Your notion of maths guiding reality rings true. The octonion algebra is new to me, but seems really complex. It would seem that one can always explain reality by adding hidden dimensions and multiverses and octonions.

But a simpler reality of just two complex dimensions seems to have all the complexity needed to explain the universe. Isn't simpler better?

    I am putting my response you wrote in my area here. I noticed you had not gotten to it or responded yet. I have have been getting caught up with work after having spent the week down with the flu.

    Thanks for the positive response. I will comment more tomorrow. It is getting a bit late for a long writing session. This is in line with the approach with Raamsdonk that spacetime is built from entanglements. I wrote an answer on stack exchange that connects with this perspective with regards to Hawking radiation.

    The open world emerges from the existence of gauge hair and BPS charge. The hair of the black hole is entangled with particles in a vast number of other black holes in the universe. In the unique situation where there are two black holes maximally entangled one would have a complete Einstein-Rosen bridge connection to the interior of the other black hole in this other world. The openness comes from the fact the spatial surface in region I has an ambiguity with respect to being connected to other cosmology or the black hole interior region. For maximal entangled Bhs one in principle can avoid the singularity and travel around to other worlds.

    I will write more tomorrow. I have been recovering from influenza, and today it the first day I feel not utterly horrible. I am not that familiar with DGP model, but I will see what I can make of it.

    Cheers LC

    PS --- I also noticed something bizarre happened. I was ranked around 17 or so from the top, but a lot of people were I think tanked with a vote of one. Now I am #4. It looks like you went down with that; too bad. I was not me who did that. I would not give a lot of papers the lowest score like that.

    LC

      The person who did this has to be an essay author. I would suspect they are somebody who wrote an essay that appeared in the group Feb 21.

      LC

      Jonathan,

      Is it possible for there to be a non-integer value for the number of dimensions associated with a space? If so, how?

      Trolls? The trolls in "Frozen" were amusing. The trolls in "Trolls" are amusing. These trolls are just small. That makes it easy to hide.

      Best Regards and Good Luck,

      Gary Simpson

        Interesting question. I have this idea cooking up about Hausdorff dimensions with Ising chains. This might be a way of working a renormalization scaling for estimating the 3-dim Ising problem.

        LC

        The correct answer is always the best one Steve...

        Effective theories need only consider the parameters they incorporate, or attempt to explain. But this is why GR and QM don't agree at the fringes. Explaining the whole universe at once requires one to work in the area of fundamental theories - which is fundamentally more difficult.

        All the Best,

        Jonathan

        I can see the relevance of the mathematical model, referencing the Mandelbrot Set, Jonathan. I do mention the repeating patterns displayed at every scale of the universe and can see that math is a compact expression of a symmetry hard to visualize otherwise. It makes England's idea of replication at every scale easier to imagine, imagining something like the Menger sponge.

        In the realm of aesthetics, I have a clearer appreciation of your piece.

        Jim Hoover

        Hi Gary,

        Non-integer dimension arises in causal structure theories of quantum gravity, which is referred to simply as a running D - compared to the case where D = n, where n = 0,1,2,3... Of course; this gives space a fractional dimension, and makes it a fractal - along the way - which is simply how surface roughness evolves into a new (whole) dimension or extent. So briefly; fractional dimension arises because of folding of space in the microscale. As Lawrence alludes, the Hausdorff dimension evolves.

        One can also think of this as relating to emergent spacetime, because if the observed properties of space and time, this means that intermediate values are accessible between the onset of geometrogenesis and the current era. One can also see this as connected with a different root dimension for the microscale and macroscale, as with Rainbow Gravity (which was explored by Magueijo, Amelino-Camelia, and others). If space is 2-d at the Planck scale and 3-d at the common scale; what is it in between?

        Lastly; this is a broad feature of what is called bi-metric gravity. There are many formulations in that family. There's too much to say simply, but as the name implies there are two co-existent descriptions of space - to deal with the weak-field and strong-field, low-energy and high-energy regime, or common scale and microscale, and so on.

        All the Best,

        Jonathan

        I can tell that your essay is very well written. I wish I could follow along on all the details, but I think it would require of me a serious commitment to read up quite a lot on the relevant mathematics. At this stage, I can only point out that the essays by Yanofsky and Simpson are also talking a lot about the mathematical terms that are mentioned in your essay. All the best and thanks for your help. Cheers!

          I am happy you enjoyed it Willy..

