Ghostly Dimensions by Niklaus Buehlmann

Dear Niklaus,

I enjoyed reading this essay. I have a prejudice as an inventor and artist, that a correct theory of nature would be very simple and should be capable of visual representation in three dimensions. I think the time dimension is illusory and a physics without it may well be possible. If so your arguments, so nicely illustrated have a ring (or sphere) of truth to them. A couple of other essays here also conjure Zeno-like arguments to prove that reality is digital. You may be on the right track. By the way your graphics reminded me o Vasarely op-art.

Ray, fractals start out as simple and increase in complexity and Nature probably works that way as I have argued in my own paper here in fqxi. Niklaus' reductionist arguments points that way too. Cheers

Vladimir

Dear Niclaus,

A nice essay with something comprehensimple, simple, and deep. It offers an explanation of how within a singularity we can have the imformation of a whole universe (if imformation density is limiteless that it isn' t).

Regards, narsep (Ioannis Hadjidakis)

Dear Sir,

I thank you very much for your interesting comment. I hope that I will find time to write also some comment to your work. So my answers will be incomplete at this time.

you wrote:

. We can not accept your logic for the simple reason that when dealing with micro lengths, any approximation cannot be overlooked or used to conceive imaginary ideas. If we are in a position to measure the length of the diameter and the circumference of a circle, we must realize that there is a difference between them that varies at a fixed ratio. Our inability to measure these precisely does not make them equal or introduce ghostly concepts.

My first remark is that my argument was not restricted to micro lengths, but rather included macro lengths (and areas) too.

In my area- example there was not the question to measure them precisely, but that for every level of inobservability we cannot know, whether the area of a real existing square, after it was tested and accepted, should be calculated simply as (side-length)^2 or if to this quantity the area of its inner circle should be added too: a very big difference, in whose interval no rationally reasonable ordering seems possible. Therefore the analog model for such cases seems inadequate for me.

My line of argumentation is the following: the theories are o.k. for me as long as they are theories.

But their claim is to explain reality. This they can do to an astonishingly high degree, but there seem to be limitations. The critical point is the anchor of the theories in reality. Here I think some aspects of classical 3-dimensional space appear. At least some postulates about forminvariances in spacetime are required. As theories with the claim to explain reality, they should also be measured in their ability to give reliable answers to more specialised aspects of objects on earth. The examples were a test for me. The formalism of modern physics is a ingenious way to incorporate all the necessary priciples e.g. Lorenz-Transforms, but they have also to be tested in everyday life: my test was in the subset of simple geometry, where the conclusion from the examples was that an analog model would be overstressed.

The expression ghostly I used to design the fact, that the limit was in a subspace which did not contain this dimension, whereas reality could include it to a very small (typically inobservable) degree.

Best regards

Niklaus Buehlmann

Dear Ray,

many thanks for your valuable comment. You wrote that I could have made my point better by filling the square with varying sizes of space-filling kissing circles and that these circles imply the importance of fractals and that this ghostly dimension may have fractal properties

With your suggestion you are probably right, but I see some problems, which my approach does not have.

I wanted to stay as simple as possible and therefore I used a uniformly convergent series of continuous functions in the plane, which converges to an interval. Ghostly means, that the limit is only in one dimension (interval) whereas the individual functions have small (even inobservable) parts in another dimension. All curves have the same length, but this length is different from the length oft he limit (Interval). Here we are clearly not in a fractal situation, as all the curves and the limit have both Hausdorff dimension 1 and Lebesgue covering dimension 1.

In your suggestion with fractals I think that we run into non-rectifiability of the limit (if it exists). Although it is interesting, it does not deliver the conclusions I wanted.

Thanks for mentioning the typing errors: they were produced implicitly by my implemented German text correcting system which does not like the English language.

I will carefully study your work and also write some comment. I am very interested to see the approach of another rocket scientist to these fundamental questions!

Best regards

Niklaus

Dear Niklaus,

Your geometrical point was well-made. I performed Cosmic Ray research with NASA's Marshall Space Flight Center for a couple of Summers while teaching College Physics during the school year. Now I manage my family's retail business in Florida. Edwin Eugene Klingman also worked at the Marshall Space Flight Center (before me - he is closer to your age - you are both 10-15 years older than me), and he has been part of California's high-tech industry for years.

Have Fun!

Dr. Cosmic Ray

Dear Sir,

We derive everything from fundamental principles that can be verified in laboratory experiments and whose results of measurement under identical conditions remain invariant at all times and places. We call this reality and the theories behind these as physics. Thus, we had derived perceptible aspects of dimension from the properties of electromagnetic fields that are responsible for ocular perception. Similarly, irrespective of whether you are dealing with a macro system or a micro system, as long as you accept that measurement is a system of comparison between similars, there will be no quantum weirdness. After all there is no precise definition as to what is a quantum system. If discreteness is the criteria, then even the galaxies or the universe as a whole will be a quantum system. If size is the criteria, then where is the dividing line and what is the basis for that division? We classify macro and quantum particles based on their nature of coupling. When two particles of different natures couple in a way so as to merge their identity to create a third particle with a different identity (like those in a compound), they are called quantum particles. When the two merged particles retain their individual identity (like those in a mixture), they are called macro particles. In any case, all measuring devices are classical instruments. Thus, the question of the area of inobservability does not arise. You can compare an observable only with another observable having similar characteristics. You cannot measure air with a measuring rod.

