# Orthogonal and parallel dimensions: independent dimensions

On the Space-Theory of Matter (1876 Proceedings of the Cambridge Philosopical Society)
by William Kingdon Clifford(1870 p157)

Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.

I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact

(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.

(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.

(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.

(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.

I am endeavouring in a general way to explain the laws of double refraction on this hypothesis, but have not yet arrived at any results sufficiently decisive to be communicated.

When generalising we sometimes have to relate the generalisation to reality. I particularly see the need when it comes to discussing n-dimensional "spaces"!
In the subjective experience we have, i have a sense that determines orthogonality. It is located in my ears and is part of a system consisting of semi circular canals and head tilt sensors. These are affected by density and so i have a sense of gravity.

Using this sense i particularly pick out right angles, perpendiculars, and by induction i find that there are only 3 perpendiculars that are independent of each other . This is what is called orthogonal but the key attribute is independence maintained by parallel motion . If i maintain these two conditions then i can have any number of independent axes.

Thus an n dimensional space refers to the rays of a sphere and the arcs of a sphere providing motion is maintained parallel to them.

The difficulty with the arcs of a sphere is though "parallel to another arc" motion is possible the arc motion itself is instantaneously changing tangents which are parallel to any of the radials, so strictly speaking an arc is not independent of the other radials. Thus an n dimensional set would indeed look like a kind of spider web, at least the polygonal shape would locally conform to the requirements of independence and parallel motion. Nevertheless the arc is still independent to its originating radial for most of its journey around the great circle, only the 2 tangents parallel to the radial are discountable.

Now in contravariant and covariant vectors usually independence is chosen by parallel motions to the two vectors and dependence is chosen by perpendicularity to the 2 vectors. This distinction disappears when the 2 vectors are themselves orthogonal, so independence is desirable but in "transform between basis" variation dependence is invoked. What this means really is that we cannot move even a minute rotation from a basis system without engaging dependently with another. Rotation has this inherent property which links everything together at one part of the motion periodically.

So changing Basis is changing dimensions and the consequences of being linked to every dimension of motion is that the spinning object remains stable in equilibrium. However an outside push can disturb the equilibrium and send the body spinning in the pushed direction. the interaction of the spin with the motion and the frictional pressures lead to an unpredictable behaviour pattern, again characteristic of some multidimensional interactions.

The tumbling of an object is a very unpredictable behaviour again because of the multidimensional interaction,but it should be noted that the dimensional change do seem to be quantized in the sense that certain trochoidal patterns dominate. This may reflect the boundary conditions of the origin of the motion or the internal dynamics of the object or both. In any case certain motions appear to be"damped" while others resonate wildly, and my previous discussion on modularity is relevant. The periodicity of a rotation supports this quantized behaviour, and retains, mangles and reconstitutes initial conditions. Thus quanta themselves reveal data about initial conditions for regional signal patterns, by an analysis of quanta distributions and spatial locations and intensities.

Why the interest in independence? Particularly when finding a solution set to resolve a situation into independent genrators means a specific solution can be found. However sometimes a specific class of solutions is sought and this is usually the kind of result for partial differential equations.

The other use of independence is in reference frames. Having independent axes means we can resolve motions uniquely. One of the things necessary for a reference frame to utilise independence is to have a flip algorithm which changes the sectors used for measuring according to the independence criteria.

When we think of reference frames we tend to be taught to think of position, static position. We are taught to ignore the action and motion we make to position or plot the point, we lose the "vector" in the geometry. Our brain is loaded with number and we do not recognise the measuring that is going on, we do not realise the rotation that is involved, So what i write down when i notate an ordered pair or triple is not even the half of it!

Motion sequent analysis makes this very clear.

Motion sequent analysis is particularly relevant to a fractal generator, which allows sequent by sequent exploration of a vector function or a dynamic magnitude and intensity function. The colouring algorithms are fundamentally iportant to the whole result of a fractal generator, and the math involved with surface detection and colouring is a direct model of spatial intensity descriptions of geometry/spaciometry. As well as surface, and colour we may use surface and sound including pressure in the haptic range.

Thus parallel motions preserve independence, and Riemann surfaces which do not preserve parallel motion except locally , mean that on them, ultimately everything is dependent on every other dimension and independent distinctions can only ever be made locally.. Thus a kind of "gravity" connecting all regions exists as part of a Riemann surface, and this exists as a relativistic phenomenon E.G. on a spherical surface all motion is tending toward the poles relativistically, and this tendency is a gravitational one.

Generalising this would lead to a possible complex surface like a Klein surface which contained local curvatures which would act like gravitational centres. Such a surface itself could be a fractal of a larger surface or a part of a spherical surface ad infinitum. Fractal Riemann geometries may be a spaciometric model of our galactic cluster or even or known universe.

Despite it being called a surface however, it is still only a rotational motion field in 3d, but by allowing generalised coordinates we can map out the relations , graph the relativistic relations on a kleinian map or mesh, each edge representing a local dimension and moving parallel to that edge represents independent resolution of the local relativistic connection. The hole Kleinian map would of course only be a motion sequent in the real motions of my experiential continuum, which as i said is based on 3d relativistid rotational motions.

"Clifford was above all and before all a geometer." (H. J. S. Smith). In this he was an innovator against the excessively analytic tendency of Cambridge mathematicians. Influenced by Riemann and Lobachevsky, Clifford studied non-Euclidean geometry. In 1870, he wrote On the Space-Theory of Matter, arguing that energy and matter are simply different types of curvature of space. These ideas later played a fundamental role in Albert Einstein's general theory of relativity.

When Clifford argued that curvature of space underlies all physical phenomenon, he meant "space" not space. He meant space as something, not space as nothing. I understand space to be something, not in some nothing or nothing containing some other substance, but the substance itself, and it is the relativistic rotational motion of that substance, that is space itself which underlies all the vast, universal computational,dynamic in which we live and breathe and have our very being.

It is also interesting to find that the debate on imaginaries was not squashed by Hamilton as page 120(1876 Proceedings of the Cambridge Philosopical Society) shows an argument in defense of them against a detractor even 5 years after Hamilton died!

http://en.wikipedia.org/wiki/William_Kingdon_Clifford#Mathematician