Is there a way to define the center of an infinite sphere?

[Amine_prince]

i was daydreaming in class today and i came through something that i was quite unable to prove on my own ...
i am not sure if it's fully related to math or physics but here it is , it's no big deal .
imagine you have a point in 3 dimensional space . and a number N of objects of Random volumes and shapes placed at random positions in the 3 dimensional space .

now imagine that there is a force that will pull all the objects to that single point as if it accumulating them on top of they each other all around that point . as if the point has infinite gravity . after all the objects are accumulated on the point and everything reaches a stable state where no object is moving . a new object S is formed , where S is the object made by the accumulation of all the objects around the point .

i want to prove that if N -> oo then S is a perfect sphere .

 

[Amine_prince]

the thing that i am interested about regarding this , is the fact that the initial position of the objects in space should not at all matter .
that is supposed to be a Propriety of "Random" and infinity being together . though i cannot figure if this is correct .or if it can be proven .

 

[Amine_prince]

well ... if you all the objects have equal volumes then the problem would become simple . also if the volumes of the objects would be limited by a Real number L it would also be simple to demonstrate . but with the volumes being fully random it turns hard .

 

[pbuk]

A perfect sphere is an imaginary mathematical construct; your question is about the real, physical world. The answer is that you don't need an infinite number of objects, you simply need enough mass so that gravity is strong enough to overcome the forces holding the objects together; apart from some surface irregularities they will then form a sphere.

This is a useful concept and is now part of the definition of a planet.

 

[micromass]

and a number N of objects of Random volumes and shapes

What do you mean with "random volume"? Do you mean that every two numbers have equal probability of being the volume of the object? In this case, a random volume doesn't exist. So your problem is ill-posed.

 

[Amine_prince]

A perfect sphere is an imaginary mathematical construct; your question is about the real, physical world. The answer is that you don't need an infinite number of objects, you simply need enough mass so that gravity is strong enough to overcome the forces holding the objects together; apart from some surface irregularities they will then form a sphere.

This is a useful concept and is now part of the definition of a planet.

interesting . i spent a long time in the past attempting to understand what a perfect sphere is . what i did get to essentially , is that a curve or any curved shape cannot exist in a limited 3 dimensional space with a unit that has a specific shape , (unlike a cube , it can exist if the unit is a cube) . for a sphere to exist in 3 dimensional space , the unit has to be infinitely small , though then distance would make no sense and you cannot construct anything out of it . though imagine an object of any shape having a real size or volume , for example a volume of 1 m*m*m in space . regardless of the shape of the object . if you have an infinite number of it and you attempt to construct a sphere out of that whole number . the object is infinitely small when compared to the sphere so then here , the singularity is not the fact that the unit is infinitely small but the fact the number of unitary objects is infinite , so i have the right to call the formed sphere a perfect one .
that's what drove me to start thinking about random shapes and volumes in the first place . what can be disregarded when dealing with infinity and shaping singular objects .
(of course i also suppose that the 3 dimensional space is infinite)

i am having a hard time attempting to translate this from my main language to english .

 

[Amine_prince]

What do you mean with "random volume"? Do you mean that every two numbers have equal probability of being the volume of the object? In this case, a random volume doesn't exist. So your problem is ill-posed.

the main reason behind this is the fact that i wanted to see if i would be able to disregard the variations of volume .i do not understand what you mean with
"every two numbers have equal probability of being the volume of the object"

 

[pbuk]

interesting . i spent a long time in the past attempting to understand what a perfect sphere is . what i did get to essentially , is that a curve or any curved shape cannot exist in a limited 3 dimensional space with a unit that has a specific shape , (unlike a cube , it can exist if the unit is a cube) . for a sphere to exist in 3 dimensional space , the unit has to be infinitely small , though then distance would make no sense and you cannot construct anything out of it . though imagine an object of any shape having a real size or volume , for example a volume of 1 m*m*m in space . regardless of the shape of the object . if you have an infinite number of it and you attempt to construct a sphere out of that whole number . the object is infinitely small when compared to the sphere so then here , the singularity is not the fact that the unit is infinitely small but the fact the number of unitary objects is infinite , so i have the right to call the formed sphere a perfect one .
that's what drove me to start thinking about random shapes and volumes in the first place . what can be disregarded when dealing with infinity and shaping singular objects .
(of course i also suppose that the 3 dimensional space is infinite)

i am having a hard time attempting to translate this from my main language to english .

