Reciprocal System Research Society

First Fundamental Postulate

The physical universe is composed of entirely of one component, motion, existing in three dimensions, in discrete units and with two reciprocal aspects, space and time.

Second Fundamental Postulate

The physical universe conforms to the relations of ordinary, commutative mathematics, its magnitudes are absolute, and its geometry is Projective.

Quote:

The physical universe conforms to the relations of ordinary, commutative mathematics, its magnitudes are absolute, and its geometry is Projective.

If you are observing from the time-space region of the material sector (as we normally do), then the geometry we see is Euclidean. However, every time a unit boundary is crossed (unit speed boundary, unit space boundary, or unit time boundary) the geometry of the frame on the other side of the boundary will also undergo the reciprocal relation, and become polar-Euclidean, metric, affine or scalar (projective or synthetic), depending upon the nature of the barrier, and how many boundaries you cross for observation.

Thus, Larson's 2nd postulate can also be restated thusly:

Quote:

The physical universe conforms to the relations of ordinary, commutative mathematics, its magnitudes are absolute, and its geometry is Euclidean in the reference frame of the observer.

LoneBear wrote:

The physical universe conforms to the relations of ordinary, commutative mathematics, its magnitudes are absolute, and its geometry is Euclidean in the reference frame of the observer.

It seems to me it is not even necessary to modify the second postulate as the idea that crossing a unit boundry results in a reciprocal is already implicit. When we "look across" from the material sector to the cosmic sector, we don't perceive speed, we perceive its reciprocal, energy. Gravity in the cosmic sector looks to the material sector to be inertia. It seems to me that given the general nature of the reciprocal relation and its manifestation when "viewed" across a boundry, the same would hold true for geometry, or at least would be a sound assumption.

danmc wrote:

It seems to me that given the general nature of the reciprocal relation and its manifestation when "viewed" across a boundry, the same would hold true for geometry, or at least would be a sound assumption.

One would think so, but since it went unnoticed for 50 years, I think it would be a wise move to include some reference to that effect, so in the future, it is not forgotten.

Unfortunately, that one concept of reciprocal geometry significantly changes the "natural consequences" that Larson came up with. And that is the bind.

bperet wrote:

The physical universe conforms to the relations of ordinary, commutative mathematics, its magnitudes are absolute, and its geometry is Euclidean in the reference frame of the observer.

The physical universe conforms to the relations of ordinary, commutative mathematics, its magnitudes are absolute, and its geometry is perceived as Euclidean. ...its geometry appears Euclidean?

Eccles wrote:

...its geometry appears Euclidean?

It does, as Bruce has mentioned, with respect to where the observer is 'standing'. We are creatures in the space/time universe, which appears Euclidean to us. Had we been in the time-region universe, motion within the unit of space, we would perceive the world as Euclidean still, but now the outside space/time region appears as polar-Euclidean.

Hence, Euclidean geometry no longer holds well when we are dealing with the motion within unit space, e.g. atoms and particles.

Gopi

bperet wrote:

The physical universe conforms to the relations of ordinary, commutative mathematics, its magnitudes are absolute, and its geometry is Euclidean in the reference frame of the observer.

Given the perceputal and conceptual ability of the mind, an observer could posit any reference frame. The "barrier" or defining "region" is ad-hoc, so I'm not sure an "observer" matters with respect to this postulate. Therefore why not state it more generally, like this:

"In a Euclidean reference frame, the physical universe conforms to the relations of ordinary, commutative mathematics and its magnitudes are absolute."

Gopi wrote:

It does, as Bruce has mentioned, with respect to where the observer is 'standing'. We are creatures in the space/time universe, which appears Euclidean to us.

As far as I can tell, being creatures of a space/time universe only provides the Euclidean (cartesion) perceptual bias only at the "classical" or extension-space/vectorial-motion level.

Gopi wrote:

Had we been in the time-region universe, motion within the unit of space, we would perceive the world as Euclidean still, but now the outside space/time region appears as polar-Euclidean.

I don't see why the space/time region would not appear as Euclidean. After all, space and time are the same thing. Therefore, regions (and the concepts of what is "inside" or "outside" of them) only exist at the whim of an observer for the purpose of defining a seperation of space and time. Or as Samuel Alexander said "...Time makes Space distinct and Space makes Time distinct... Space or Time, may be regarded as supplying the element of diversity to the element of identity supplied by the other."

Gopi wrote:

Hence, Euclidean geometry no longer holds well when we are dealing with the motion within unit space, e.g. atoms and particles.

This is not true. Euclidean geometry may hold when we are dealing with motion within unit space. It is just that it is not as amenable to the description of time-based phenomena (such as within a defined "unit space" region). Same reason legacy physics started using wave equations for it. There is absolutely no reason we can not use Euclidean geometry - it is just more complex to do so - like using a summation instead of an integral.

MWells wrote:

"In a Euclidean reference frame, the physical universe conforms to the relations of ordinary, commutative mathematics and its magnitudes are absolute."

That might be a better way to state it, because it is known that the polar operators, i, j and k, are NOT commutative, so this, indeed, may be a restriction of linear, Euclidean geometry only.

And we also have to deal with the fact that quantized motion appears continuous in the reciprocal reference frame.

MWells wrote:

Gopi wrote:

Had we been in the time-region universe, motion within the unit of space, we would perceive the world as Euclidean still, but now the outside space/time region appears as polar-Euclidean.

I don't see why the space/time region would not appear as Euclidean. After all, space and time are the same thing.

Space and time are the same thing--from TWO different perspectives. When you have ONE perspective, as in the case of an observer, they appear different. Just a question of where you put the dichotomy--on the reference frame, or the objects in the reference frame.

So with the single perspective, the time-space region (aka space/time region) will appear polar to the time region observer, since the reference frame has to accomodate the dichotomy.

With two, concurrent perspectives--an eye in each realm--both will appear Euclidean, but then you may have difficulty correlating the motions with each other, since space and time would be the same thing and your consciousness could not distinguish them from one another.

MWells wrote:

Gopi wrote:

Hence, Euclidean geometry no longer holds well when we are dealing with the motion within unit space, e.g. atoms and particles.

This is not true. Euclidean geometry may hold when we are dealing with motion within unit space. It is just that it is not as amenable to the description of time-based phenomena (such as within a defined "unit space" region). Same reason legacy physics started using wave equations for it. There is absolutely no reason we can not use Euclidean geometry - it is just more complex to do so - like using a summation instead of an integral.

Yes, you can use any geometry in any region, but each comes with its own set of problems. You can treat the unbounded angle of the Turn as a distance of an unbounded Euclidean line, but then you have to account for the fact that each "step" in the line is shorter than the last--they are not uniform. If you deal directly with the counterspace Turn, all the steps are uniform, because the reference system has the correction in it.

It is simply a "best fit" choice. As Scotty would say, "the right tool, for the right job!"