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It is well worth the time. I've printed it out and gone over it several times now, and am finally catching on. It turns out that there are 4 "strata" of geometry: projective, affine, metric and Euclidean. Each stratum has fewer degrees of freedom (15 for projective, only 6 in Euclidean--3 translate, 3 rotate). The tutorial uses homogeneous coordinates, so it is not that difficult to understand.I hope to go thru the Tutorial on CS then.
I've also purchased and read Rudolf Steiner's book, "Theosophy." It was quite an eye-opener. I'm reading a couple other books now that describe how to apply Steiner's descriptions to projective geometry.
What "counterspace" appears to be is a mathematical description of the reciprocal of Euclidean space -- namely, the time region, minus the assumption of a "plane at infinity". I've noticed this in the polyhedral geometry, as well. The tetrahedron is the reciprocal of itself (connect the center of the faces of a tetrahedron, and it forms another tetrahedron). The cube and octahedron are reciprocals, as are the dodecahedron and icosahedron.
But what it took me a while to understand is that all these geometric strata (which they call "ambiguities") are just methods used to remove our "stereoscopic" perception of the external world. Each stratum has its "invariant" concept, and degrees of freedom. I believe the "strata/ambiguities" relate thusly: Projective = scalar, Affine = Time Region, Metric = Equivalent Space, Euclidean = Time-Space Region. I will be interested in your observations of this when you go thru the tutorial.
Here is a quick summary:
Projective:
15 degrees of freedom (4x4 homogeneous coordinate matrix, with one non-zero element), where the cross-ratio is the ONLY invariant. What is interesting about this, is that the description of the cross-ratio is basically your 2-gear and pinion system, where the speed of each gear form one set of ratios and the pinion forming the cross-ratio. This may be the basis of Larson's scalar dimensions and scalar motion, since there is ONLY "ratio" (two aspects in reciprocal relation), and no type of vectorial or coordinate information.
Affine:
12 degrees of freedom; 3 degrees removed by the assumption of a plane at infinity. Points and planes are duals at this stratum, and one can be transformed to the other. By defining "infinity", the invariants are: cross-ratio, relative distance along a direction, parallelism, and the plane at infinity. Counterspace uses the "point at infinity" transformation. (Note that "relative distance" only applies ALONG a specific direction, not between directions).
Metric:
7 degrees of freedom, invariants are: relative distance and angle (no longer just "along direction," since "angle" is introduced to relate direction), and an "absolute conic" (still having some trouble with the absolute conic concept). Rotation is also introduced at this level (but it is not the Euclidean concept of rotation as much as it is a concept of 3-dimensional shear.)
Euclidean:
6 degrees of freedom, invariant: absolute distances.
I'm somewhat excited about this, because it provides the basis for transforming scalar motion directly into vectorial extension space -- something missing from the RS.
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I've been holding off on further energy level study until I understand the nature of "projection". But I do believe that my findings may be correct -- that the "electron" is actually the cosmic positron, and that there are no "electrons" in the atom, but just "shear" with the characteristics of electrons.Is there any progress on the ?atomic energy levels? study? I posted a message on the inconsistencies of the concept of ?displacement.? Nobody has responded!
I do have a question for you: If we assume that the electron "particle" is actually the c-positron, then it will exist in the S-frame of the space region, not the T-frame of the atom. This would mean that the electron would interact non-locally with the local, temporal motions of the atom, correct? It would also infer that the c-positron would appear "local" in the S-frame of equivalent space or the time-space region, correct?
Could you explain how a non-local motion would effect a local motion? Would the wave-like structure of non-local motion induce a vibratory motion in the local rotations?
Re: Your post to ISUS-Discuss
I think everyone pretty much agreed with what you said, and it did not open any discussion (though I thought Ron Satz might jump to the defense of Larson's system).
You need to remember that most of the group are Americans, and Americans are not very well educated. Our school systems are a joke, and getting worse. (And I have taught High School.) I will admit that your posts can be "intimidating", because your command of the written English language is far better than most Americans, and you present ideas clearly and strongly. I know that many people in ISUS consider you to be the "RS Guru", and as such, elevate you to the "ascended master" status, and it makes you a bit unreachable. It comes from the impressive work you have done with the RS. I know, because when I first read your work, I had the same impression.
If you wish to get an exchange going, I would recommend that you put less information in a post, and pose it in a question format, so it leaves the post open to inquiry. You can use multiple posts to present multiple pieces of information. I've been posting on forums for many, many years, and have found that people don't have the attention span to handle more than a screen-full of text (2-3 paragraphs). As they say, "sad, but true." Once a conversation is going, then you can expand upon it. I, too, wish we had a more interactive ISUS group.
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No recent news. It has been very quiet. Dan McCann also wrote to Mr. Thomas a few years ago, and did get a reply. It said something like he was too busy to learn a "new system". I think the way to approach this, and perhaps the entire Anthroposophic society behind the "counterspace" concept, would be to present them with a paper -- like the atomic energy levels -- that use counterspace concepts along with RS concepts. After all, they are missing half the universe, with no "cosmic sector" in their theory. It would be nice to get us working together. I think it would take understanding a long way towards truth.Any news of Doug and other person?s research of the RST? I received no reply from Mr Thomas.
Bruce