Different Interpretations

[Horace]

I'd like you to note that there are different interpretations of RS Theory circulating about.

I have come up with a classification system according to Geometric Algebra, which can systematically and succintly compare them.

( If you are unfamiliar with the Geometric Algerba jargon, please see the attachment. )

1) RS2 conceptualizes RS Theory as a RATIO of 2 bi-vectors and/or vectors, each with 1 or 2 degrees of directional freedom, respectively. (a bi-vector below unity, and a vector above unity)
2) Doug Bundy conceptualizes RS Theory as a RATIO of 2 tri-vectors, each with 0 Degrees of Directional Freedom (DDF).
3) Ron Satz conceptualizes RS Theory as a RATIO of 2 vectors, each with 2 Degrees of Directional Freedom (DDF).

All of the above interpretations assume that the absolute magnitudes of the blades |b|, associated by the RATIO, must be equal to each other and represent unity.

This is a scalar constraint.

All of the above interpretations define a 3D system, because a vector w/ 2 DDF, a bivector w/ 1 DDF, and a tri-vector w/ 0 DDF (a pseudoscalar), all define a 3D system.

This is as succint comparison of the different interpretations, as I can muster.

[bperet]

I read thru the paper on Geometric Algebra that you provided. I've been using the same techniques in my RS2 models for years now. I just don't call them "blades"... I've been calling them "densities" because I didn't know anyone formalized it. A density refers to a level of complexity... a tri-vector or 3-blade is just what I refer to as a "3rd density" object or function.

Horace wrote:

1) RS2 conceptualizes RS Theory as a RATIO of 2 bi-vectors and/or vectors, each with 1 or 2 degrees of directional freedom, respectively. (a bi-vector below unity, and a vector above unity)

RS2 is based on homogeneous space, not Euclidean space, so you have to up the blade grade by 1 on everything. It is a multivector approach, as defined in your attached paper. Scalar motion is represented as R(16), a 4-blade, having 15 DF. The basis blades of that 4-blade create the projective layers of perception, such that Euclidean space is the basis bivector (lines as bivectors, not point/vector pairs), and the polar counterspace of the time region is the H+H basis trivector. The cosmic sector is also modeled as a trivector. Of course, these statements are based on a material sector point of view.

Horace wrote:

3) Ron Satz conceptualizes RS Theory as a RATIO of 2 vectors, each with 2 Degrees of Directional Freedom (DDF).

This is Larson's RS. Motion would be more accurately represented as a multivector of {1,vector=space,bivector=time}, where space is translational (electric vector), and time is rotational (magnetic bivector). Using the A-B-C notation, and atom would be: {{1,C,A},{1,C,B}}.

Horace wrote:

All of the above interpretations assume that the absolute magnitudes of the blades |b|, associated by the RATIO, must be equal to each other and represent unity.

This is a scalar constraint.

All of the above interpretations define a 3D system, because a vector w/ 2 DDF, a bivector w/ 1 DDF, and a tri-vector w/ 0 DDF (a pseudoscalar), all define a 3D system.

Larson deals strictly with Euclidean space; RS2 upgraded that to Homogeneous space, so the DDF increases in RS2 from Larson, because of the additional dimension.

[Horace]

This is just the type of the feedback I was hoping for.

Your correction of the GA definition for RS2 is exactly what I was trying to provoke with my quick&dirty definition.

Please help me distill this down to a succint statement. "RS2 conceptualizes RS Theory as a RATIO of ..."

Bruce wrote:

RS2 is based on homogeneous space, not Euclidean space, so you have to up the blade grade by 1 on everything.

Could you bring me up to speed on this "homogeneous space" ?

Bruce wrote:

It is a multivector approach, as defined in your attached paper. Scalar motion is represented as R(16), a 4-blade, having 15 DF.

I don't understand how you get 15 DF. Are these the dimensions of DF (as in the binomial expansion) or the poles of DF (as in trinomial expansion)? For the context of this question see the attachment.

Also, what are the different blades of the grade-4 multivector ?

Is there a RATIO between the component blades of the multivector ? How does the ratio of space and time fit into the multivector ?

Is it a ratio of two grade-4 multivectors (one for space & one for time) or is there one multivector that handles both magnitudes ?

