely precisely given a radius,
does not mean that the circle doesn't have an exact area.
What if you placed a circle with an area of 1 square centimetre inside a square of 4 square centimetres?

a square centimeter in a circle.... eghad!!!!

 

[Stryder]

I consider the hypothesis of the universe being a simulation.

If a simulation was built there would likely be a cryptographic "Seed" to the server(s) [or construct a identification key] which would be a value that everything relies upon, that wouldn't change but would still be capable of developing strong encryption depending on the technological development of the equipment (via moores law). PI being an Irregular number is brilliant for this, as depending on the depth of progression of technology defines the number of digits that can be handled at any one time.

It takes into consideration that to build a simulation doesn't mean spamming out revised numbers and leaving a bunch of unsupported "alpha" models slowly decaying [after all I haven't seen any blocky people have you?], in essence such a simulation development would include all previous versions of legacy based upon their placement in decimalisation of an irregular number.

Incidentally this hypothesis means that our universe might have a different value of PI to another universe, which might well explain why we only see our one.

 

[someguy1]

The outside volume of the circle inside the box would be an infinite resolving number as will the volume inside the circle....the difference between the two when compared to the whole box, should be 1/infinity as far as I can work out... I'm making up a diagram to show the thought experiment [ double reduction absurdo - or what I interpret this to mean ]

The circumference of the circle has zero area in 2D space. You're correct about that. But it certainly exists. For example the unit circle in 2D is the set of points whose distance is exactly 1 from the origin. That set of points exists, even though it has zero area in the plane.

Perhaps you're confusing math with physics. In the physical world there are no perfect circles or dimensionless points. But in math there are.

 

[CptBork]

@CBork
I imagine that Archimedes would have given everything he owned and lived in rags
to have thought of Taylor's Series. Simple, and beautiful.
Why does it work?

If you studied it hard enough,
I expect you would find that it has its basis in Geometry,
and is cookie cutting by another route.

Taylor's series is about expressing various mathematical functions (i.e. $$\sin x$$ ,$$e^x$$, $$x^{8.26}$$) in terms of infinite-degree polynomials. For instance we can write $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\ldots$$ when angle $$x$$ is given in dimensionless radian units. Obviously you can't calculate infinitely many terms to get the exact answer in general, but normally you only need to take the first few terms in order to get a good approximation, and then adding more terms serves as a small correction to make the approximation more exact. There's also an error term that allows you to put limits on how far off your approximation is at worst, so if you wanted to calculate it accurately to a certain number of decimal places, you'd know that you're guaranteed at least that level of accuracy or better after adding a certain minimum number of terms (or more).

It's a concept that arose with the formal development of calculus in Isaac Newton's time, useful for geometric applications (like calculating trigonometric functions for arbitrary angles), but it's also useful for just about any other kind of mathematical function under certain conditions, and I don't think it was geometrically motivated anymore than any other development in calculus. Actually, Taylor's series gives us a way of defining certain mathematical functions like sine, cosine, inverse tan, etc., and even $$\pi$$ itself, in a purely algebraic way that requires no reference whatsoever to shapes or anything involving geometry and trigonometry.

Using the double reduction absurdo method mentioned earlier

Your methodology here is not "double reductio ad absurdum", I'm afraid it's not really math at all to be honest.

The reason why this is exciting for me is graphically shown in this image of Baileys beads.
https://upload.wikimedia.org/wikipedia/commons/0/04/Бриллиантовое_кольцо.jpg


especially when you compare with the image posted earlier:

http://zeropointtheory.com/images/a_this_site/sundry/sqcircle2.png
which relates back to what I posted earlier:

The fuzzy glowing edge of a photographically enhanced black hole event horizon looks like the fuzzy edge of a shaded circle drawn in a graphics program. So what?

I consider the hypothesis of the universe being a simulation.

If a simulation was built there would likely be a cryptographic "Seed" to the server(s) [or construct a identification key] which would be a value that everything relies upon, that wouldn't change but would still be capable of developing strong encryption depending on the technological development of the equipment (via moores law). PI being an Irregular number is brilliant for this, as depending on the depth of progression of technology defines the number of digits that can be handled at any one time.

It takes into consideration that to build a simulation doesn't mean spamming out revised numbers and leaving a bunch of unsupported "alpha" models slowly decaying [after all I haven't seen any blocky people have you?], in essence such a simulation development would include all previous versions of legacy based upon their placement in decimalisation of an irregular number.

Incidentally this hypothesis means that our universe might have a different value of PI to another universe, which might well explain why we only see our one.