A (perfect) sphere is a 3 dimensional surface containing every point that has distance r (the radius) from a single point (the centre). Note that in maths, a sphere is not a solid object, we call the solid bounded within a sphere a ball.

In the real, physical world it is fine to call a solid object which has this shape a sphere. But in the real world objects are composed of particles of finite size (which are constantly moving in an unpredictable way) and so it is never possible for them to exactly form a cube, sphere or any other perfect shape.

 

[Amine_prince]

i know , with "sphere" i mean the solid object bounded by the external surface of the sphere or the sphere itself .
here i suppose that i have a unit for 3 dimensional space that i construct things with .
it's not the physical world we are speaking about here .

 

[pbuk]

i do not understand what you mean with "every two numbers have equal probability of being the volume of the object"

Remember this is the "General Math" forum and we deal with things mathematically here. Micromass was stating that your problem is not well posed mathematically; explaining why would probably be a diversion from the main topic of your thread.

 

[Amine_prince]

Remember this is the "General Math" forum and we deal with things mathematically here. Micromass was stating that your problem is not well posed mathematically; explaining why would probably be a diversion from the main topic of your thread.

i was lost at first , couldn't figure where to post this. i didn't post this in "physics" section because the 3 dimensional space that i am mentioning here is not the real one .

 

[pbuk]

it's not the physical world we are speaking about here .

Oh in that case there is no problem. We consider the limiting case as the size of the objects tends towards 0 - this overcomes problems associated with infinite numbers of objects.

Unless constrained, objects will move from a position of higher (gravitational) potential to lower. It can be shown that total gravitational potential is minimised when the objects form a sphere. As the size of the objects tends towards 0 the sphere becomes perfect.

 

[Amine_prince]

Oh in that case there is no problem. We consider the limiting case as the size of the objects tends towards 0 - this overcomes problems associated with infinite numbers of objects.

Unless constrained, objects will move from a position of higher (gravitational) potential to lower. It can be shown that total gravitational potential is minimised when the objects form a sphere. As the size of the objects tends towards 0 the sphere becomes perfect.

gread ! as the size of the objects tends to 0 the object tends to become a perfect sphere . now what about if the number of objects tends to infinity ?

 

[pbuk]

gread ! as the size of the objects tends to 0 the object tends to become a perfect sphere . now what about if the number of objects tends to infinity ?

As the number of objects tends to infinity their size must tend towards zero, otherwise the solid becomes infinite and its shape is not defined.

 

[Amine_prince]

As the number of objects tends to infinity their size must tend towards zero, otherwise the solid becomes infinite and its shape is not defined.

we need to apply more singularities to this . now l'ets go back to the same 3 dimensional sphere . if you get the 3 dimensional sphere and infinitely enlarge it , it will be an object of infinite size and that's right but it will still be a sphere , it just happens to be infinitely big .
in fact , since the sphere is formed with an infinite number of infinitely small objects it must be acceptable that for an object to go from being infinitely small to having a real volume like 1 m*m*m it has to be infinitely enlarged , so a singularity has to be applied to break the first one .

there is a certain singular enlargement that would turn the infinitely small unit to a unit that has a real size , the same enlargement would turn the perfect sphere to a sphere (that i like to call perfect) but that also happens to be infinite in size .

so essentially we reversed the singularity , first the sphere had a real size and the objects forming it were infinitely small , but then the sphere became the one that's infinitely big (the singular element) and the constructing object became the real element due to it's new real volume .

both of them should be called perfect spheres.

 

[pbuk]

we need to apply more singularities to this.

