Different Interpretations

Discussion concerning the first major re-evaluation of Dewey B. Larson's Reciprocal System of theory, updated to include counterspace (Etheric spaces), projective geometry, and the non-local aspects of time/space.
Gopi
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dof and dimension

Post by Gopi »

I am lost at this point due to one main trouble...

How is a degree of freedom different from the dimension?

Dimension: The number of values required to fully determine the scalar motion.

Degree of freedom:?

Quote:

I would like to understand how 3 dimensions of speed emerge from a cross-ratio that has 15 DOF.
Don't the three dimensions of speed create the degrees of freedom in the first place?

Cheers,

Gopi
Horace
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Different Interpretations

Post by Horace »

Good question.

One school of thought defines the degrees of freedom as the dimensions that are "free" to vary. For example 2D motion in a 3D system has 1 degree of freedom. In other words - the difference between two numbers of dimensions, a.k.a. dimensions of freedom.

Another school of thought allows two directions of translation per dimesion, e.g. left & right, and these two directions are often called two degrees of freedom.

With such interpretation, a point in 3D Euclidean space has 6 degrees of freedom (2 per dimension).

You can have rotations in addition to translations, when you get to objects larger than dimensionless points and if the system has more than one dimension. (In 1D system rotation is becomes reflection). Such rotations can be considered as additional degrees of freedom. However in Euclidean system rotation is not a primary motion, so I am not sure it can be counted as a degree of freedom in this system.

This is slightly off topic, but note that it is impossible to continuously vary direction with 1 dimension of freedom, since the change of direction along a line must be discontinous.

At least 2 dimensions are necessary to reverse a direction without discontinuity, for the same reason a car cannot go back on a narrow road without a reverse gear (a discontinous direction device)

Let's hear how Bruce conceptualizes "degree of freedom".

All of this should be in a FAQ somewhere...
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bperet
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Ratio and speed

Post by bperet »

Horace wrote:
The stratification of 3D geometry is clear to me. Thanks. I understand that the cross-ratio is invariant under the transformations of Projective Geometry and all the geometries that are a subset of it.
Fantastic! That is one of the biggest hurdles to overcome with RS2. It is something that Larson never even considered.

Horace wrote:
However, I am still unclear how you get from a cross-ratio to the 3 dimensions of scalar speed.
Examine the 4 strata of geometry, and remember exactly WHAT a 'scalar' is... a change of SCALE. Not a translation, not a rotation... a SCALE change only. Euclidean is scale-invariant, it's scale is fixed at unity, so you cannot have scalar motion in a Euclidean framework. Metric has only ONE scale variance--the same scale applies to everything, so you can't have three scalar dimensions. In the projective stratum, everything is "scale" and nothing else. That leaves the Affine geometry, where the plane at infinity becomes defined, and the orthagonal relationships of dimension occur.

Of the 12 degrees of freedom in the Affine stratum, 6 of them are scalar. This is Larson's realm of scalar speed. He sets the translational scales at unity (removing 3 dof), then varies the rotational scales to create the atomic and sub-atomic structures.

Now I'm sure you've run into the concept of how ONE scalar dimension manifests as THREE coordinate dimensions, right? It is one of those concepts that RS students had a really hard time with. Examine the transitions from Affine to Metric to Euclidean... 3 independent scalar dimensions in the Affine, the projection of which creates ONE variable "scale" (single, scalar dimension) in the Metric, which then adjusts the linear and rotational components so that scale becomes Unity in the Euclidean (no scalar motion, at all). This is exactly how the projection from "scalar motion" into "extension space" (coordinate space) actually works.

This point was never addressed in the RS, even by Larson. He treated the scalar and coordinate dimensions as being separate, but with RS2, the coordinate dimensions are a projective transformation of the scalar dimensions.

Horace wrote:
Quote:
In a scalar sense, it (the cross-ratio) relates two scalar speeds thru a ratio.

