The Infinitesimal - In a 3 d environment.

Am i right in thinking the infinitesimal is equal to 0.0recuring1?

 

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Quantum Heraclitus:

How?

Heraclitus, listen to me: You are making an analytic error. You CANNOT have a zero-volume sphere! It is a contradiction.

A sphere is DEFINED as having volume. Zero volume = zero radius, diameter, area, et cetera. In essence: It isn't there.

You're talking about square-circles here.

The volume can be infinitesimal, but the radius and diamter would not be. The diameter or radius cannot be an infinitesimal.

And once again: If you have a sphere with zero volume, you do not have a sphere.

You're really b utchering geometry here, my good man. You are throwing in terms which are meaningless and constructing shapes which are contradictions.

Here, look:

4/3 * pi * r^3

Radius = 0.

4/3 * pi * 0

4/3 * 0

0

Aka: You don't got no sphere...

But this is flawed: It isn't. There is no sphere here. You're making up one and putting it there with no reason. You're talking about something non-existent as if it was existent and then confusing yourself by bending it back on into itself and using it again.

 

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ok...enough...

If you agree that it is the nature of a 3 d environment to create an infinitesimal that has a curvature then YOU work out an answer to the problem that poses when magnifying a sphere to infinity.

You tell me how a sphere has a center of zero and you can tell me how this is derived. [ when zero is non-existant.]
You tell me what forms the zero distance between 4 cones touching at center with curved, concave infinitesimal points in a 3 d environment?

 

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Tell me the logic behind a mobius strip or a Kleine bottle?

 

Quantum Heraclitus:

I know very little about mobius strips or Kleine bottles. Mobius strips are essentially a twisted piece of paper taped into a loop. Kleine bottles are bottles bent back in on themselves. That's about my extent of knowledge concerning them. Why do you ask?

What kind of answer are you looking for specifically? You want to know the smallest segment?

What do you mean precisely a center of zero?

The act of "touching" is the zero-distance to begin with: To say something touches, is to say there is no space between them.

 

Steve100:

Basically, yes. An infinitesimal is defined as a number which is above zero (or zero - techniaclly zero IS an infinitesimal...but it is a useless one for what we're discussing and significantly different from the other infinitesimals) which is nonetheless beneath any other number. In essence: It is the smallest one can go before reaching zero.

 

in the case of a 3 d environment does the infnitisemal have a curve or is it flat and straight [ regarding spheres]
or :
if i reduce a sphere infinitely does the resulting infinitesimal retain the form of a sphere?

 

A Mobius strip is the quotient space made by taking the 2 dimensional region \((x,y) \in [0,1] \times [0,1]\) with the equivalence relation \((0,y) \sim (1,1-y)\).

The Klein bottle is taking this quotient space and then imposing the additional equivalence relation of \((x,0) \sim (x,1)\). It too is unorientable and cannot be put int 3 dimensional space without self intersecting.

It is another example of an unorientable surface, only this time it doesn't have a boundary.

It's not. The notation isn't well defined. If you want to do infinitesimals, then you basically do as you do with complex numbers, you add a new basis element which spans your system. For instance, the hyperreals can be generated by considering an infinitesimal e, where 0<e<1/n for any n in R, e*x < 1/n for any x and n in R. Immediately you get a heirarchy such that since e*x<1/n for any x, n in R you have that e*(e*x) < e*(1/n), so \(e^{2} < x*e\) for any x in R. Similarly \(e^{n} < x*e^{m}\) for n>m in Z.

Given the heirarchy I just outlined, you can go smaller than infinitesimals, you have infi-infitesimals, which are spanned by \(e^{2}\). Then there's infi-infi-infitesimals, spanned by \(e^{3}\). An infinite tower.

Why, pray tell?

 

AlphaNumeric, thanks for posting that..very interesting!
If you were to describe an infinitesimal sphere how would you and what would be the results of a standed set of dimensions such as diameter and radius etc etc.....would it retain it's spherical shape?

and what aspect is infinitesimal?
I believe this would pose a connundrum but not impossible...

 

Quantum Heraclitus:

Yes. A sphere with an infinitesimal volume, as this would be the smallest sphere that one could make.

 

Ok make the sphere a hollow sphere and ask the same question?

 

Spheres are hollow. The term used for a 'filled in sphere' is a ball.

I'll address your post tomorrow. It's 2am and I need some sleep, too much Mathematica coding.....

 

so if the volume of the solid sphere is infinitesimal then does a zero point exist within the sphere or is it non-existant?

Also if we have a 3 d volume of infnitesimal space does it have a centre and what would the radius be ?

 

Thanks Alpha, the distinction between sphere and ball is obviously important.

 

Quantum Heraclitus:

All spheres are hollow mathematically speaking.

The volume is still infinitesimal.

What do you mean "zero point"?

No. It does not have a center. The center would indicate the ability to be further divisble.

 

So we have a spherical volume with out a center...yes?

Is this not a contradiction in terms?

 

Think of it this way:

We might think of a 1-dimensional ball; the 2-dimensional ball is the interior of a circle, and the 3-dimensional ball is the interior of a sphere.
Note that the "open ball" is not really a ball but an open interval on a larger space, or area or line. It has to do with a set (of points) being open or closed.

 

You cannot contract a sphere to a point. This is because the surface of a sphere and it's centre at distinct. The n-ball includes the centre.

Mathematicians make this disctinction in a pretty straight foeward way, an n-ball is 'contractable'. An n-sphere isn't. The proof for an n-sphere and an n-ball is pretty straight forward, but for other shapes it's not.

For instance, an n-torus (so a donut in n-dimensions) is not because it must have a hole to be a torus. Without it it's not a torus, it's a different shape.

The n-sphere and n-ball are building blocks of other shapes. Suppose I stick the n-hemi-ball (so half an n-ball, like a mound) on the side of a shape. I don't change it's topological properties because it's a smooth deformation, I can contract it back to it's original shape without breaking the surface. But if I stick an n-hemisphere on it, it's like putting a cap over a portion of the surface. In order to return it to it's shape, I must 'pop' it, like bubble wrap.

All of this can be done in any number of dimensions for any shape in a rigorous way.

 

Surely the infinitessimal is equal to 0, in the same way as 0.9recurring equals 1.
Therefore the sphere doesn't exist.
Am I right?

 

Infinitesimals have nothing to do with 0.9r=1. The Real numbers do not include infinitesimals other than 0. 0.9r is a well defined concept without infinitesimals and it's equal to 1.

If you do work in a space with infinitesimals then 0.9r still equals 1, but 'to within an infinitesimal amount' but 'to within a zero amount'.

Infinitesimals are trotted out by the "0.9r isn't equal to 1" crowd when typically they don't understand the Reals, never mind hyperreals.