Adel,

at first to your question: a line is continuous i.e. it has a continuous number of points. It doesnt't matter how long the line is.

You need an uncountable number of points to form a line nothing less.

More later

Torsten

Dear Torsten,

Seems I am reading some of the best essays last! Very nice entry.

Your arguments are quite sound from basic physical principles and GR viewpoint.

My own arguments are more from a philosophical perspective and not as quantitatively argued but I think there are still similarities in our picture. Like you I agree time will bring out the discreteness in continuous space picture, if that is what you mean by foliation. I also discuss a linkage between Time and Existence at a discrete level, although from a philosophical view not from that of a physicist.

I very much agree with your plan to derive matter from the space, i.e. the geometrization of matter. This should be one of the next goals of physics. I myself have started thinking along this line.

Following additional insights gained from interacting with FQXi community members on my essay, I posted on my blog the judgement in the case of Atomistic Enterprises Inc. vs. Plato & Ors delivered on Jul. 28, 2013 @ 11:39 GMT.

A deserving above the average score!

Best regards,

Akinbo

    Torsten,

    Thanks for your reply. Of course the line segment has uncountable points, that's elementary. But My question was (more clearly) that if I pick infinite random points on a that line uniformly, would I cover all the points on the line? My guess is that it will not since you have irrationals and maybe some other problems. Is that correct? I hate to make this a forum, I won't feel bad if you don't answer.

    Thanks

    Adel

    Adel,

    you have to choose uncountable real random numbers uniformly. Every real number has the probability zero to choose.

    But you are right, it sounds impossible to do.

    Now to my further questions:

    There are gaps in the explaination. So, I tried to fill these gaps by thinking about. But your answer showed me, I was wrong.

    My main problem is on page 3, the red part. Up to this place everything is clear to me. But how did you get the Schrödinger equation and more importantly what is the wave function. Before you spoke about random lines etc. (and I assumed you have a probability distribution for these random lines, then the dynamics is given by a Fokker-Planck equation etc. etc.)

    Interestingly, your simulation results (Fig 3, 4 and 5) support my assumption: you simulate the probability distribution of a Fokker-Planck equation (with constraints, i.e. you put it in the box). This Fokker-Planck equation has the same ground state then the Schrödinger equation (but a probability distribution has to be positive everywhere).

    I wrote my PhD thesis about this connection (using it in the evolutionary algorithms). The correct name is Fisher-Eigen equation (a reaction diffusion equation)

    Show me where I'm stupid to follow you.

    Best

    Torsten

    Dear Akinbo,

    thanks for your words. Yes my intention is to uncover the geometric origin of matter. In particular, I try to obtain it from simple assumptions like the use of exotic smoothness structures.

    Unfortunately, I had only time to skim over your essay. There are parallels to my view and I'm glad that you notice it. I have to read it more carefully because it is more philosphically.

    Best wishes

    Torsten

    Dear Torsten,

    I thought to have rated (highly) your paper at the time I red it. But my mark seems to have been lost, may be when the system was interrupted.

    Did you have a chance to read my own essay? Any way I will give you the rate I had in mind and possibly more because I learned during this contest.

    Best wishes,

    Michel

      Dear Michel,

      yes, I read your essay but was on the vacation before I had the chance to write you. I like your geometric model very much (I rated your essay long ago with maximum score).

      Now after a second reading I have some questions:

      - You used the dessin d'enfants to visualize the contextuality. I understood the Mermin square but how did I see it in dessin d'enfant (Fig. 3b). Is it the number of half-edges (odd number) which produces the contradiction?

      - Why is the transitive action so important? In case of a non-trivial orbit, you can check every point seperately.

      One remark about the triple 0,1,infty: In the projective geometry, you always have the invariance w.r.t. the inversion operation. In the context of your model it means you have the operator and its inverse operator. Then 0 is related to infty and 1 is related to itself via inversion.

      Thanks in advance for the answers.

      Torsten

      Torsten,

      Thanks for the elucidation. I need to get more up to date with sub manifolds as gauges, but find the 'simplest idea' to be a kinetic interaction with particles with structure, not the QM assumption breaching the Law of the Reducing Middle. Now applying points, and yours now done.

      Best of luck

      Peter

      Dear Torsten,

      Thanks for your careful reading.

      1) It is not straight to see the contradiction in the dessin of Fig. 3b, I failed to see it in general (for other contexts). Also there is not a single dessin leading to Mermin's square but many, why is it so? More work is necessary. This non-bijection is general for most geometries I have tried to reconstruct from the n-simplices to projective configurations such as Desargues, Cremona-Richmund (i.e. the doily W(2) of two-qubit commutatitivity) and others.

      2) You are right that transitive action may not be a necessary condition. The geometry is constructed by having recourse to the stabilizer of each point in the permutation group relevant to the dessin.

