Can pi be reduced to a discrete number?

[bgivens@xtra.co.nz]

Imagine a polygon with sides that are only a Planks Length long. Since no physical quality can be shorter than a planks length then this polygon could also qualify as being a circle. Using its sides as a polygon one may be able to calculate it area of the circle/polygon without using the constant pi. Then once is area is know the pi can be derived from the equation pi = A/r2.

Anything smaller than smaller than a Plank length becomes meaningless. The maximum number of sides of a polygon/circle are then = c/plank length. This is the absolute maximum number of sides for a polygon and so perhaps it is the point at which the shape could be called a circle.

The problem here is getting an exact value for Plank lenght. I don't that this is possible. But it may be possible to prove that a circle with sides that are Plank lenght long will produce a exact value for pi. This then could explain why pi is an infinite series of nonrepeating numbers - i.e. because it is based on a circle with a infinite number of sides. This then would imply that circles are not really round but have sides - that circles are actually factual.

[HallsofIvy]

Imagine a polygon with sides that are only a Planks Length long. Since no physical quality can be shorter than a planks length then this polygon could also qualify as being a circle. Using its sides as a polygon one may be able to calculate it area of the circle/polygon without using the constant pi. Then once is area is know the pi can be derived from the equation pi = A/r2.

Anything smaller than smaller than a Plank length becomes meaningless. The maximum number of sides of a polygon/circle are then = c/plank length. This is the absolute maximum number of sides for a polygon and so perhaps it is the point at which the shape could be called a circle.

The problem here is getting an exact value for Plank lenght. I don't that this is possible. But it may be possible to prove that a circle with sides that are Plank lenght long will produce a exact value for pi. This then could explain why pi is an infinite series of nonrepeating numbers - i.e. because it is based on a circle with a infinite number of sides. This then would imply that circles are not really round but have sides - that circles are actually factual.
That would imply that you are not talking about circles! Circles are mathematical objects which are defined as the set of points at a given distance from a given point. Circles cannot have "sides". Plank length is a physical quantity, not a mathematical entity.

What you have shown is that mathematical circles cannot exist in the physical world- but that's true of all mathematical objects and has nothing to do with your claim.

[bgivens@xtra.co.nz]

That would imply that you are not talking about circles! Circles are mathematical objects which are defined as the set of points at a given distance from a given point. Circles cannot have "sides". Plank length is a physical quantity, not a mathematical entity.

What you have shown is that mathematical circles cannot exist in the physical world- but that's true of all mathematical objects and has nothing to do with your claim.


Thanks for your reply.


1. The area of other shapes such as squares can be known exactly. This raises the question of why shapes using the quantity pi cannot be known exactly
2. The definition of a “point” is inadequate in that one cannot measure from a dimensionless location.
3. Plank length is the smallest possible real measure of space and is a space where quantum complementarity effects dominate.
4.
From wikipedia.org/wiki/Complementarity_%28physics%29

“A profound aspect of Complementarity is that it not only applies to measurability or knowability of some property of a physical entity, but more importantly it applies to the limitations of that physical entity’s very manifestation of the property in the physical world. All properties of physical entities exist only in pairs, which Bohr described as complementary or conjugate pairs (-which are also Fourier transform pairs). Physical reality is determined and defined by manifestations of properties which are limited by trade-offs between these complementary pairs. For example, an electron can manifest a greater and greater accuracy of its position only in even trade for a complementary loss in accuracy of manifesting its momentum. This means that there is a limitation on the precision with which an electron can possess (i.e., manifest) position, since an infinitely precise position would dictate that its manifested momentum would be infinitely imprecise, or undefined (i.e., non-manifest or not possessed), which is not possible. The ultimate limitations in precision of property manifestations are quantified by the Heisenberg uncertainty principle and Planck units. Complementarity and Uncertainty dictate that all properties and actions in the physical world are therefore non-deterministic to some degree.”

The complementarity properties of a “point” are location and size.

5. The above suggests that the exact value of pi cannot be determined because defining a point as a dimensionless object at a exact location violates the principle of complementarity by defining both aspects of size and location exactly. This would be like exactly defining the velocity and location of an electron.
6. All of the above suggest that “points” have a dual nature.

[HallsofIvy]

Thanks for your reply.


