What Happens With Form As Size Goes To 0, Another question of limits

I have a following puzzle.

Let us have a simplice of volume-tetrahedron. Its volume is some number. Let us have some process where we require this volume in limit to reach 0 ( the volume) - e.g differential dP/dV.

Question: what happens with the form? what will be the form of the 0 volume?
Tetrahedron, sphere, point (what is that?) etc?

The same question applies to triangle as area simplice as its area is reduced to 0, and straight line as 1D simplice as its lenght is contracted to 0.

Will it still be a line with 0 length, or a point with 0 dimensionality?

Thank You in advance.

AFAIK, when you shrink something to zero size it cannot have any geometric properties that distinguish it from other 'types' of points.

For example, we can define a process where you take a triangle (or a tetrahedron, it doesn't really matter) and consider the vertexes getting closer together until they meet. When this happens the sides will have zero length and the vertexes will be on top of one another.

Now think about a square or cube in the same way. In the limit that the length of the sides goes to zero they are exactly the same object.

In this way you can argue a point with zero dimensionality and a line in the limit that the length goes to zero are identical.

Now this isn't actually very physical, because of the uncertainty principle so objects like strings can never shrink to zero size. I can't claim to know much about string theory though. Ask me again in a few years.

You have two situations

1. The shape has non-zero length sizes. It has form and volume regardless of how samll you make it.

2. The object has 0 length sides, so has no form. It is a point.

The limit of the volume is 0, but that is meaningless to the physical property of an ever shrinking object. The limit cannot be reached for any real value. There is no "transition" as such.

Just my opinion.

An interesting example of infintesimal form is the mandelbrot set. Fractint can be used to zoom in on the boundary of the set. When one has zoomed to 10^26, the original set has expanded to the side of the Universe. there are animations of zooms to 10^89. There is no end. Of course, there is no concept of limit that can be applied to the set and the region around it as a whole. Just the trajectories of points within the set (which are either cyclic, converging, or chaotic).

http://www.fractal-animation.net/ufvp.html

Search for E+89 (the original wmv is the best, with sound)

So You are all saying the form will vanish together with dimensions.

hmm. How about law of e.g. conservation of information and derivatives/integrals?

The size of the volume does not contain information about its form, neither do the length of its edges.

It is the functional dependency between the variables involved in defining form that is important.

Now we know that the limit of a function depends on its form - e.g if function is x^2, than d(x+dx)^2)/dx when dx reaches 0 is 2x.

That means, even when size vanish, the ratio related to the properties of the curve x^2 remains.

Now, the same must be true in case of reducing 1 dimension, 2 dimension 3 dimensional from some volume to 0.

It is like writing : df(x,y,z) / dxdydz = ???? This something that is left will be related to form, so I can not accept that form vanishes without trace.

if we apply integration to it, we may even reconstruct the previous form with accuracy limited by constant, so in fact, we can reconstruct the FORM exactly, while we can not reconstruct the VOLUME - we do not know from which volume we started decreasing the tetrahedron; but we know it was tetrahedron from the derivative which is left after volume goes to 0.

Once we also know initial conditions, we are able both to regain the Volume or other parameters together with FORM by the means of remembering the integration interval and start point- that is definite integral.

So I answered my question. Form remains, volume does not. Thank You for input

This post has been edited by Ivars on Nov 8 2007, 01:06 PM

@Ivars
I guess for me to understand that I will need a differentiatible form for a tetrahedron.

y = x^2 is a continuous function.

A sphere is a continuous function x^2 + y^2 + z^2 = 1

A cube is not continuous.

So, while your analogy is intriguing, I wonder how to implement it. Maybe you transform it into another space first?

Anyway, if you think about the derivitive as the operation as dxdydz APPROACHES zero, you still can have form and the volume also APPROACHES zero. In the limit, it is zero, but it seems like taking a limit is not a conserving operation.

For example, what is the limit of x^2 as x goes to zero. All form is lost. What is the limit of the volume of a cube as xyz all go to zero. It's a different sort of operation.

So, I think a cube can shrink forever, and will never go to zero. It cannot shrink to zero.

The 0.9r = 1 argument is similar. The limit of the sequence 0.9 0.99 0.999 ... is 1. But, 1 is not a member of the sequence. The limit of a shrinking cube is not a cube.

My math sucks, I'm just guessing.

hej MeBigGuy

I guess You make it up from from continuous Fourier transform, and than take derivative from that as tetrahedron or cube volume goes to 0.

In 2 D, you can take triangular function , its Fourier transform and than derivative from that as dA ( area) go to 0.

I do not know what is Fourier transform of dA, but something will be definitely left, as Fourier transform of e.g triangular function involves sin^2 (pi w)/(piw)^2 which definitely does not vanish after differentiation.

So the form of shape with sharp corners is preserved in it Fourier transform derivatives as its volume or area goes to 0.

Which means its FORM actually is phase dependent superposition of waves.

Now Fourier transform is not the only way to represent some shape, there could be other orthogonal basis - which leads to a conclusion that form of cube or tetrahedron is actually hidden in phase information between its constituents, not the actual shape of orthogonal functions used.

Ivars if you think of it as being of 'scales' all of it disappears :)

form defines function.
A shell is as much the emptiness 'inside' it as the 'form' defining it.

so while i find your thoughts really cool i would like it to go one step further.

Welcome to the house of illusions

hej yor_on

Welcome:)

I think all this exactly definable in mathematical terms still but may require some twisting of maths.

e.g. taking of futher derivatives of x^2 eliminates form after 2 steps, while x^3/2 has infinite depth of forms. This seems strange, not to say wrong. On other hand, by taking fractional derivatives if x^2 which are not multiples of 2 we again can continue till infinity.