I'm afraid $$\pi$$ isn't up for negotiation as it doesn't depend on the universe in which we live. It's entirely based on the idealized concept of a perfect circle on which each point of the boundary is exactly the same distance from a central point, on a perfectly flat surface where parallel lines never meet.

 

[Quantum Quack]

Taylor's series is about expressing various mathematical functions (i.e. $$\sin x$$ ,$$e^x$$, $$x^{8.26}$$) in terms of infinite-degree polynomials. For instance we can write $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\ldots$$ when angle $$x$$ is given in dimensionless radian units. Obviously you can't calculate infinitely many terms to get the exact answer in general, but normally you only need to take the first few terms in order to get a good approximation, and then adding more terms serves as a small correction to make the approximation more exact. There's also an error term that allows you to put limits on how far off your approximation is at worst, so if you wanted to calculate it accurately to a certain number of decimal places, you'd know that you're guaranteed at least that level of accuracy or better after adding a certain minimum number of terms (or more).

It's a concept that arose with the formal development of calculus in Isaac Newton's time, useful for geometric applications (like calculating trigonometric functions for arbitrary angles), but it's also useful for just about any other kind of mathematical function under certain conditions, and I don't think it was geometrically motivated anymore than any other development in calculus. Actually, Taylor's series gives us a way of defining certain mathematical functions like sine, cosine, inverse tan, etc., and even $$\pi$$ itself, in a purely algebraic way that requires no reference whatsoever to shapes or anything involving geometry and trigonometry.



Your methodology here is not "double reductio ad absurdum", I'm afraid it's not really math at all to be honest.



The fuzzy glowing edge of a photographically enhanced black hole event horizon looks like the fuzzy edge of a shaded circle drawn in a graphics program. So what?



I'm afraid $$\pi$$ isn't up for negotiation as it doesn't depend on the universe in which we live. It's entirely based on the idealized concept of a perfect circle on which each point of the boundary is exactly the same distance from a central point, on a perfectly flat surface where parallel lines never meet.

never heard of Bailey's beads?

Certainly not an enhanced image of a fuzzy black hole..thingo

 

[Quantum Quack]

The circumference of the circle has zero area in 2D space. You're correct about that. But it certainly exists. For example the unit circle in 2D is the set of points whose distance is exactly 1 from the origin. That set of points exists, even though it has zero area in the plane.

Perhaps you're confusing math with physics. In the physical world there are no perfect circles or dimensionless points. But in math there are.

All math is devoted to some form of physical construct or at least exploring the path to some form of physical construct.

To me there is no distinction between math, physics, cosmology, philosophy... as ultimately that are all a part of the same human quest for understanding himself and that which surrounds him.
Pi which is a mathematical number, demonstrates, using the "logic" symbol-ogy of mathematics [language] some of that which surrounds us.
To me Pi is just a short hand symbolic, logical representation of a very important part of 3 dimensional space.

Math is after all the science of logic is it not?

In the physical world there are no................. or dimensionless points. But in math there are.

To me the above statement is highly debatable and unfortunately not one for this thread...

As proved using the badly described method previously an example of possible application:


You have a moon that has a "matter to space" boundary of (<1/infinity) thickness that allows light to stream through it causing, under certain conditions, the effect known as Baileys Beads. [during a solar eclipse]

 

[someguy1]

All math is devoted to some form of physical construct or at least exploring the path to some form of physical construct.

Even if I granted you that, it would still be the case that math is math. When you are in math class and they ask you to consider the unit circle, namely the set of points (x,y) in Euclidean 2-space satisfying x^2 + y^2 = 1, did you raise your hand and say "Oh but that circle can't exist because it has no area." Or did you silently seethe, waiting for the chance to express that point of view on an Internet forum? Do you mean to claim that the mathematical concept of the unit circle doesn't exist because it has no area?

To me there is no distinction between math, physics, cosmology, philosophy... as ultimately that are all a part of the same human quest for understanding himself and that which surrounds him.

I wouldn't disagree with you that "everything is related" or even "all is one," if I'm in that kind of metaphysical mood. But still, math is math. Do you deny the mathematical existence of the unit circle? How did this belief manifest itself when you were in math classes in high school or college? When you took high school geometry and they said, "Euclid says a line is composed of points," did you believe that there are no lines and there are no points, because lines and points don't have a nonzero area?

Pi which is a mathematical number, demonstrates, using the "logic" symbol-ogy of mathematics [language] some of that which surrounds us.
To me Pi is just a short hand symbolic, logical representation of a very important part of 3 dimensional space.