You say: "whether the area of a real existing square, after it was tested and accepted, should be calculated simply as (side-length)^2 or if to this quantity the area of its inner circle should be added too". Area is non-linear accumulation of length, where the length closes in on itself. This is because linearity of length is derived from the properties of positive charge while closing in the length in non-linear (opposite - confining) fashion is derived from the properties of negative charge. For using as a unit, we take the simplest length of a unit measure. With this unit of area, if we compare the area of any square or for that matter any other closed area, the result will follow an invariant pattern. This can be proved in any laboratory experiment. Circle belongs to a different category, where the relationship between the diameter and circumference is related to the orbital theory and which manifests as pi, whose value is linked to the properties of electromagnetic radiation. We have theoretically derived the value of pi from the properties of orbitals and the mechanism of perception. The mechanism and units in both cases being different, (briefly discussed in our essay) there is no reason to mix up both.

You say: "the theories are o.k. for me as long as they are theories". We fail to understand this. As long as they are not proved, they are only postulates. They become theory only after they are proven, which means they correspond to reality irrespective of when, where and how many times the experiment is repeated. Then where is the confusion? We agree that examples are tests for postulates for making them theories. But we are questioning you other assumptions. What is the basis for such assumptions? There are alternative explanations for each of these without involving quantum weirdness and exotic manipulation of numbers or logically inconsistent manipulations like renormalization or conversion of Schrödinger's equation in so-called one dimension to three dimensions.

Regards,

basudeba.

Dear Vladimir,

many thanks for your comment. I see it as you do: a theory which will interprete reality should also allow tests within our everyday phenomena. Originally I experimented with sequences of functions

of the type: f[x]=c*Sin[d*x]

whose arc-length over an inteval [a,b] can be calculated with some formulas that contain the elliptic integral of the second kind. With some sequences of such functions, which converge uniformly to zero on some interval [a,b] you can obtain simlar results.

But there comes your argument: simplicity is better for the explanation of reality: therefore I have chosen the sequences of circles and spheres which require only few elemetary mathematics. As these illustrations produce some esthetical effects, I thought it would be more interesting for the readers, than a lot of formulas of the above type.

For me the examples are interesting as we cannot know e.g. the area of a real quadratic plane mirror, even after checking it with the most advanced instruments: we cannot know, whether its

area is s*s or s*s plus the area of the inscribed inner circle. Whith this very bad situation of reality the analog model is very questionable for me!

Best regards

Niklaus

Dear Ioannis,

many thanks for your comment. I think the examples show some kind of singularities oft he following kind: we have a sequence of uniformly convergent functions which converge to zero on some interval (or square) and whose arc-lengths (or areas) are equal, but substantially larger than the arc-length (or area) of the limiting function. Then it is a simple translation of the definition of limes into speech, to describe what happens: we can not measure the lengths (or areas) of very simple real existing objects, even after having checked them with the best instruments we have!

Best regards

Niklaus

Dear Niklaus,

Interesting geometric analysis of the distinction between digital and continuous systems. The illustrations were well-constructed.

Best wishes,

Paul

Paul Halpern, The Discreet Charm of the Discrete

Niklaus

I have only just come to your essay and wish I has sooner. I love your geometrical approach and proofs. A top score coming! When you refer to the;

"paradoxical examples where two lengths and two areas cannot be compared."

I have derived the same in application to two speeds. i.e. two reference frames.

This proved to be at the heart of our misunderstanding on relativity. I really hope you can read my essay (and score it!) as I believe you will have the conceptual ability and understanding to see the solution.

I've also just posted the logical analysis in the string.

Please do comment. http://fqxi.org/community/forum/topic/803

Very many thanks and best wishes.

Peter

Dear Peter,

many thanks for your interesting and valuable comment.

I have to refresh some parts of the theories, as I had studied physics 1968-75, but I will try to understand also your contribution and give a good rating for it.

Many thanks and best wishes

Niklaus

Dear Sir,

thank you for your interesting comment. I see still a problem:

You wrote:"In any case all measuring devices are classical instruments."

I think that these instruments have to be considered within a historical process of refinement.

Therefore, given e.g. the testing and measuring possibilities a hundred years ago, an object that looked like a piece of a stright line could have passed as a piece of a stright line and its legth measured accordingly, whereas today, with the refined possibilities, it could be identified as a certain member of the sequence of semicircles which I have mentioned and its length would be completely different.

Here I see the problem: How can we define a length in the real world, such that it is not changing during history?

Best regards

Niklaus Buehlmann