In maths we use the term "singularity" when something is not defined. I don't understand what you mean by "apply a singularity".

now l'ets go back to the same 3 dimensional sphere . if you get the 3 dimensional sphere and infinitely enlarge it , it will be an object of infinite size and that's right but it will still be a sphere , it just happens to be infinitely big .

That is not correct. How would you define the centre of such a sphere?

in fact , since the sphere is formed with an infinite number of infinitely small objects it must be acceptable that for an object to go from being infinitely small to having a real volume like 1 m*m*m it has to be infinitely enlarged , so a singularity has to be applied to break the first one .

there is a certain singular enlargement that would turn the infinitely small unit to a unit that has a real size , the same enlargement would turn the perfect sphere to a sphere (that i like to call perfect) but that also happens to be infinite in size .

so essentially we reversed the singularity , first the sphere had a real size and the objects forming it were infinitely small , but then the sphere became the one that's infinitely big (the singular element) and the constructing object became the real element due to it's new real volume .

both of them should be called perfect spheres.

None of this makes any sense mathematically. In order to develop a better understanding of how it is possible to deal consistently with infinite and infinitesimal quantities you should study calculus.

 

[Amine_prince]

In maths we use the term "singularity" when something is not defined. I don't understand what you mean by "apply a singularity".

That is not correct. How would you define the centre of such a sphere?

None of this makes any sense mathematically. In order to develop a better understanding of how it is possible to deal consistently with infinite and infinitesimal quantities you should study calculus.

thank you . i am not taking "Topology" but i really like it .
do you think it's possible for me to study it on the internet ? thank you

 

[pbuk]

thank you . i am not taking "Topology" but i really like it .
do you think it's possible for me to study it on the internet ? thank you

I am not sure that there is a good toplogy course on the internet, although what you are probably looking for is differential geometry rather than topology.

But before you get there you need a sound understanding of analysis. Again I don't know of a great foundation for this on the internet, but I suppose https://www.khanacademy.org/math/precalculus/limit-topic-precalc goes most of the way.

 

[Amine_prince]

i was listening to a mathematician , he called himself a "Model theorist" i don't know what that means , and he said it's ok to define a number k . that is superior than any other number in N or R , but that is not superior than k itself or k+1 ... etc .

the thing is , people these days allow a sphere to be placed in a 3d space , even though the smallest unit to construct that sphere is infinitely small . relative to the smallest unit in the sphere the Radius of the sphere would be infinite .

so that if i let the radius of the sphere be k . then the smallest unit that constructs the sphere would be a finite number that is not infinitely small .
one can ask how will the center of that sphere be defined . well you allow a sphere of radius R to exist while R is relatively infinite relative to the used unit same as K .
and both the spheres en-capsule an infinite amount of points whether the radius is real or not , since if the radius is real the unit is a singularity and if the radius is K then the unit is finite .

with unit i mean the smallest thing the space allows to exist .

 

[jbriggs444]

with unit i mean the smallest thing the space allows to exist .

You really really need to learn some real analysis. None of this is sensible. Most is just word salad.

In the real numbers there is no "smallest thing". For every x > 0 there is a y > 0 such that y < x. The infinite and the infinitesimal do not work the way you think they do.

 

[Amine_prince]

You really really need to learn some real analysis. None of this is sensible. Most is just word salad.

In the real numbers there is no "smallest thing". For every x > 0 there is a y > 0 such that y < x. The infinite and the infinitesimal do not work the way you think they do.

Sorry for wasting your time sir , i am just a high-school student and i don't know the right terms , the thing is whenever i ask my teachers they ask me to ask a university professor and when i ask the professional people they don't understand me .

 

[pbuk]

Sorry for wasting your time sir , i am just a high-school student and i don't know the right terms , the thing is whenever i ask my teachers they ask me to ask a university professor and when i ask the professional people they don't understand me .

That must be very frustrating. Perhaps you could try changing the way you ask questions and listen to answers?

i am having a hard time attempting to translate this from my main language to english .