But pi is a dimensionless point on the real number line. It has no nonzero area. By your logic pi doesn't even exist. How do you reconcile your statements?

Math is after all the science of logic is it not?

No, not really. But even if for sake of discussion I agreed with you, I'm more interested in your denial of the mathematical existence of the unit circle. Am I representing your opinion fairly on that matter? And how do you account for the fact that pi is a dimensionless point on the real number line?

I had originally written:

In the physical world there are no perfect circles or dimensionless points. But in math there are.

And you responded:

To me the above statement is highly debatable and unfortunately not one for this thread...


You believe in pi. You think it has cosmic significance. Yet pi is a dimensionless point on the mathematical real line. By your own logic you don't believe in pi.

Explain the contradiction, please.

 

[Quantum Quack]

Even if I granted you that, it would still be the case that math is math. When you are in math class and they ask you to consider the unit circle, namely the set of points (x,y) in Euclidean 2-space satisfying x^2 + y^2 = 1, did you raise your hand and say "Oh but that circle can't exist because it has no area." Or did you silently seethe, waiting for the chance to express that point of view on an Internet forum? Do you mean to claim that the mathematical concept of the unit circle doesn't exist because it has no area?




I wouldn't disagree with you that "everything is related" or even "all is one," if I'm in that kind of metaphysical mood. But still, math is math. Do you deny the mathematical existence of the unit circle? How did this belief manifest itself when you were in math classes in high school or college? When you took high school geometry and they said, "Euclid says a line is composed of points," did you believe that there are no lines and there are no points, because lines and points don't have a nonzero area?



But pi is a dimensionless point on the real number line. It has no nonzero area. By your logic pi doesn't even exist. How do you reconcile your statements?



No, not really. But even if for sake of discussion I agreed with you, I'm more interested in your denial of the mathematical existence of the unit circle. Am I representing your opinion fairly on that matter? And how do you account for the fact that pi is a dimensionless point on the real number line?

I had originally written:

In the physical world there are no perfect circles or dimensionless points. But in math there are.

And you responded:

To me the above statement is highly debatable and unfortunately not one for this thread...


You believe in pi. You think it has cosmic significance. Yet pi is a dimensionless point on the mathematical real line. By your own logic you don't believe in pi.

Explain the contradiction, please.

Firstly, you have asked some hard (as in solid] questions and ones I would love the proper opportunity to address. Secondly if I do that here this thread will be moved to another fora. [ and I will probably be unjustly accused of trolling]

We are talking about the circles line [ boundary ] not it's area per see.

Let me ask you a counter question with the view to going towards answering your questions:

Do you believe in zero?
Does it exist?
If it doesn't exist then why do you believe in it?

I believe Pi proves the existence of a value [a circles line] that is (< 1/infinity) thick. [In math and physics]
Is that value zero or is it simply (< 1/infinity)?

Is the circles line zero in thickness [re: Euclid] or is it (< 1/infinity) in thickness or is it paradoxically both?

(A value and a non-value simultaneously as is the case with zero)
If I am not mistaken zero does NOT and can NOT equal an infinitesimal. (1/infinity)

But pi is a dimensionless point on the real number line. It has no nonzero area. By your logic pi doesn't even exist. How do you reconcile your statements?

Every number is a zero dimensional point on a real number line....do they exist?

Edit: An appropriate thread in pseudo science about "Reductio ad absurdum" may prove interesting.....do you want to start it or do you want me to...?

 

[Stryder]

I'm afraid $$\pi$$ isn't up for negotiation as it doesn't depend on the universe in which we live. It's entirely based on the idealized concept of a perfect circle on which each point of the boundary is exactly the same distance from a central point, on a perfectly flat surface where parallel lines never meet.

That is precisely the point, what is the perfect circle? I mean if you draw a circle on a computer while indeed you can have curvatures vectored in, the base of it is a polar axis with a number of vertices that are connected. The more "circular" the circle, the more vertices used. You could apply a concept of how Moore's Law could be applied to drawing an ever perfectionistic circle, constantly upgrading the level of available definition by increasing definition with each build/render. That is the point however as what would define the rate of technological progress to ascertain as to how many vertices are used with each build? would it just be "adding 4 vertices each build due to asymmetry" or would those numbers change/alternate based upon the available technological parameters of the time or perhaps something more entropic in origin (like a programmer deciding not to get out of bed to do that build a particular day because it's raining in Gibraltar)?

If you could observe the timeline of such a perfectionistic venture, you'd understand where things can diverge and where values you take for granted could well be different elsewhere. (I'm not suggesting that the values are different here)

 

[Captain Kremmen]

@CBork
Seems that Brook Taylor had great ideas, but lacked the ability to express them clearly. His work was only recognised as important 40 years after his death.