I just recently joined. It's my 2nd day on this forum and I haven't understood everything yet...

[Alluvion]

lol horace, don't worry - i've been here for over a year and a half now and I am still so lost ; )

-a

[bperet]

Horace wrote:

Your correction of the GA definition for RS2 is exactly what I was trying to provoke with my quick&dirty definition.

Please help me distill this down to a succint statement.

"RS2 conceptualizes RS Theory as a RATIO of ... RATIOS".

In RS2, the fundamental "building block", analogous to Larson's "scalar speed" is the cross-ratio, a ratio of ratios. No aspects, no directions, no nothing... just the concept of "ratio" in and of itself.

Perceptual assumptions are then added to the cross-ratio (center, plane at infinity, cone of vision, etc) to construct the various geometric strata that results in how we view these ratios from different perspectives (vectors, bivectors, trivectors, etc). That is why I changed the "Fundamental Postulate" to "...and its geometry is Projective" for RS2.

Horace wrote:

Bruce wrote:

RS2 is based on homogeneous space, not Euclidean space, so you have to up the blade grade by 1 on everything.

Could you bring me up to speed on this "homogeneous space" ?

See: http://forum.antiquatis.org/fileattachm ... models.doc for a summary that I wrote for Nehru a couple years ago, when we first started doing the re-evaluation of the RS. (Attachment from the "Time Region Speeds" post in the forum).

Horace wrote:

Bruce wrote:

It is a multivector approach, as defined in your attached paper. Scalar motion is represented as R(16), a 4-blade, having 15 DF.

I don't understand how you get 15 DF. Are these the dimensions of DF (as in the binomial expansion) or the poles of DF (as in trinomial expansion)?

Neither. They are ratios, in a 4x4 matrix (R16 from your Bott Periodicity reference). In homogeneous space, there needs to be at least ONE anchor point, and in RS2, that is "unity" (unit motion), represented in the 4x4 matrix as:

Code: Select all

| 0 0 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
| 0 0 0 1 |

With unity "fixed", the other numbers (zeros in the matrix) represent 15 degrees of freedom for the function, since they can all take on any value, independent of each other.

If you want the multivector representation...

R(0) = {1}

R(16) = {1; 0,0,0,0; 0,0,0,0,0,0; 0,0,0,0; 0}

It isn't until much later on, after you've compounded a few assumptions like the plane at infinity, zero, and orthogonal relationships, that the numbers have fixed relations to each other, and the degrees of freedom drops.

This is the "projective stratum" of geometry, 15DF; the affine stratum has 12 DF, the metric, 7 DF, and Euclidean, 6 DF.

Horace wrote:

For the context of this question see the attachment.

Bundy wrote:

Zero-dimensional scalars have one "direction" and one pole. ...

A "scalar" does not have the properties of "dimension", "direction", nor "pole"... that's what makes it a SCALAR -- magnitude ONLY! Conclusions derived from this false premise are therefore false.

IMHO, don't pay much attention to the Bundy System of theory (or the "BS Theory", as I refer to it). He will just confuse you, as his "theories" are usually not grounded in logic, nor common sense. Stick with Larson.

Horace wrote:

Also, what are the different blades of the grade-4 multivector ?

{1; e1; e2; e3; e4; e12; e13; e14; e23; e24; e34; e123; e124; e134; e234; e1234}

1 = scalar (1)

e_i = vector (4)

e_ij = bivector (6)

e_ijk = trivector (4)

e_ijkl = 4-Blade (1)

You can see the degrees of freedom in the multivector... the scalar is fixed at one; the DF is therefore 4+6+4+1 = 15.

Horace wrote:

Is there a RATIO between the component blades of the multivector ?

The ratios ARE the components. Each scalar or vector represents a ratio, what Larson calls "motion". The n-blades define compound motion, and how that motion can break down into "basis" motions.

Horace wrote:

How does the ratio of space and time fit into the multivector ?

In RS2, space and time are derived aspects of the Euclidean stratum, which we call on this forum "space" (linear) and "counterspace" (polar) geometries. The aspects come into play when the multivector is transformed into vectors (space) and bivectors (counterspace).

Larson only dealt with the vector/bivector components, and the remainder fell under "scalar motion". He never did really explain the connection between scalar motion and "extension space", the 3D space of our ordinary existence. RS2 addresses that thru projective geometry.