So the depth of FORM becomes dependant on space we deal with it in and the way we reduce the VOLUME when taking limit.

The question is then is there an absolute mathematical space where each unique form will have its ultimate description, or , from other point of view, are there some fundamental non-reducible forms every other form consists of.

Math seems to be illusion anyway, but FORM and VOLUME seems to be reality. So perhaps math is not an illusion afterall, or it is real illusion, or reality is illusion, whatever pleases us best as long as it works to explain logic behind workings of Nature.

Sorry, but I don't buy any of your fourier transform stuff. If you believe it, show me with the math.

I repeat

If you think about the derivitive as the operation as dxdydz APPROACHES zero, you still can have form and the volume also APPROACHES zero. In the limit, it is zero, but it seems like taking a limit is not a conserving operation.

For example, what is the limit of x^2 as x goes to zero. All form is lost. What is the limit of the volume of a cube as xyz all go to zero. Taking the limit is a different sort of operation from taking a derivitive..

I am saying that a cube can shrink forever, and will never go to zero. It cannot shrink to zero. It cannot lose its form.

If you perform an operation that makes it go to zero, it loses its form.

The only way to get something to go to zero is to multiply by zero or subtract. Taking a limit has no physical analogy. It is not a process to make something go to zero (as the original problem stated)

You mean [(x+dx)²-x²]/dx.

You mean 'differentiable'. For instance, the sphere always has a smooth shape, while a sphere has sharp points and edges.

y = |x| is continous but it isn't differentiable at x=0.

Ivars, don't even get me started on how much BS you're saying with that crap about Fourier transforms!! You complain that I push you too much when you talk about maths, that you're trying to learn, yet you persist in talking about things you have absolutely no clue about. Why do you pretend you know anything about Fourier transforms? Or simplex constructions of 'volumes' in various dimensions?

You've bearly said anything correct or viable in this thread, you're just spouted nonsense.

Ok,Ok

But the original question remains:

What happens with arbitrary FORM when volume (area, length) goes to 0? In continuous mathematical space?

Do not say it disappears as it obviously does not in case when form is defined by a differentiable function.

E.g. in 1 D FORM would be defined by a curve. A derivative of this curve is obtained by reducing both variable ( unit) and function depending on variable defining curve to 0. So length of curve and measure used goes to 0 simultaneously. But the ratio/functional dependence between these 2 remains, and allows to reconstruct curve (FORM) by integration but not the exact positioning of it.

The other question I posed is why do some functions disappear ( nth derivative =0 ) to a non-re-construable form after finite number of steps, while others does not?

Some, like e^x even does not notice the consecutive differentiations. I know mathematically why, but where does the info about FORM is stored that it is impossible to loose it no matter what?

Some, like sinx , cos x, e^ix behave cyclically, shifting phase by 90 degrees after each differentiation.

Why should some FORMs behave like that, while others disappear leaving no trace? (e.g d^3/dx^3 (x^2) = 0, and no one knows that it was x^2 this time).

This post has been edited by Ivars on Nov 10 2007, 11:03 AM

sin, cos, tan, exp are all defined by infinite series, they can never be differentiated away. Polynomials, by definition, have only finitely many terms in their Taylor expansion (they are their Taylor expansion) and so will terminate.

There's no 'conversation of information', when you differentiate something, you are asking how it smoothly changes at a particular point. Consecutive derivatives then compute how that change changes. Some functions eventually have constant changes of changes of changes of ..... of changes.

And dV is not "volume is zero", but the change in volume is very small. dP/dV means "What's the ratio of the small amount pressure changes by when I change the volume by a small amount?", not "What's the pressure when volume is zero".

Shrinking a ball to a point make it a point. There doesn't need to be any way of reconstructing it from a sphere, though at least a solid ball and a point are topologically the same (both solid, compact, simply connected spaces). A sphere and a point are not, so you, technically, cannot morph one into the other in a well defined way.

This post has been edited by AlphaNumeric on Nov 10 2007, 11:11 AM

Kind of fun to speculate around isn't it :)

" The only way to get something to go to zero is to multiply by zero or subtract. Taking a limit has no physical analogy. It is not a process to make something go to zero (as the original problem stated) "

I guess that is a mathematical statement MBG?

Because in QM everything seems to lose its form when magnified, does it not?
Or do you know any 'real' particles, keeping its 'form' under magnification?
What we see as 'form' seems more to be a matter :) of scale to me.
Which doesn't mean that there can't be 'forms' that will be intact under all circumstances.

But then you will have to introduce some 'other' factor that sort of create that 'stability'. Fractals are funny that way. Could there be a universe solely built on fractals? I don't think so, because then we would have to accept ' from nothing comes all' which frankly makes me somewhat uncomfortable ::))

This post has been edited by yor_on on Nov 10 2007, 11:27 AM

Physically perhaps dV can be very small, but when Euler defines derivatives in his Foundations of Differential Calculus he is very specific that for correct application of differentials one must understand that (p.vii):

And

So, in differentiation, the only valid way to apply it correctly is by looking at vanishing completely, not just small or relatively small increments.

I was not asking about discrete approximations to FORM, but what happens to geometrical FORM of curve, shape, volume when its length, area, volume is reduced to 0.

It says 'vanishing INCREMENT'

By your logic, y=x^2 only has a derivative when x=0. Wrong, it has a derivative when the dx in (x+dx)^2/dx goes to zero. See the difference?

There's a difference between f'(x) and f'(0). f'(x) is the derivative, f'(0) is the derivative evaluated at x=0.

If you can't understand your own quotes, why are you even bothering to read such books?

This post has been edited by AlphaNumeric on Nov 10 2007, 02:20 PM