      3) Last remark, the geometry is of the projective type not the dessin. Here you have to refer to the theory that is well explained, for example in Lando and Zvonkin (my ref. [6]).

      Torsten, please check that you vote was recorded.

      Michel

        Dear Michel,

        I voted for you a longer time ago (Mid June) and it must be recorded because I'm unable to vote again.

        But I had the same problem: many unfair votes.....

        Best Torsten

        Dear Torsten,

        I am trying to better understand your deep essay but it turns out to be quite difficult accounting for my poor knowledge of differential geometry.

        I have a naive question. The (first) Hopf fibration S^3 can be seen as the sphere bundle over the Riemann sphere S^2 with fiber S^1. Could you explain what is the sphere bundle S^2 x [0,1] that you associate to the gravitational interaction? May it be considered as some sort of lift from dessins d'enfants on S^2 to S^2 x S^0, and the latter object lives in circles on S^3, right?

        I have in mind Matlock's essay as well.

        All the best,

        Michel

        4 days later

        Dear Torsten,

        I haven't yet rated your essay and I want to know whether you have rated mine. If so/not, feel free to inform me at, bnsreenath@yahoo.co.in

        Best,

        Sreenath

        Dear torsten,

        We are at the end of this essay contest.

        In conclusion, at the question to know if Information is more fundamental than Matter, there is a good reason to answer that Matter is made of an amazing mixture of eInfo and eEnergy, at the same time.

        Matter is thus eInfo made with eEnergy rather than answer it is made with eEnergy and eInfo ; because eInfo is eEnergy, and the one does not go without the other one.

        eEnergy and eInfo are the two basic Principles of the eUniverse. Nothing can exist if it is not eEnergy, and any object is eInfo, and therefore eEnergy.

        And consequently our eReality is eInfo made with eEnergy. And the final verdict is : eReality is virtual, and virtuality is our fundamental eReality.

        Good luck to the winners,

        And see you soon, with good news on this topic, and the Theory of Everything.

        Amazigh H.

        I rated your essay.

        Please visit My essay.

          Dear Amazigh,

          interesting essay. I agree that duality is important and for me it is a cornerstone in philosophy too.

          Thats the reason why I rated you rather high.

          Best

          Torsten

          Dear Michel,

          the Hopf fibration is a non-trivial bundle but I had a trivial bundle in mind. So it is the simple cross product S^2x[0,1] but with a non-trivial foliation.

          But it has some parallels to Matlock's construction in his essay.

          All the best

          Torsten

          Dear Torsten,

          Thanks for drawing my attention to your essay. You're right that I'm interested in a geometrical explanation of accelerated expansion, and I'm glad that you picked this up from my comment on Sean Gryb's essay, and that it drew your attention to my essay. I see that we have some common interests, and will therefore read your essay with interest. In the meantime, I thought I'd direct you to the discussion thread I opened up on Ken Wharton's page (near yours at the top), because that pretty much outlines how I think a geometric description of the observed expansion rate should be handled.

          Anyway, thanks for reading my essay. I'll comment again when I've read and rated yours. I hope you do rate mine as well before tomorrow night (since you've said you found it interesting ;)).

          All the best,

          Daryl

            Dear Torsten,

            I found your essay intriguing in many ways, yet highly technical (unfortunately, I think beyond the scope of this contest in that respect). I was also puzzled why, when you've already assumed a model that is spatially homogeneous and isotropic, you would be interested in recovering inflation? It doesn't seem to fit.

            But those two things aside, you have some very interesting results, and I see a lot of overlap with what I'm thinking about, although we're approaching the problem in some ways differently. I wonder if, when the dust settles here, you'd be interested in reading through the discussion thread I began on Ken Wharton's essay page and emailing me your thoughts on what I've put there. I think I see a possibility from your essay that would really be of mutual benefit, and I imagine you'd pick that out as well from my posts.

            As I said, interesting and intriguing essay. I rated it accordingly. I look forward to hearing more from you.

            Best of luck, here and always,

            Daryl

            Greetings Torsten,

            I found your essay deeply meaningful and engaging. It was not light or easy reading, but I found myself learning something meaningful with each paragraph I read. I think I was able to understand most of your technical points, although the depth of your coverage was astounding, which attests to the clarity of your exposition. I especially like the observation that the smooth and triangulated or PL constructions of the manifold are equivalent, and find greatly satisfying the idea of a tree-like branching spacetime.

            There is much to like about your essay, and I gave you a high rating. I had started to read it at least twice before, but it was so dense with content as to take as much time as 3 or 4 lighter ones. However; I felt I needed to come back to it, before the deadline, and give you your due. You might enjoy my essay as well. I'll say more later, if there is time. Good Luck!

            All the Best,

            Jonathan