Taylor's Methodus Incrementorum Directa et Inversa (1715) added a new branch to the higher mathematics, now designated the "calculus of finite differences." This work is available in translation on the web [1]. Among other ingenious applications, he used it to determine the form of movement of a vibrating string, by him first successfully reduced to mechanical principles. The same work contained the celebrated formula known as Taylor's theorem, the importance of which remained unrecognised until 1772, when J. L. Lagrange realized its powers and termed it "le principal fondement du calcul différentiel" ("the main foundation of differential calculus").
http://en.wikipedia.org/wiki/Brook_Taylor

 

[Fednis48]

That is precisely the point, what is the perfect circle? I mean if you draw a circle on a computer while indeed you can have curvatures vectored in, the base of it is a polar axis with a number of vertices that are connected. The more "circular" the circle, the more vertices used. You could apply a concept of how Moore's Law could be applied to drawing an ever perfectionistic circle, constantly upgrading the level of available definition by increasing definition with each build/render. That is the point however as what would define the rate of technological progress to ascertain as to how many vertices are used with each build? would it just be "adding 4 vertices each build due to asymmetry" or would those numbers change/alternate based upon the available technological parameters of the time or perhaps something more entropic in origin (like a programmer deciding not to get out of bed to do that build a particular day because it's raining in Gibraltar)?

If you could observe the timeline of such a perfectionistic venture, you'd understand where things can diverge and where values you take for granted could well be different elsewhere. (I'm not suggesting that the values are different here)

The perfect circle is better-defined than I think you're giving it credit for. It's not just something that we can approximate ever-better with improved computers. A perfect circle is the shape defined by all points in a given plane at a particular distance from a given center. This definition does not invoke any particular properties of our universe, so it would be exactly the same in all possible universes.

On the other hand, considering your response did give me one idea for how $$A=\pi r^2$$ might not be true in all universes. That formula relies on the Euclidean distance metric: $$d=\sqrt{x^2+y^2+z^2}$$ (or more generally for $$n$$ dimensions, $$d=\sqrt{\displaystyle\sum_{i=1}^n x_n^2}$$). In a universe with a different distance metric, a circle might look different from what we think of as a circle, even if it was defined exactly the same as above. As an example, if we were in a universe where distance was given by the 1-norm $$d=\left |x+y+z\right |$$, a circle would be identical to a square.

Actually, since General Relativity uses non-Euclidean distance metrics, does that imply that $$A=\pi r^2$$ is not strictly true in our universe? It sounds weird, but I know that GR implies the internal angles of a triangle don't have to sum to $$180^\circ$$, so I'm wondering if the area of a circle works in a similar way.

 

[someguy1]

Firstly, you have asked some hard (as in solid] questions and ones I would love the proper opportunity to address. Secondly if I do that here this thread will be moved to another fora. [ and I will probably be unjustly accused of trolling]

I haven't been around here long enough to have a sense of the moderation standards. I'm just trying to clarify your misunderstandings regarding the nature of the real numbers. To that extent, this conversation seems on-topic. I'll have to trust the moderators.

We are talking about the circles line [ boundary ] not it's area per see.

Agreed. In your "circle in square" demonstration you pointed out that a circle, being a 1-dimensional shape living in 2-space, has no area. I quite agree. I've been careful to use the phrase "circumference of the circle" to indicate that I'm talking about the circle itself, not the region it encloses. This is a standard convention in math. The unit circle is the set of points satisfying x^2 + y^2 = 1. It's just the circumference we're talking about.

Let me ask you a counter question with the view to going towards answering your questions:

Do you believe in zero?
Does it exist?
If it doesn't exist then why do you believe in it?

I indicated earlier that your basic point of confusion is that you are not distinguishing between math and physics. I'm just talking about math, this being the math section of this website. Zero is a perfectly well-defined real number. It has mathematical existence. Whether it exists in the physical world is no concern of mine in this context. That's a question of philosophy, or physics

[But, FWIW ... I don't even believe that the number 3 has physical existence. 3 books or 3 tables or three planets, yes, those are physical manifestations or instantiations of the number 3. But the number 3 itself does not exist in the physical world. For one thing, the number 3 is not bound by gravity. It's not a physical thing. The number 3 is an abstract mathematical object, just as the number zero is.]

But in the world of modern mathematics, zero is a real number. It's a particular point on the number line. It's the additive identity of the integers considered as an Abelian group. There's no controversy or confusion on this point.