          Even some hardcore Math folks shy away from the octonions, but they are worth the effort for those with the skills. My main point is that if you go high enough up the chain, evolutive properties in Math are easy to observe.

          All the Best,

          Jonathan

          I thank you Lawrence,

          I am not worried about you being a vote rigger. It does appear there must be folks who enter the contest on a pretense, then vote for their friends, while casting down those who might be serious competition. It is discouraging to see such childish behavior.

          Sorry to hear about the flu. I have had lingering cold symptoms myself. I think it might be worth elaborating on the hairy ball theorem, as an explanation of the above. The conventional sphere S2 has hair that can't all flow one way, so is non-trivial, while S3 can be combed all the way around - because it is parallelizable.

          More later,

          Jonathan

          Thanks for this follow-up Jim..

          The Mandelbrot formula is very compact, compared to the complexity of the object. And it elegantly displays the replication of similar forms at different levels of scale. I think it is the progression of form, which is the most relevant, but the regularity of its repetitive form is impressive.

          I am glad I gave you something to chew on.

          All the Best,

          Jonathan

          Always a pleasure to hear from you Steve..

          This time, I do feature spheres of various dimensions, in my essay. I think it is a very useful construction that nature views as fundamental.

          All the Best,

          Jonathan

          For those visiting here..

          I thank you for your interest in my work and I'll try to reply to all comments eventually, to most as soon as there is time. I will also attempt to read as many as possible of the other essays in the time allotted.

          I will be systematically working from the earlier submissions to the present, while trying to honor those who visit my essay with an earlier reading and feedback. I'll be trying to also look for topics of my own special interest and authors whose work I respect. Lastly; I will look to the community and public ratings to find essays to read, during the final run up.

          While I don't often give out a rating of 10, I have not given a score lower than 5 so far this contest. I will give out a decent rating if an essay is either well-written or makes a powerful point, with extra points for essays that do both. Mostly; I will be looking for quality of writing, strength of premises, and compelling logic. Essay that have all of that will be rated highly.

          Good Luck to Everyone!

          Jonathan

          Hi, Jonathan. This is a great essay and I agree on the role of the octonions etc. It's good that you concentrated on that theme.

          You mentioned consciousness a few times but did not go into much detail. What is your view on consciousness? Is it an illusion, some new physics or what?

            Thanks greatly Phil,

            I could have written a very different essay, emphasizing the progression of consciousness angle. But I wanted to show that those Maths shape Physics in a definite way. The octonions generate a hierarchy in levels of abstraction, though, such that a pattern like what Young describes can be spelled out in many different ways to form sentences. I've written out several dozen at this point.

            One, open, as multiplicity and formless nothingness, finds peace in true relation, and knows all as self. Oneness, leads to openness, as-ifness, multiplicity, and so on. The rest reflect the theme of a book by Briggs and Peat 'Turbulent Mirror' that there is a far shore to chaos where order re-emerges, and life exploits that border region greatly. This one is a sort of personal credo. But you could also put it in religious terms.

            One being, Goddess and God, begat manyness, then complexity beyond reckoning, to find the missing pieces of themselves, and become as one again. So then it becomes a love story featuring the male and female persona of the divine, arising from the order found in the octonions. But it's a bit controversial that the divine feminine emerges first; don't you think?

            I know that sounds pretty wild, but I think the stages in the progression within octonion algebra have moods, so that the octonions could be used to craft a fully subjective and qualitative relational database, as a counterpart to the powerful objective and quantitative data classification systems we have today. But this only works because it is also something emergent in the properties of non-associative Maths.

            All the Best,

            Jonathan

            I wanted to add this..

            If viewed as a generator of conceptual hierarchy; the octonions appear to spell out the entire arc of learning - from knowing only that you exist, to knowing everything about the universe and seeing it all as a part of yourself. This last stage is sort of like the experience described in T.S. Eliot's classic poem "Little Gidding" to return back home and see it anew, as if for the first time, but now in fullness with the perspective gained from exploring the world.

            For the record; I don't think I am making this up, but re-discovering an ancient truth. The same message appears again and again in Mythology, where the Zen Ox fables and the Hero's Journey tell the same tale. Arthur Young expounds somewhat on this Math-Mythology connection, but I think that he never got to make the explicit connection with the octonions - which is what jumps out for me.

            All the Best,

            Jonathan

            Jonathan,

            If you get the time, I would be interested in your thoughts on my essay.

            Jim