WITHIN THE OCTONIONS

In the second to last post, I outlined a theory of quantum gravity which utilises hyperbolic geometry. But there are a few interesting elements to this that need to be looked at in more detail. On his site, John Baez gives a very accurate breakdown of what hyperbolic geometry is and how it can be constructed from simple, everyday Euclidian geometry;

Take a bunch of equilateral triangles. Glue them together so that 3 meet at each corner. You get a regular tetrahedron.

Then take a bunch of squares. Glue them together so that 3 meet at each corner. You get a cube.

Next, take a bunch of regular pentagons. Glue them together so that three meet at each corner. You get a regular dodecahedron.

This is fun! We're getting a series of Platonic solids.

Next, take a bunch of regular hexagons and glue them together so that three meet at each corner. Now the angles of the hexagons add up to 360 degrees, so we don't get a Platonic solid. Instead, we get a tiling of the plane. It looks like a honeycomb that stretches out forever in all directions.

Next, let's take a bunch of regular heptagons and glue them together so that three meet each corner. Now the angles add up to more than 360 degrees, so we get a tiling of the "hyperbolic plane". The hyperbolic plane is like the opposite of a sphere, since it's saddle-shaped at every point instead of bulging out at every point. In fact the sphere and the hyperbolic plane are the two most symmetrical forms of non-Euclidean geometry. The sphere is "positively curved", while the hyperbolic plane is "negatively curved".

We can input this series into a table (of sorts) to get a better sense of how the geometry develops;

Tetrahedron;……3D, 4 triangles,…...3 meeting at each vertex

Cube;……………3D, 6 squares,……3 meeting at each vertex

Dodecahedron;...3D, 12 pentagons, 3 " " "

Hexagonal tiling; 2D, ∞ hexagons,…3 at each vertex

Hyperbolic plane;4D, ∞ heptagons,..3 at each vertex

What is interesting here is that we go from 3 dimensions, back to 2, and then straight to 4. This corresponds with what I wrote in Direct Relativity, where an object that is accelerating beyond the speed of light goes from 3 dimensions, to 2 and finally to 4. In this sense we can think of the above series as representing what happens to space as an object accelerates, and shows that it essentially "turns itself inside out". Acceleration is equivalent to an increase in energy, this energy reaches its limit at the hexagonal 2D plane. But, were it possible to go beyond this limit more dimensions would have to be added. This means that extra-dimensions equal an increase in energy/information.

7,3 hyperbolic tiling shows 7 linesconverging on an axis.

When we think of 4-dimensional space, we often think of the three dimensions of length, breadth and height, along with a fourth spatial dimension at right angles to each of these. But what the above series shows us is that when we introduce the fourth dimension this breaks the continuity of the first 3 dimensions, into something more like six. This break in continuity occurs so that each axis remains equidistant from one another around a central point. We must therefore think of each of these 3 dimensions as being comprised of two separate values, much as we do in daily life. Height is broken into; up and down, breadth into; left and right, and length; into forwards and backwards. In reality there is no break in continuity. But from the point of view of an outside observer, like ourselves, the inclusion of a fourth spatial dimension results in an apparent disconnection of these dimensions.

The 4/7 dimensions of hyperbolic geometry

 and the quantum scale

Time is often thought of as the fourth dimension, but it should also be noted that, in some sense, time is just another spatial dimension. So the heptagonal (7,3) tiling is an accurate description of 4-dimensional space-time and the hyperbolic curvature that creates gravity. We should also note that whereas the 3 spatial dimensions each have 2 components; up/down, forwards/backwards etc. the time axis only has 1. This lack of symmetry may account for the reason why time only moves in one direction; from past to future, and never the other way.

If we were to remove one of the seven dimensions, it does not matter which, the other six will rearrange themselves into the infinitely tiled hexagonal plane of only two dimensions. Lets say, for argument sake we can remove the "up" dimension. This results in time taking the place of the up dimension, and therefore time no longer serves the function of duration in this instance. Without time we also lose the use of the 3rd dimension, because as I stated in Direct Relativity, our conscious is a 2-dimensional plane, and can only access the third dimension by moving into it (and time is a function of motion). Despite this, the hexagonal plane will continue to possess the appearance of being 3-dimensional, due to the fact that its structure still contains all the necessary data for l, b and h implicit in its geometry.