Zero has mathematical existence.

I believe Pi proves the existence of a value [a circles line] that is (< 1/infinity) thick. [In math and physics]
Is that value zero or is it simply (< 1/infinity)?

The expression 1/infinity has no meaning in mathematics. It's not defined. There are no infinitesimal quantities in the real numbers. The thickness of a line in 2-space is zero. This is not in dispute in any way.

But pi is a particular real number on the real number line. It has length zero. Do you understand that? It's just a point. It's a zero-dimensional object. It has no length or width or thickness. Zero. A mathematical point has zero length, zero width, zero area. Pi is a real number that represents a point on the real number line.

Is the circles line zero in thickness [re: Euclid] or is it (< 1/infinity) in thickness or is it paradoxically both?

It's zero. The expression 1/infinity has no meaning. It's undefined. It's true that in calculus we often speak of the limit of 1/x as x goes to infinity. But in that context, the phrase "limit as x -> infinity" has a very specific technical meaning. It's irrelevant here. The thickness of a line or circle in 2-space is zero.

(A value and a non-value simultaneously as is the case with zero)
If I am not mistaken zero does NOT and can NOT equal an infinitesimal. (1/infinity)

There are no infinitesimals in the real numbers. Zero is a perfectly well-defined and well-understood value in mathematics. It's a point on the number line and it's the additive identity of the integers, as I said. One may certainly argue against the existence of zero in physics and/or philosophy. But in mathematics, zero exists and there's no dispute or disagreement about that.

Every number is a zero dimensional point on a real number line....do they exist?

Edit: An appropriate thread in pseudo science about "Reductio ad absurdum" may prove interesting.....do you want to start it or do you want me to...?

One thread like this is plenty for me, thanks. To the extent that you're interested in understanding mathematics, I'm happy to continue. To the extent that this is going to be an unproductive crank discussion, I won't be able to continue much longer. It's up to you. You're the one whose handle identifies you as a quack. I'm trying to talk math with you.

I would like you to directly respond to the question I asked you earlier. You say that pi exists, but that a circle having zero thickness does not exist. But pi is a real number representing a point on the number line that has no length, width, or area. By your logic, pi doesn't exist.

I would answer that pi has perfectly well-understood mathematical existence as a point on the number line. And that the unit circle, the set of points in Euclidean 2-space satisfying x^2 + y^2 = 1, has perfectly well-defined mathematical existence.

 

[CptBork]

Certainly not an enhanced image of a fuzzy black hole..thingo

I assumed that's what it was because you were calling it an "event horizon", which would be like calling a block of cheese a "hammer". Anyhow, the ring of light around the moon during an eclipse has finite nonzero thickness, it's not a perfect circle or a line.

The perfect circle is better-defined than I think you're giving it credit for. It's not just something that we can approximate ever-better with improved computers. A perfect circle is the shape defined by all points in a given plane at a particular distance from a given center. This definition does not invoke any particular properties of our universe, so it would be exactly the same in all possible universes.

On the other hand, considering your response did give me one idea for how $$A=\pi r^2$$ might not be true in all universes. That formula relies on the Euclidean distance metric: $$d=\sqrt{x^2+y^2+z^2}$$ (or more generally for $$n$$ dimensions, $$d=\sqrt{\displaystyle\sum_{i=1}^n x_n^2}$$). In a universe with a different distance metric, a circle might look different from what we think of as a circle, even if it was defined exactly the same as above. As an example, if we were in a universe where distance was given by the 1-norm $$d=\left |x+y+z\right |$$, a circle would be identical to a square.

Actually, since General Relativity uses non-Euclidean distance metrics, does that imply that $$A=\pi r^2$$ is not strictly true in our universe? It sounds weird, but I know that GR implies the internal angles of a triangle don't have to sum to $$180^\circ$$, so I'm wondering if the area of a circle works in a similar way.

If you draw a circle on a spherical surface, the definition of distance is different than what we normally conceive of, and the ratio of circumference to diameter is no longer $$\pi$$. Similarly, the angles of a triangle on a spherical surface no longer need add to $$180^\circ$$. So the defining relationship for $$\pi$$ does indeed depend on having a flat 2D surface where parallel lines never meet, a perfectly idealized situation which probably can't be found in nature.

 

[Quantum Quack]

@someguy1

I indicated earlier that your basic point of confusion is that you are not distinguishing between math and physics. I'm just talking about math, this being the math section of this website.

Just to clarify, this forum category is called "Physics and math" And I am not confusing physics and math as I am discussing both with the threads OP.
to the rest of your post later...