RS2 incorporates several concepts that are omitted in Larson's RS (which was published in 1959, so some where just not known at the time):

1. The Observer: Larson's RS does not include the assumptions made by the observer and their instruments when trying to observe and measure phenomenon. RS2 includes the observer "assumptions" in order to back-calculate what is really being observed and measured.
2. Reciprocal Geometry: Larson's use of Euclidean geometry was "linear", so he had to construct devices such as a "rotational base" to account for rotational systems. RS2 recognizes that geometry also has its "reciprocals", and the inverse of linear geometry is polar geometry and that rotational motion is a natural consequence of a polar geometry.
3. Projective (Synthetic) Geometry: though Larson postulates "Euclidean geometry", his use of scalar motion as the basis of his theory is not Euclidean in nature--it falls into a different geometric strata known as "affine geometry." RS2 adopted a technique used in the virtual reality systems of computers called "projective geometry", which explicitly defines the relationships and transformations between geometric strata--and how scalar motion (affine) can transform into coordinate motion (Euclidean, which remains a mystery in the RS).

Though some of the natural consequences change with these additions, the bulk of the RS2 system is the same as Larson's original work. It just simplifies a lot of concepts, since conceptual devices can be reduced.

No, RS2 does NOT use 15 dimensions... it uses 3 dimensions with 15 "degrees of freedom". Since RS2 incorporated information determined from "virtual reality" models in computers, some of the techinques used in computer modeling were adopted for use with RS2:

Homogeneous Coordinates

Are the conventional X, Y, Z coordinates that includes a 4th coordinate that is a scale factor, w. Use of this form allows the plane at infinity to be modeled, thus eliminating the problem of division by zero or reaching infinity. The coordinates are represented in matrix form as: [ X/w Y/w Z/w w/1 ]. Note that in terms of "motion", it represents 3 SPEEDS: x/t, y/t, z/t, which is much closer to Larson's concept.

Quaternions

One of the big problems with rotation in computer models is the old "Gimbel Lock" problem, which happens when trigonometry is used solely for rotation. There are circumstances where the equations cancel each other out, causing a rotation to jump to an unexpected orientation. The use of quaternions, another 4-coordinate system that uses a scale factor and 3 "imaginary" operators to represent the rotation on X, Y and Z axes, eliminates the problem. Quaternions are also expressed in matrix form: [w/1 iX/w jY/w kZ/w]. In terms of motion, we are looking at the inverse of homogeneous coordinates: iX/s jY/s kZ/s.

Transformation Matrix

After a couple decades of thinking about "motion", Larson eventually concluded that he was dealing with nothing more than "abstract change in three dimensions" (from Video). "Abstract change" is represented in virtual models by a concept known as a "transformation matrix", a 4x4 matrix that is a composite of rotation, translation and scaling, with the "4th coordinate" being the scaling factor representing the point and plane at infinity. The requirements for a transformation require that ONE of the 16 cells be defined; so normally 1 cell is fixed at Unity, with the remaining 15 representing the degrees of freedom of motion. This is where the misconception of "15 dimensions" originates.

References

An excellent reference on Projective Geometry and how matrix mathematics are used in it can be found at: Marc Pollefeys Visual 3D Modeling from Images

The primary assumptions are taken from how we view the external world. Imagine you are carrying a camera... what are the components involved with getting the "outside" into your brain?

• The "camera" -- what is detecting the image (your eyes are also a "camera" on the world)
• The focal/projective plane, which determines what you see clearly.
• The type of lens used (orthographic, projective, stereoscopic, fisheye)
• The environment that determines what is "coordinate" through its geometry (in RS2, "space" is rectilinear and "time" is polar).
• The object under observation, with all its structure (rotations, locations, etc).

The concepts can be summarized by this figure:

RS2 postulates are based on Larson's originals, just edited down a bit for a more general application:

• The universe is composed of one component, motion, existing in three dimensions, in discrete units, and with two reciprocal aspects.
• The universe conforms to the relations of ordinary mathematics, its primary magnitudes are absolute, and its geometry is Projective.

Compared to Larson's original postulates:

• The physical universe is composed of one component, motion, existing in three dimensions, in discrete units, and with two reciprocal aspects, space and time.
• The physical universe conforms to the relations of ordinary commutative mathematics, its primary magnitudes are absolute, and its geometry is Euclidean Projective.

The reasons for the changes are:

• Removing "physical": Larson's last book, Beyond Space and Time, had to invent a number of metaphysical postulates to explain certain things he ran across during his research, that could not be addressed in the physical theory. By removing "physical", and generalizing other terms, these metaphysical postulates are not needed, as the original postulates apply to the metaphysical.
• Removing aspects of "space and time": Though applicable for the mechnical view of the physical universe, the aspects--though working in the SAME fashion--tend to receive different names in the metaphysical, such as "yin and yang". The names may change, but the reciprocal relationship holds true to form.
• "Commutative" was dropped because mathematics are only commutative in a 1-dimensional, linear geometry (the number line). With the inclusion of polar geometry--the reciprocal of linear geometry--commutative laws no longer apply since we are dealing with multi-dimensional geometry, such as that expressed in "imaginary" numbers. Ordinary mathematics still holds true for the respective geometries.
• "Euclidean" was changed to "Projective" to include all geometric strata. Larson's "scalar motion" model actually uses affine geometry--not Euclidean--which is included in Projective geometry. The bottom layer of Projective geometry IS Euclidean. By using the rules of projection, RS2 can transform scalar motion to coordinate motion in the appropriate frame (something Larson was never able to accomplish).

If you are an authenticated user, you can post and reply on the Forum, but not much else. If you are honestly interested in trying to understand and contribute to the RS2 research project, you can apply for the "Researcher" role by clicking on "My Account" and selecting the tab, "Apply for Role" and apply for the researcher role. Researchers can create blog entries, books, stories, polls and videos in addition to forum posting.

Researchers can create different types of content for the site:

Forum topic -- used to hash out ideas with other researchers or ask questions about the theory.

Story -- used for informal papers. Will need to be connected with a Menu so it can be found! Contact the Administrator for menu additions.

Book page -- used for formal papers on the theory.

Video -- used for a video presentation, slide show or lecture. Note that due to server limits, upload size is limited. The file will automatically be converted to FLV (Flash Video), so some resolution may be lost. If you have a large video, contact the Administrator to FTP it up to the site, which will create a video page for you.

Poll -- used to survey the membership at large on a specific question.

Blog entry -- for your personal use. The latest blog entries are listed on the front page, in a last-in, first-out order. The blog should concern research regarding the Reciprocal System and RS2 extensions.

You can apply for a researcher role by clicking on "My Account" then "Apply for Role". Research roles are only granted to people who are actively participating in the RS2 project, so unless you've made some posts to the Forum, don't expect to be approved.

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