Hi Marina,

You wrote:

> I'm giving you a high rate it deserves.

Thank you!

> I know that topologists call it a 4-sphere, emphasizing the 4-dimensionality of the object as a whole, while mathematicians and physicists call it a 3-sphere, being mainly interested in its 3-dimensional surface.

Yes, I apologise to the topologists. I had to pick one convention or the other for the essay. Since Wikipedia calls it a 3-sphere I went with that.

> 1. we live in 4 spatial dimensions, while being aware of only 3. I discussed how this can be explained in my last year essay (did not do too good of a job, I'm afraid).

I will have a look at that after the contest.

> I wonder, would it be possible, using your Landscape Test, to *prove* that matter is actually 4-dimensional and that we live in a 4D universe, crawling on its 3D surface?

If the highpoints of the landscape are geometrically aligned (as I suggest the statistics show) then the question is: how is this possible if the points are moving great distances relative to each other, over geologic time?

The simplest answer is that what we are seeing when we look at the familiar 3D world is a projection from a higher dimensional system, akin to Plato's shadows on the cave wall. This is because, when shifting such a geometric projection, cocircularity and coincidence are preserved while position of vertices is not.

You can see an analogy by imagining a wire frame cube (3D) projected by a light source onto a plane (2D). As you rotate the cube, the corners will move around in the 2D image and the lengths of the line segments connecting them will grow and shrink. But the line segments will stay straight and the coincidence of lines at the corners will still be apparent in the 2D image. So if we see a system that behaves like such an image, we can guess that there is a higher dimensional structure behind it.

> Also, with your amazing knowledge and expertise in this area, where did you see such an explicit description of such a 4D universe model?

The model is my own. For many years I have been working on the observational side... getting better and better data related to highpoints, and coaxing the statistical analysis to suggest the underlying geometry. I began seriously thinking about the theoretical side in January this year, and writing the essay in May and June helped me to flesh out the simulation model with supporting papers. I have had a habit of collecting and organizing interesting papers for many years, so I could go to my collection for most of what I needed.

> The other *proof* I was looking for is that 4-space is unique among all N-spaces (N>2) in the sense that it has the highest degree of all conceivable symmetries. Because this would serve as yet another rationale why our universe is 4D.

I think the hypersphere plays an important organizing role, but there are other structures involved. For example, even though any system of great circles is symmetric (the structure at antipodes are mirror images), the Earth itself is not. In fact highpoints do not ever occur at antipodal points. And the cosmos as a whole does not appear to have a mirror symmetry. So there is another geometric factor that breaks such symmetries.

> Thank you very much again for inviting me to read your very interesting essay.

You are quite welcome. Thanks for asking about the Landscape test.

> I understand how difficult it was for you to cut it down to 9 pages, after the first 30-page draft -- and yet to took a risk with the last section. Why?

To me, cosmology is more than what physicists study under the name of "physical cosmology". I felt it was important to remind the reader that the effort to understand our cosmos is very ancient and that traditional views are not, by necessity, wrong. Yet I think the way forward is careful mathematical analysis of observable data, and I hope I have suggested a methodology and a picture that can ultimately reconcile traditional and modern perspectives.

Hugh

I also appreciate the importance of the 3-sphere while I come at things in an entirely different way [basically I take the opposite stance] :)

I'd appreciate your thoughts on my essay.

Cheers and best of luck,

Jennifer

    Hi Chidi,

    I will have to take another look after the contest... I still have many essays to rate before tomorrow, since I saved that task until I had read as many as I could.

    Hugh

    Hi Charles,

    Thanks for doing this... it is great to have a summary post and a kind of index!

    Hugh

    Hi Helmut,

    You wrote:

    > It is clear, that a different transformation between explicate space and implicate information occurs if spacetime is seen differently. And I do that...

    Over time, I hope to model several different possibilities for the transformation, within the general picture I described. I will take another look at your work on this.

    > Despite these objections, I think it is a meaningful and important concern to have a conscious look at these transformative processes. So, I scored your paper very high.

    Thank you!

    Hugh

    Hi Torsten,

    You wrote:

    > Also from the topological point of view, the 3-sphere is the root of all compact 3-manifolds (one can obtain every compact 3-manifold by surgery -or cut and paste- along a knot or link).

    I wonder if we might be able to see dynamical processes in the 3-sphere induce such knots... and the knots will take us a long way: Lou Kauffman has described the basic connections between knots and physics in several papers and his book.

    > PS: So, you got a high vote more than one week ago.

    Thank you!

    Hugh

    Hi Jennifer,

    Thanks... I made some comments on your blog back on July 22.

    Hugh

    Hugh,

    I like your idea of a digital simulation model in which you view the implicate cosmos as "a dynamical fluid on the surface of a hypersphere". Good food for thought!