1. The area of other shapes such as squares can be known exactly. This raises the question of why shapes using the quantity pi cannot be known exactly
I have no idea why you think pi is NOT "known exactly". It certainly is known exactly. It is just that in a base 10 numeration system it cannot be written in terms of a finite number of decimal digits, like most (in a technical sense, "almost all) numbers.

2. The definition of a “point” is inadequate in that one cannot measure from a dimensionless location.
What DEFINITION of "point" are you talking about? "Point" is an UNDEFINED term. The existence of undefined terms is a crucial part of mathematics.

3. Plank length is the smallest possible real measure of space and is a space where quantum complementarity effects dominate.
Did you really mean to say the "Plank length" is a space? In any case this is physics and has nothing to do with mathematics. Do you understand the difference?

4.
From wikipedia.org/wiki/Complementarity_%28physics%29

“A profound aspect of Complementarity is that it not only applies to measurability or knowability of some property of a physical entity, but more importantly it applies to the limitations of that physical entity’s very manifestation of the property in the physical world. All properties of physical entities exist only in pairs, which Bohr described as complementary or conjugate pairs (-which are also Fourier transform pairs). Physical reality is determined and defined by manifestations of properties which are limited by trade-offs between these complementary pairs. For example, an electron can manifest a greater and greater accuracy of its position only in even trade for a complementary loss in accuracy of manifesting its momentum. This means that there is a limitation on the precision with which an electron can possess (i.e., manifest) position, since an infinitely precise position would dictate that its manifested momentum would be infinitely imprecise, or undefined (i.e., non-manifest or not possessed), which is not possible. The ultimate limitations in precision of property manifestations are quantified by the Heisenberg uncertainty principle and Planck units. Complementarity and Uncertainty dictate that all properties and actions in the physical world are therefore non-deterministic to some degree.”

The complementarity properties of a “point” are location and size.
Again, that is physics and has nothing to do with mathematics.

5. The above suggests that the exact value of pi cannot be determined because defining a point as a dimensionless object at a exact location violates the principle of complementarity by defining both aspects of size and location exactly. This would be like exactly defining the velocity and location of an electron.
The above suggests nothing about pi. One more, time: "pi" is a mathematical concept and mathematics is NOT physics.

6. All of the above suggest that “points” have a dual nature.
All of the above suggests that you simply do not know what a "point" is, what "pi" is, or, indeed, what mathematics is.

[bgivens@xtra.co.nz]

Hi thanks for your reply

I have no idea why you think pi is NOT "known exactly". It certainly is known exactly. It is just that in a base 10 numeration system it cannot be written in terms of a finite number of decimal digits, like most (in a technical sense, "almost all) numbers

Why is this? In what system does it have a finite number of digits? I think you are talking waffle.

What DEFINITION of "point" are you talking about? "Point" is an UNDEFINED term. The existence of undefined terms is a crucial part of mathematics.

I was using the one I was taught at school:
Point - a zero-dimensional figure; while usually left undefined, has four main representions - the dot, the node, the location, and the ordered pair of numbers.


What is your mathematical definition?

[bgivens@xtra.co.nz]

The following is a plank length circle. Its radius is one Plank length and its sides are one Plank length. It is then made up of six Plank length equilateral triangles. The area of each triangle is 1/2bh. (Plank length = 1.6.163 x10-23m). The area of the circle is 6 times the area of each triangle = 7.8372 x 10-23m2.

Area of a circle = pi r2
Pi = A/r2
Pi = 7.8372/2.612 = 3.00

[HallsofIvy]

Hi thanks for your reply



Why is this? In what system does it have a finite number of digits? I think you are talking waffle.
No, I'm talking mathematics. Clearly to you "mathematics" and "waffle" are the same thing. In a numeration system with base pi, pi itself would be represented by "1". But why are digits important to you? Numbers do not depend solely upon how they are represented.

.

I was using the one I was taught at school:
Point - a zero-dimensional figure; while usually left undefined, has four main representions - the dot, the node, the location, and the ordered pair of numbers.
None of those are "points', they are, as you say representations. And, "a zero-dimensional figure" only makes sense if you have already defined "dimension" and "figure". How were they defined?


What is your mathematical definition?
What part of "undefined" do you not understand?

[bgivens@xtra.co.nz]

None of those are "points', they are, as you say representations. And, "a zero-dimensional figure" only makes sense if you have already defined "dimension" and "figure". How were they defined?

I was hoping just to throw some ideas around.