So while this site says they aren't finding anything they keep finding more "proof"? What am I missing?Forums:
I noticed from some of the posts that there seems to be some confusion regarding dimensionality in the Reciprocal System. Let me attempt to clarify, because Larson's use of "dimensions of motion" do not match the conventional definition of a dimension being a single, scalar magnitude of measure.
In conventional parlance, a "dimension" is a measure of height, width or depth, in one of those directions. When it is applied to math or physics, a "dimension" is a magnitude attached to some unit of measure, which can be "3 inches," "5 seconds," "spin-1/2," etc. Multiple dimensions are just a list of numbers used to describe some structure or behavior.
For example, if you see Z3, it has a dimensionality of 3, because it is Z x Z x Z -- three "Z's" that happen to all have the same value. If Z=2, Z3 = 8 just as 2 x 2 x 2 = 8. A "dimensional power" is just the same magnitude, repeated. X.Y.Z is also 3-dimensional, but can have different magnitudes for X, Y and Z.
When it comes to the "dimensions of motion," or "scalar dimensions" (Larson tends to use the terms interchangeably), confusion sets in because in the RS, you cannot have space without time, nor time without space--the "dimensions" are actually ratios of s/t or t/s, composed of TWO magnitudes--not ONE.
Conventional thought would consider a "dimension of motion" to be 2-dimensional, because s/t = s1t-1. That's two variables, like X and Y on a graph and hence would be 2-dimensional.
The confusion with dimensionality in the RS stems from applying the rules of the conventional frame of reference (space only, with width, height and depth) to a universe based on the ratio of motion--three dimensions of speed, (s/t, s/t, s/t).
So when dealing with RS/RS2 concepts, remember that a "dimension of motion" is considered a single dimension, even though it is composed of two, scalar magnitudes, and that the datum of the system is UNIT SPEED -- a one-dimensional ratio.
In order to extract conventional dimensions from the RS dimensions of motion, three things are needed (described in detail elsewhere): an observer, something to observe and a second "something to observe" to act as a reference to define which way is "up." Once you have defined these absolute locations, a conventional coordinate system can be created from the scalar dimensions of motion as a projection, much like the sun casts a shadow of an object on the ground. Note that these coordinate dimensions have no independent existence--they are a shadow, only. If you remove the observer, observed or reference, coordinates can no longer be determined--and cease to exist.
The dimensions of motion are static, hence Larson's use of the term, "absolute location" to describe them. The coordinate dimensions (material sector s3/t or cosmic t3/s) are dynamic, in the sense that they are created by the observer.
If you are interested in a better understanding of how we create coordinate systems, study the learning process in infants--when a baby, new to the world, tries to figure out how to see what is going on around them, and reproduce it. The first things that show up are the line and circle... linear and angular velocity, tied to clock time to produce lengths and arcs. They see mom and dad as "stick figures" -- it takes a lot more to learn surfaces, colors, textures and the thousands of other factors that go into the properties of an object's projection. (If you've ever played around with face recognition software, it makes stick figures out of images--looking for the eye, mouth and nose "circles" at specific angles from each other.)
Larson's Reciprocal System is comprised of three dimensions of motion. It is the "projective stratum" of projective geometry, where the dimension of motion is actually a cross-ratio, with one of the ratios set to the unit speed datum. And the rules of projective geometry are followed, to produce the Euclidean projection by adding assumptions (affine, metric and Euclidean). There are two Euclidean projections in the RS, the material sector and the cosmic sector.
In RS2, there are technically four scalar dimensions, two forming a projective duality in space and two in time, making the system completely symmetric. However, because only a single dimension can be transmitted across the space-time boundary (again, a ratio--two magnitudes), we only observe three dimensions: the two in the aspect where our observer/observed is located and the net motion from the other two in the reciprocal aspect.
This projection of two dimensions into one is most noticeable in Larson's atomic displacement model of A-B-C, where A-B are the two, "magnetic" dimensions and C is the single, projected dimension from the other aspect, the "electric." In RS2, we have updated this to be A-B--C-D to define the full motion. (It turns out that C-D, when the electric motion is seen as 2-dimensional, defines the quantum energy levels--exactly.)
So if that did not totally confuse you, I don't know what will! :-)Forums:
I am working on updating the RS2 site and forum to Drupal 8. If you find the site or forum inaccessible or giving weird errors, please wait a day and try again. Even though this new version of Drupal has been out for a while, most of the contributed modules are missing or buggy... I won't know what the problems actually are, until I try the upgrade.
I don't know if this is any help to RS theory but......
Simple Set Game Proof Stuns Mathematicians this is the part that caught my eye
could this help with your projective geometry?
Game, Set, Match
To find an upper bound on the size of cap sets, mathematicians translate the game into geometry. For the traditional Set game, each card can be encoded as a point with four coordinates, where each coordinate can take one of three values (traditionally written as 0, 1 and 2). For instance, the card with two striped red ovals might correspond to the point (0, 2, 1, 0), where the 0 in the first spot tells us that the design is red, the 2 in the second spot tells us that the shape is oval, and so on. There are similar encodings for versions of Set with n attributes, in which the points have n coordinates instead of four.
The rules of the Set game translate neatly into the geometry of the resultingn-dimensional space: Every line in the space contains exactly three points, and three points form a set precisely when they lie on the same line. A cap set, therefore, is a collection of points that contains no complete lines.