    You wrote that the "the explicate world of 'It' arises from the implicate world of 'Bit'. For Bohm, knowledge of the implicate order is acquired by insight. What we perceive is implicate order unfolded as explicate order. If so, it is explicate order, as the contents of our consciousness, that is epistemic (composed of "bits") and the underlying implicate order that is ontic ("it").

    Also see my response to your comments to my essay "A Complex Conjugate Bit and It".

    Best wishes,

    Richard

      Hi Richard,

      > If so, it is explicate order, as the contents of our consciousness, that is epistemic (composed of "bits") and the underlying implicate order that is ontic ("it").

      Rather than a binary epistemic/ontic distinction, I would break these up into three categories: (1) an observer's physical environment, the explicate, (2) an objective, shared reality, the implicate, and (3) the contents of our consciousness.

      My category (2) may be best described by "ontic", but notice that (1) is the only thing we can measure, and what we commonly take for "it". Categories (1) and (2) are closely linked by via a (mathematical) transformation, which I think of as (1) from (2) or "It from Bit".

      To properly account for the phenomenology of consciousness, I see Mind as an architectural layer (in the sense of software architecture) below Matter (i.e. not emergent from Matter as in the conventional view). This is what I call "Us" and is what informs (2). Thus my category (3) would correspond to the epistemic.

      (1) Explicate, It

      from (2) Implicate, Bit, ontic

      from (3) Consciousness, Us, epistemic

      Hugh

      Hi Hugh,

      I found this to be an interesting essay. I'm not really sold on the more mystical overtones at the end, but the idea of a search for systematic effects that might arise from the source code of the Universe is a good one. Maybe such a search is a low-likelihood, high-payoff undertaking, like SETI. Rather than looking for evidence that highpoints on Earth are in some way correlated, it seems more reasonable to me to search for correlations at galactic or super-galactic scales. If the idea that most computational power is given to the places close to observers, then I would expect the CMBR to be quite "crude", and the detail in the planet I live on to be very high.

      Cheers,

      Sundance

        Hi Hugh

        thank you for taking time to answer my questions. I really appreciate it!

        In regard to this:

        > The other *proof* I was looking for is that 4-space is unique among all N-spaces (N>2) in the sense that it has the highest degree of all conceivable symmetries.

        you wrote:

        "I think the hypersphere plays an important organizing role, but there are other structures involved. For example, even though any system of great circles is symmetric (the structure at antipodes are mirror images), the Earth itself is not. In fact highpoints do not ever occur at antipodal points. And the cosmos as a whole does not appear to have a mirror symmetry. So there is another geometric factor that breaks such symmetries."

        I'm afraid I meant it in a different context. Here I meant it precisely just as a Euclidean flat 4-space. 4D houses the highest number of regular polytopes (6), while all higher spaces have only 3. Based on this fact and perhaps some others --I'm looking for them-- I want to find a proof that 4D corresponds to the lowest energy state for a... N-dimensional vibrating structure that seeks to conserve its energy. Here dimensionality is one of the properties of this structure -- i.e. it can dynamically be 'compressed' into higher dimensional state or 'relax' by expanding into a lower-dimensional structure/state.

        I visualize it as a dynamic vibrating wire frame of N-dimensions, N>4. I believe that if one takes such a N-dimensional structure, undisturbed, it will naturally settle by expanding into a 4D configuration, precisely because, topologically, it offers the highest degree of symmetries.

        I wonder if you know where I could find the answer to this :)

        Thanks a lot for all your feedback,

        -Marina

        Hi Sundance,

        > Rather than looking for evidence that highpoints on Earth are in some way correlated, it seems more reasonable to me to search for correlations at galactic or super-galactic scales.

        There was interest in this sort of analysis of the CMB about 5 years ago. Jean-Pierre Luminet and also Jeff Weeks have described techniques and results. They basically look for regions of space that appear to be replicated in more than one direction, suggesting the shape of space is, for example, dodecahedral.

        If we want to extend the fractal creasing Landscape Test to the cosmological scale we must decide what constitutes a "highpoint". For example, we might use density, or energy. We also have to decide what we want to use as a "center". Using the Earth as a center is convenient but seems overly anthropocentric.

        Nevertheless, about ten years ago I tested a gamma ray burst catalog for evidence of alignment, but found that source positions were not sufficiently resolved to tell anything. The digital elevation data on Earth had much higher resolution and has a natural geometric center so that is what I ended up using for later analysis.

        > If the idea that most computational power is given to the places close to observers, then I would expect the CMBR to be quite "crude", and the detail in the planet I live on to be very high.

        Whether or not it is true in some objective sense, this is true in terms of our state of knowledge. There is a lot of interest now in finding ways to explain CMB anisotropy of various sorts. But the resolution of current CMB data is good enough to find complex creasing patterns, while the detail we have of our planet's terrain is much higher.

        Hugh

        Sorry.. the last sentence should read:

        But the resolution of current CMB data is *not* good enough to find complex creasing patterns, while the detail we have of our planet's terrain is much higher.