This is why in mathematics we define terms precisely so that everyone agrees they mean the same thing. But you have used words which are precisely defined to mean different things, even when you have been told what they actually mean - see for example my explanations my explanations of how a sphere is defined and the difference between a ball and a sphere in post #8, and my pointing out what the term "singularity" means in post #16.

Will it help if I explain a little more where you are going wrong in your post #19?

i was listening to a mathematician , he called himself a "Model theorist" i don't know what that means , and he said it's ok to define a number k . that is superior than any other number in N or R , but that is not superior than k itself or k+1 ... etc .

Yes you can define such numbers - one set that includes them is the hyperreals. They have some interesting properties, but you can't use them to describe a geometry in which something you would understand as a sphere (or ball) exists. In order that you have a geometry that works in a way consistent with our everyday experience of the real world, you need to restrict lengths to those that can be measured by the real numbers $\mathbb R$: this is known as the Archimedean Property.

, even though the smallest unit to construct that sphere is infinitely small .

But you have already been told that a ball is not constructed mathematically by adding pieces together. Instead a sphere is defined as the set of all points equidistant from a single point, the centre.

relative to the smallest unit in the sphere the Radius of the sphere would be infinite .

so that if i let the radius of the sphere be k . then the smallest unit that constructs the sphere would be a finite number that is not infinitely small .

But $k \notin \mathbb R$.

one can ask how will the center of that sphere be defined.

When someone who is trying to help you learn says something like "How would you define the centre of such a sphere?" it is because trying to answer the question will help you will realize where you are going wrong: do not ignore the hint. In this case I hoped that you would remember that I told you that the centre of a sphere is the point that is equidistant from all of the points on the surface of the sphere. If the surface of the sphere is at infinity, then the centre could be where I am sitting because every point on the surface would be infinitely far from me - or equally the centre could be on the Moon. If we can't define where the centre of a sphere is then it is not a sphere; we must therefore discard the notion of an infinite sphere (in Euclidean space).

. well you allow a sphere of radius R to exist...

A sure sign that someone needs to listen more and talk less is when they use language like this - do not put words into other people's mouths.

with unit i mean the smallest thing the space allows to exist .

As jbriggs says, there is no smallest thing (greater than 0) in $\mathbb R$.

 

[Amine_prince]

That must be very frustrating. Perhaps you could try changing the way you ask questions and listen to answers?This is why in mathematics we define terms precisely so that everyone agrees they mean the same thing. But you have used words which are precisely defined to mean different things, even when you have been told what they actually mean - see for example my explanations my explanations of how a sphere is defined and the difference between a ball and a sphere in post #8, and my pointing out what the term "singularity" means in post #16.

Will it help if I explain a little more where you are going wrong in your post #19?

Yes you can define such numbers - one set that includes them is the hyperreals. They have some interesting properties, but you can't use them to describe a geometry in which something you would understand as a sphere (or ball) exists. In order that you have a geometry that works in a way consistent with our everyday experience of the real world, you need to restrict lengths to those that can be measured by the real numbers $\mathbb R$: this is known as the Archimedean Property.But you have already been told that a ball is not constructed mathematically by adding pieces together. Instead a sphere is defined as the set of all points equidistant from a single point, the centre.But $k \notin \mathbb R$.When someone who is trying to help you learn says something like "How would you define the centre of such a sphere?" it is because trying to answer the question will help you will realize where you are going wrong: do not ignore the hint. In this case I hoped that you would remember that I told you that the centre of a sphere is the point that is equidistant from all of the points on the surface of the sphere. If the surface of the sphere is at infinity, then the centre could be where I am sitting because every point on the surface would be infinitely far from me - or equally the centre could be on the Moon. If we can't define where the centre of a sphere is then it is not a sphere; we must therefore discard the notion of an infinite sphere (in Euclidean space).A sure sign that someone needs to listen more and talk less is when they use language like this - do not put words into other people's mouths.As jbriggs says, there is no smallest thing (greater than 0) in $\mathbb R$.

thank you very much sir , that helps a lot .