The Infinitesimal - In a 3 d environment.

When discussing the use of the infinitesimal one thing seems absolutely certain about it.
That is the infinitesimal must be > zero and never equal zero.
That it is the smallest possible dimension available mathematically.

Assuming that this is fundamental to it's use and a correct assessment it can be claimed that when using a single dimension whether that be distance, time there appears to be no problem however when it comes to 3 dimensional volume it does seem to get a bit sticky.

The gendanken shown uses a 3 dimensional sphere.
The centre of this sphere can be either zero dimensional or of an infinitesimal dimension.
Now as infinitesimal is the smallest we can go in math it appears that at the centrer of our sphere is a infinitesimal volume.
By virtue of 3 dimensional geometry that volume must be spherical

So in essense we have an infinitesimal sphere at the center of our spherical object.
The first diagram show a segment taken from the sphere and demonstrates a segement of our infinitesimal spherical shape at "end a"

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The second diagram shows it included in our sphere:

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Keeping mind that the surface area of this inner sphere must incorporate the infinitesimal.

How could we express the volume of this inner sphere as a ratio.
And what would that ratio be?

note:
As it is impossible to grant a value to the volume I would assume it can only be described as a ratio so that regardless of how much the sphere is infinitely magnified the ratio of volume to inner surface would stay the same.
I am anticipating a result like 1: 4 or 1 : 3

I guess using a sphere that is hollow with a 2 dimensional skin on it would achieve the same ratio.
say a sphere of this type that is 10 meters in diameter.
can you express the volume of such a sphere as a ratio to surface?

My apologies if my terminology is terrible and possibly someone else could word the question better and care to share with the board.

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As theoretically the inner spheres volume should be zero the ratio eventuallly derived will show zero as it's solution.

 

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A sphere with a radius of 0 has a volume of 0, and a surface area of 0, it's called a point.

It's also known as some other thing - like a 0-ball, or a 0-dimensional sphere or something.

 

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Quantum Heraclitus:

Here is the formula for the volume of a sphere: 4/3 * pi * r^3

Here is for the area of a sphere: 4 * pi * r^2

Let me put in that for 10 metres. I'll show my work so you can follow.

Volume: 4/3 * 3.1415 * 5^3
4/3 * 3.1415 * 125
4/3 * 392.6875
523.58333333333333333333333333333

Area: 4 * 3.1415 * 5^2
4 * 3.1415 * 25
4 *78.5375
314.15

So the ratio is approximately 1:.60 roughly

 

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well how about that:
The golden ratio is approximately 1.6180339887
greek "phi"

 

Well, that's certainly an odd coincidence...

I'll have to check my book ont he Golden Ratio that I bought to see whether they referenced spheres in it. I don't think they did, only the Golden Rectangle.

 

From Wikipedia, the free encyclopedia
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Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes; Golden numbers, an indicator of years in astronomy and calendar studies; or the Golden Rule.

The golden section is a line segment sectioned into two according to the golden ratio. The total length a+b is to the longer segment a as a is to the shorter segment b.In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887.[1]

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter (phi). The figure of a golden section illustrates the geometric relationship that defines this constant.

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Given that this ratio is decribing a space or volume that is non-existant this is quite amazing.
So non-relative zero or absolutely nothing could be described by using the Golden ratio...hmmmmmm


and the Golden mean in philosophy is the "the felicitous middle between two extremes"
something and nothing....hmmmmm

 

Quantum Heraclitus:

It's also notable that the infinitesimal aspect of the sphere would have to be the radius, not the diameter, surface area, et cetera. As a diameter can by definition be divided in half to find the radius. So the smallest sphere would have a diameter of 2(infinitesimal).

 

As an aside:
You will also note with some interest that traditionally the Golden ratio has linked mathematics to philosophy and the arts. Da Vinci was an expert and a genius in it's use.

mixing science with asthetics and that which is subjective such as beauty and so on.
Organic Symetry are the words I use to explain the Golden ratio.
"A cloud appears whole and complete yet non-symmetrical yet fully symmetrical in an organic sense"

 

Quantum Heraclitus:

Check out Mario Livio's "The Golden Ratio". It is a good, layman's book on the subject, which discusses the things we're discussing here on it.

 

disagrees with approach...sorry

the only thing existing is the surface area, everything else is non-existant in this context.
The sphere actually has zero volume so therefore it's diameter and radius is zero yet we are left with an infiniteismal surface area....

And thats the twist and why it works.

nothing can be smaller than an infinitesimal and as non-relative zero doesn't exist it cannot be smaller.

 

yet it can be [after confirmation of hypothesis is achieved] expressed by default using the golden ratio.....by golly what a story...the irony

 

Quantum Heraclitus:

This is contradictory. If you have zero volume, it's not 3d. Thus it is not a sphere. It's a circle.

A sphere is defined as having volume. You can not have a sphere whose volume is 0.

 

Quantum Heraclitus:

Give me the volume of a cube which has only 3 sides.

 

sorry I need to think about that one...at this point no idea..?

btw I have just skimmed the wiki inclusion on Phi... and wow this may be a rather interesting coincedence, I was expecting pi I must admit but phi will do nicely. apparently pronouced [phee or fee]
The great Pyramid of Giza Egypt.....hmmmmmm....

 

exactly! The sphere does not exist but only the surface does.
remember nothing can be smaller than the infinitesimal.....nothing....

 

we can only imagine that the sphere exists....

 

Quantum Heraclitus:

If Wiki says this, then it is false. Livio refutes that claim easily. Phi is not found in the pyramids in any meaningful sense.

I agree that nothing is smaller than an infinitesimal. But you can't have a sphere with no inside volume. It would not be a sphere: It'd be a circle.

The reason why I asked you the area of a cube with only three sides is that is an absurd question: A cube has six sides, not three. You can't give me answers to objects which are contradictory.

 

well tell me what your solution is? the questions raised in this thread...

 

And I thought you had sometniing meaningfull to state....ha

Of course a sphere has to have volume but we are not talking about a real sphere but we are talking about real inner surface of the external surface of a sphere...