        Hi Marina,

        I am not sure that a lower dimension would provide a more relaxed environment for high-D structures (wouldn't they feel "squashed" by the restriction?), but the lower dimensions seem to provide more opportunities for interesting structures to appear. I can think of some links that might give you some ideas for your research. As far as dimensions go, the ones that get the most attention seem to be 4, 8 and 24.

        In my essay, I discuss the 4D case, which corresponds to S3 and the quaternions. One interesting structure in 4 space that comes to mind is the very symmetrical 24-cell which is discussed by Frans Marcelis on several pages. If you are thinking about models for QM, Philip Gibbs has written about how the 24-cell is related to systems of 2-qubits and also systems of 4-qubits.

        The 8D case is related to S7 and the octonions. John Baez discusses 8 dimensions in his Aug 3, 2013 post here, and a related structure here.

        He previously made the case for 24 dimensions here, including an argument related to harmonic oscillators.

        When it comes to a wire frame compressing and expanding, I think of Robert Gray's jitterbug motion between the cuboctahedron and octahedron, but his all takes place in 3D (i.e. does not change dimension). It is worth mentioning that the cuboctahedron has many nice properties and was a favorite of Bucky Fuller. Perhaps you can generalize this motion for the 24-cell, using the Hopf fibration that Frans Marcelis discusses.

        Hugh

        Hugh

        Good to see you're soaring high on the list -- and I suffered a series of 1's :(.

        As usual, you are a source of wonderful info. Thank you! Still, I have my doubts. I have a very strong gut feeling --and I wish I had something more that this-- that 4D is magic. Think about it, why would Universe be 4D? Why not 5? or pick any other number? I am sure that this is because, topologically, it offers the max symmetries. Mathematicians don't get it, because they are mostly dealing with vectors or fields but an nD _object_, which is, essentially a chunk of n-space, has inherent limitations on number of dimensions it lives in and remain... I forget now, unfortunately, but maybe you will remember, there is some kind of limit, something to surface ratio, that reaches infinity already at n=7. There are topological constrains like this that limit the number of 'real' dimensions -- as in reality to actually a rather small number, Of them, 4D is it. There gotta be many good reasons why universe is 4D.

        Thank you for your feedback and good luck in the finals,

        -Marina

        Hi Jonathan,

        Thanks, good to see you in the top 40, as I "rated you highly". (I guess that's the catch phrase in the contest...)

        The process in the last week has been interesting to watch... it gave me the impression of 183 turtles in a bucket, clambering over each other to get to the top.

        But many of my favorites made it, and I think the organizers have at least 20 good essays to choose from.

        Hugh

        Hi Hugh,

        thank you for your reply. I'm afraid I did not explain myself clearly. You wrote: "I am not sure that a lower dimension would provide a more relaxed environment for high-D structures (wouldn't they feel "squashed" by the restriction?)"

        Not at all. Let me explain. I'm talking about a dynamic, vibrating structure, for which dimensionality is one of its attributes. Going from higher to a lower dimension does not 'squash' it but *unfolds* it. Take an example of a tesseract. Its surface is 8 cubes. You can stack these 8 cubes in 3D, in effect rearranging the 4-volume into a 3-volume; and the length the edge of this 'dimensionally reduced' object becomes twice as long. That's how, in fact, I understand the expansion of space, i.e. the higher-dimensional structure 'relaxes' into a (n-1) structure, which increases the length of its edges.

        Also, when I spoke about your Landscape Test, I had quantum theory in mind, not the macro world. I am not clear yet --have to reread your essay-- how exactly you apply it to the Earth surface.

        .

        I'm pursuing the answer to the question: Why the universe is 4D? Why not 5, or pick any other number. I strongly believe that there is a good reason for this; and my hunch is that, topologically, 4D offers the maximum number of symmetries => in 4D the structure of space finds its lowest energy state.

        When mathematicians and physicists deal with higher dimensions, the objects they consider are limited mostly to points, vectors and fields consisting of points and vectors. Those are 0 or 1-dimensional objects. But a real object, in real n-space, is essentially a segment of that space. For example, a cloud is a familiar object in 3D and so is a billiard ball. Their densities may differ, but ultimately, both are just segments of space with clearly delineated boundaries. These boundaries is what makes them 'real objects' as opposed to points and vectors -- even when a boundary wraps 'emptiness'.

        Now, regarding 'real objects' in n-D, there is a topological theorem, the details of which I can't recall now, but hope you could remember -- and it says that some important ratio of... surface to.. 'something' -? reaches the limit of infinity already at n=7. This implies that there are not that many 'real spaces' that can contain 'real objects', and 4D is very special.

        Again, there _is_ a clear, logical and unambiguous answer to the question 'Why 4D?' I am looking for it.

        Thank you very much for all your input,

        -Marina

        Hi Hugh

        congratulations on making the cut. I left a post in the end of my thread in your blog above and hope very much that we could continue our discussion about 'Why 4D?'

        Thanks a lot for all your input,

        -Marina

        Write a Reply...