Scalar Motion is therefore the projectively invariant cross-ratio, with specific aspects of space and time.
...so from the quotes above it logically follows that:

1) cross ratio is a ratio of two Scalar Speeds

2) cross ratio is Scalar Motion

...THUS

The Ratio of two Scalar Speeds is Scalar Motion

Do you agree with such odd sounding statement ?
No, the concepts are not commutative, since you are dealing with a projection. It is easy to go from a object (ratio) to a shadow (scalar speed), but you can't accurately reproduce an object by starting WITH the shadow. The projection of the cross-ratio is scalar speed/motion. The projection of scalar speed/motion is NOT the cross-ratio.

Horace wrote:
What's the title of the topic on this forum?
I think it is buried in the "Time Region" topic. See Section 9 of Nehru's paper, "Some Thoughts on Spin".

Horace wrote:
I would like to understand how 3 dimensions of speed emerge from a cross-ratio that has 15 DOF. Where did the remaining 12 degrees of freedom disappear ?
Hopefully, I just answered that above. All the DOF disappear thru assumptions imposed by our physical senses.
Every dogma has its day...
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bperet
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Re: dof and dimension

Post by bperet »

gopiv wrote:
How is a degree of freedom different from the dimension?
"Dimension" is derived from the Latin "dimensio", which means "to measure."

"Degree of Freedom" comes from the concept of the "degree", derived from the Latin "densus" (density), which means "frequency of occurence" (how close things are together). From that, we derive the mathematical concept of "degree", which are small graduations of measurement.

I've never really examined the origins of these words before, but do find it amusing that it comes down to that basic concept put forth in the Law of One material of "density" versus "dimension." In essence, "dimension" and DOF are inverses of each other, though commonly confused to be the same thing.

"Dimensioning" is counting the number of degrees of measure. If you have no "degrees of freedom" to count, then you can't have a dimension. Therefore, the "dimension" is the "shadow" of the degrees of freedom.

I consider the "degrees of freedom" to be a more generic term than "dimension", like "ratio" compared to "speed." When I use DOF, then I am just talking about an independent, algebraic variable, with no other assumptions. When I use Dimension, there is an implied one-dimensional, axial assumption with orthagonal relationships to other dimensions.

Thus, three "scalar dimensions" can be viewed as the common 3-axis set of measure, connected at a point so they are an aggregate of relationships, but otherwise act independently.

Three "degrees of freedom", however, can mean anything. When I use a paint program to pick a color, I have 3 DOF in my selection, "hue", "saturation" and "lightness", or "red", "green" and "blue". They are independent variables, but are not considered "dimensions", even though they measure steps, because the color wheel does not have an orthogonal relationships between its degrees of freedom.

gopiv wrote:
Don't the three dimensions of speed create the degrees of freedom in the first place?
When it comes to projective geometry, I've noticed that they use DOF when talking about the transformation, itself (the cells in the projective matrix), with the result of the applied transform as having x,y,z "dimensions."

That makes the "dimension" the shadow of the projection of "degrees of freedom". What you call it, depends on which strata you place the screen to cast the shadow on.

So, the initial 15 degrees of freedom create Larson's scalar dimensions of speed.

I'll have to re-evaluate this a bit, because there ARE 6 degress of freedom regarding scale in the affine stratum. Larson apparently splits the linear and polar geometries, interpreting linear in Euclidean (fixing scale at unity) and polar in Affine (atomic rotations).
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bperet
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Different Interpretations

Post by bperet »

Horace wrote:
This is slightly off topic, but note that it is impossible to continuously vary direction with 1 dimension of freedom, since the change of direction along a line must be discontinous.

At least 2 dimensions are necessary to reverse a direction without discontinuity, for the same reason a car cannot go back on a narrow road without a reverse gear (a discontinous direction device)
Noticed that, huh? The only way you can do it, is thru accelerated motion (t2, not "t"). It is why Larson's photon model does not work when you try to model it.

Horace wrote:
All of this should be in a FAQ somewhere...
Good idea. I'll start one.
Every dogma has its day...
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