Horace wrote:

Is it a ratio of two grade-4 multivectors (one for space & one for time) or is there one multivector that handles both magnitudes ?

Space and Time form a ratio; those ratios ARE the components of the multivector.

It appears you may be thinking "inside-out"... the multivectors don't create the ratios... the ratios create the multivectors.

Horace wrote:

I just recently joined. It's my 2nd day on this forum and I haven't understood everything yet...

Good to have you on board. Nice to get some fresh ideas!

[Horace]

Bruce wrote:

A "scalar" does not have the properties of "dimension", "direction", nor "pole"... that's what makes it a SCALAR -- magnitude ONLY!

OK, no geometric direction but what about the sign of the magnitude which is often called a "direction".

For example the "inward" or "outward" displacement of scalar speed from the default unit ratio, needs to be named and differentiated somehow.

What's wrong with the idea of naming the sign of a real number, the "direction" ?

Also any pseudoscalar has no degrees of freedom & hence - no geometric direction. But, what's wrong with naming the direction of a pseudoscalar, as such ?

e.g. inward or outward "direction" of a pulsating volume in a 3D system.

In english, words can have many meanings. I don't like it, but the only alternative is to coin another word, e.g.: "pseudodirection"

.

[Horace]

Bruce wrote:

RS2 conceptualizes RS Theory as a RATIO of RATIOS

Fine, but the last ratio in this statement is a quotient of: what divisor and dividend ?

In Projective Geometry the cross-ratio is a set of four distinct points on a plane.

http://en.wikipedia.org/wiki/Cross-ratio

What is the relation of this plane to RS Theory ? Is this plane an analog of scalar speed ?

Does the plane defined by the cross-ratio have 3 DOF? If not, how can there be three-dimensional scalar speed, then ?

Larson in http://www.reciprocalsystem.com/ce/dimmot.htm wrote:

We may therefore identify the gravitational motion as three-dimensional speed

[bperet]

Horace wrote:

Bruce wrote:

A "scalar" does not have the properties of "dimension", "direction", nor "pole"... that's what makes it a SCALAR -- magnitude ONLY!

OK, no geometric direction but what about the sign of the magnitude which is often called a "direction".

Magnitudes are the counting numbers; they do not include zero or negative numbers, so there is no "sign."

I wrote a paper on scalar motion, which you should read. It is on the RS2 Theory, Prerequesites, "Scalar Motion" sub-section. The direct link is: http://www.rs2theory.org/theory/scalar_motion.html

Horace wrote:

For example the "inward" or "outward" displacement of scalar speed from the default unit ratio, needs to be named and differentiated somehow.

I know it is confusing, but "inward" and "outward" are illusions, created by an assumption of a reference point. Ratios (speeds) are absolute, since they are magnitudes. You cannot have an "inward speed" until you have a point in which to measure it from. That point is NOT unity, however. You can't go "in" from unity, because there is no smaller counting number than 1. But you can change aspects... you can increase the amount of space (numerator), or increase the amount of time (denominator). From the assumed reference point of space, increasing the amount of time gives the illusion of moving "inward", because of the reciprocal relationship between space and time.

Horace wrote:

What's wrong with the idea of naming the sign of a real number, the "direction" ?

Also any pseudoscalar has no degrees of freedom & hence - no geometric direction. But, what's wrong with naming the direction of a pseudoscalar, as such ?

e.g. inward or outward "direction" of a pulsating volume in a 3D system.

You'll find it much easier to understand the RS if you can conceptually separate the various stages of "motion" in your mind, so that you know what assumptions are being added at which stage. Yes, you can create names, and assign concepts such as sign and direction, but I find that tends to add to confusion, rather than reduce it. You may notice the lack of acronyms in my writing... I do that on purpose. It is more typing, but requires less effort for the reader to comprehend... compare:

"within the US of the TR, the NRS of the TF forms a PCS"

to

"within the unit space of the time region, the natural reference system of the T-frame forms a polar coordinate system."

Which is clearer?