The hexagonal tiling still has all the informationof 3-dimensions implicit within it. In fact, becausethe cubes can be viewed in two different waysit may even imply a 4th dimension.

It is obvious that since time is the fourth spatial dimension it needs the other three to build on. But what is less obvious is that without time, space, as a 3-dimensional construct, also ceases to exist in a useful manner. Therefore, time and space are inextricably linked in this geometric fashion. The more intense the curvature of space the less time is seen to pass, because time is becoming space.

The hyperbolic plane shows a description of space-time viewed from the quantum scale. At this magnitude space appears flat, but in the distance the accumulative effect of matter is shown to curve space-time. Since Einstein related space-time curvature to acceleration, it appears that from the point of view of the quantum world, space is being accelerated at the macro scales. Acceleration is further linked to time dilation and therefore time ceases to exist in the meso-macro scale for all particles and hadrons in the quantum world. Could this account for some of the strange behaviour, as seen by us, on the quantum scale? I think it must.

As already noted, the hyperbolic plane is a description of how matter interacts with space-time and not anything else. One of the key principles of matter is that it is always changing. Time is the rate of change. Therefore, matter is time, or more accurately an expression of space-time. We could redraw our seven axial model of space-time, in which the primary spatial dimensions are preserved and the 4th/7th dimension becomes bent into a loop (see Fano plane). This looped time is essentially matter. In this regard, we could ask does matter create the curvature of space-time or does space-time curvature create matter? It is a chicken-and-egg phenomenon.

The Fano Plane Order 2could be used to describe thesix directions of space along withthe looped dimension of time,which is also matter.

Just like with normal perspective, when we zoom in on a particle that particle grows in size making all the other atoms and particles appear smaller. By zooming out again, all their relative sizes are made uniform, making all of space appear flat again. Continuing to zoom out at would warp space in the opposite direction, giving space the appearance of one of the Platonic Solids or simply a sphere. Again this is related to Scale Relativity where being zoomed in on an object at the quantum scale is equivalent to stopping, and zooming out the equivalent of accelerating. From this perspective flat space-time i.e. what we normally perceive space as, is equivalent to space traveling at the speed of light (see Direct Relativity). 

The heptagonal hyperbolic plane is linked to the number 168, and by extension to the Klein Quadric, PSL (2,7), the Fano Plane and Octonions. We have seen already that the Fano plane relates to space and time, as does the number 168 (24 x 7). The fact that the numbers 24 x 7 (as in 24 hours, 7 days a week) should crop up in our formulas for space time implies that our local description of time, as denoted by the rotation of the Earth around the Sun and the geometric division of the circle into 360º, is somehow intrinsic to the nature of higher-dimensions and the universe on a cosmological scale. How could this be, when we have so long been told that Earth and our solar system are no longer the centre of the universe? The answer, I think, lies within the Octonions.

We have already looked at space as a 7-dimensional construct, at least at very small scales, but the 8-dimensional algebra of the Octonions would ask us to look again. Octonions have 7 dimensions plus 1; this extra dimension is a projective point, which I maintain may be linked to consciousness. Reality, as a whole, must be projected from the conscious mind of the individual perceiving it, just as the Buddhists and mystics have been saying for thousands of years. This in itself would explain why the structure of time and space in our locale are inextricably linked with the underlying mathematical laws that the human mind has been developing over the ages.


We already spoke of time as resulting from the asymmetry of the 7 dimensions of hyperspace. But, if consciousness represents the eighth dimension then shouldn't it provide the symmetry needed to prevent the forward motion of time? Well in some ways it does; consciousness provides us with a memory with which we can skip through the events of our lives without a great deal of difficulty.


However, the arrow of time is still very much a persistent factor in our daily lives, and cannot be dealt with so easily. A more comprehensive view then would suggest that the eighth dimension is a fractal dimension containing all of the previous 7 dimensions on repeat down towards infinity. This makes sense because consciousness is the only place where we can be aware of dimensionality and of the passage of time. It is also clear that consciousness, and life in general, is a self-referencing organism that would sit well alongside our definition of a fractal system.

For more information on how the number 168 relates to time and the speed of light see my first post; Scale Relativity.