What I would find "wrong" with naming the sign of a real number as a "direction", would be that it is misleading. A speed of 1/3 has a displacement of +2. A speed of 3/1 has a displacement of -2. That's what Larson uses, and led to all sorts of confusion from RS students. Why? Because in creating the concept of "displacement", Larson converted an inverse to a polarity -- two different, and distinct, mathematical concepts. True, it is simpler to use a displacement in atomic relationships than the actual speeds, but this is no longer 1955, and we have computers to do the conversions for us.

If you are reading the 4th density science forum, Tulan is also talking about a similar point, regarding the effectiveness of communicating. I suggest you use any convention you desire, that helps you to better understand the system. But just know why you made the assumptions that you did, so when it comes your time to explain it to someone else, you can say, "I did this, for this reason", so they know where you are coming from.

[bperet]

Horace wrote:

Bruce wrote:

RS2 conceptualizes RS Theory as a RATIO of RATIOS

Fine, but the last ratio in this statement is a quotient of: what divisor and dividend ?

Whatever you would like to call them... they are unnamed in the projective layer, since there aren't any assumptions yet in place to even give the concepts such as numerator and denominator.

Horace wrote:

In Projective Geometry the cross-ratio is a set of four distinct points on a plane.

http://en.wikipedia.org/wiki/Cross-ratio

IMHO, that definition is unnecessarly complex, for such a simple concept. "Points on a plane" is just a way to visually represent 4 related magnitudes. Each point is just a number; two points pair to form a ratio, the two paired points then pair to form the cross-ratio.

If you know anything of matrix algebra, read this site: http://www.cs.unc.edu/~marc/tutorial/node25.html on the stratification of 3D geometry. It is all done in homogeneous space, so your questions regarding that may also be answered with this paper.

Horace wrote:

What is the relation of this plane to RS Theory? Is this plane an analog of scalar speed?

The plane is just a "device" for understanding. It does not actually exist. Larson has not analog for it, since he starts with the concept of "scalar motion", which is a subset of the general concept of the cross-ratio (a magnitude with the specific aspects named "space" and "time"). RS2 starts with the ratio, then after the concepts of direction and dimension have been introduced, derives "scalar speed". This is where Larson and the RS pick up.

Horace wrote:

Does the plane defined by the cross-ratio have 3 DOF? If not, how can there be three-dimensional scalar speed, then ?

By compounding assumptions of a reference point, direction, and dimension upon the cross-ratio. The reason for 3 dimensions is discussed elsewhere in this forum, from Nehru's work on quaternions, resulting in the dimensional equation: n(n-1)/2=n with the only stable solution of n=3.

Note that there is a big difference between "three dimensions of speed" (a,b,c) and "three-dimensional speed." (a³); the former being three, independent speeds and the latter being ONE speed operating in three dimensions.

Once you have compounded enough assumptions to bring the cross-ratio into 3D, Euclidean space, it has 6 DOF, 3 translational (linear: x,y,z) and 3 rotational (polar: ix,jy,kz).

Using the Wikipedia definition, any cross-ratio in the framework of a 3-dimensional system in homogeneous space has 15 degrees of freedom (a 4x4 matrix). The only reason it doesn't have 16 degrees, is that the "complex plane" they use as a device for interpreting the math needs to be defined in the system as well, and that takes one DOF to do it. Mathematically, you just have to stick a "1" somewhere in the 4x4 matrix to give it a non-zero scale factor, and all the other 15 numbers are free to vary independently.

A discussion of gravity in the context of projective geometry would be best in a separate topic, since it differs from Larson's view slightly, since there is no "inward" motion--it is all relative, outward motion (the Earth expands faster than the surrounding space, and therefore "runs" into it, giving the illusion of things falling to Earth).

[Horace]

Bruce,

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.

However, I am still unclear how you get from a cross-ratio to the 3 dimensions of scalar speed.

For example:

1) http://www.rs2theory.org/theory/scalar_motion.html wrote:

In a scalar sense, it (the cross-ratio) relates two scalar speeds thru a ratio

2) http://www.rs2theory.org/theory/scalar_motion.html wrote:

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 ?

Bruce wrote:

By compounding assumptions of a reference point, direction, and dimension upon the cross-ratio. The reason for 3 dimensions is discussed elsewhere in this forum, from Nehru's work on quaternions, resulting in the dimensional equation: n(n-1)/2=n with the only stable solution of n=3.

What's the title of the topic on this forum? 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 ?

.