You are right. I started calling this description of the eight 3d "directions," which Larson gave us to answer those who mistook the six directions of a 3d coordinate system as six 3d directions, Larson's cube, in honor of him, many years ago.I have read every book and paper published on the Reciprocal System and nowhere in them do I find a "Larson cube," outside of your posts. The ONLY reference Larson made with a cube was to clarify that in a 3-dimensional system, there were EIGHT possible directions, not SIX as would be presupposed (± on each axis). That's IT. There is no magic cube to solve the mysteries of the Universe.
The reason I named it that is because of its central role in connecting the magnitudes, dimensions and "directions" of numbers and geometry. Numbers have two "directions" for each dimension as does geometry and Larson's illustration of a 2x2x2 stack of 8, 1-unit cubes, is the starting point for understanding that.
Consider a 1d scalar expansion: It expands in two opposite "directions" from a point. Numerically expressed, this length is a 21 = 2 magnitude, when the space/time expansion is 1 unit of space per unit of time. Granted, the unit of space is one unit in each "direction," but that is what makes it "magnitude only;" that is, a scalar magnitude differs from a vector magnitude in that sense.
Now, expanding this 1d scalar magnitude into the two opposite "directions" of a second dimension, the expansion produces a plane, with 22 = 4 "directions," a plane in each "direction." Finally, a subsequent expansion in the two "directions" of the third dimension produces a stack of eight cubes, one in each of 23 = 8 "directions."
Now, this is not a "magic cube to solve the mysteries of the Universe," but it does represent the mathematical and geometric truth of scalar expansion. If nature expanded by the numbers, the unit number of a 3d scalar expansion would be eight, in the RST, given the fundamental postulates of the system.
I'm not sure how you arrive at the confusion that I am "confusing structure with probability." The geometry of a 3d expansion is not the structure of the eight unit stack. Nature expands/contracts in all directions simultaneously. If we analyze 3d scalar motion, we have to pick a starting point, just as we do in analyzing vector motion. The vector is not a structure of some kind, but we can construct, or graph it, mathematically to understand it better.Larson's work is based on motion, not little boxes. One must learn to think in terms of motion--contours and speedometers--to understand what Larson was trying to explain. Most of your posts confuse structure with probability, which is why you cannot solve many of your issues.
It's no different in our studies of scalar motion. The trouble is, in both cases, numbers don't correspond to circles and spheres, so we search for ways in which we can find correspondences between them. Most of our current science and technology is based on the correspondence between the angles of rotation and the geometry of the circle. The changing sines and cosines of rotation map the numbers of a coordinate system to the motion of rotation, but that is not the only possibility. In fact, for scalar motion we can't use them.
Pray tell, what are the other three roots that can be calculated here? Remember (or perhaps you didn't realize), these three unit numbers are the radii of the three balls defined by Larson's cube: The radius of the inner ball, the radius of the ball just containing the 2d slice of the stack and the radius of the ball just containing the stack. It's just a physical fact that the radii of these three balls is √1, √2 and √3. Expressing that truth using the Pythagorean theorem in no way changes the expansion/contraction of the balls from scalar to "Euclidean," whatever that means. The geometry of the universe of scalar motion is postulated to be Euclidean in Larson's system, so I don't follow the logic here.dbundy wrote:
"The bottom line is that the scalar system, thus defined, gives us three unit numbers, one for each non-zero dimension, by the Pythagorean theorem:
1) √(12 + 02 + 02) = √1
2) √(12 + 12 + 02) = √2
3) √(12 + 12 + 12) = √3"
First, a square root has TWO solutions, a positive and negative one, meaning you have SIX "unit numbers," not three. Second, you used the Pythagorean theorem--geometry--to obtain them. This is a COORDINATE system, not a "scalar system" (scalar = magnitude only, NO geometry). This is all based on the hypotenuse of triangles--Euclidean, not scalar. Perhaps you should study some projective geometry to get your terms correct.
Real numbers don't exist in Larson's discrete system. In his system, motion is postulated to exist in discrete units, with two reciprocal aspects, space and time. I guess we could agree with Kronecker on this: "God made the integers. All else is the invention of man."dbundy wrote:
"Adding, multiplying, subtracting & dividing these numbers is as straightforward as doing so with the so-called Reals; that is to say, the numbers are ordered and their operations are distributive, commutative and associative, regardless of their dimension."
"So-called Reals?" What do you call them?
A "real" number defines an ordered set, with properties of distribution, commutation and associativity. Not the other way around. "Imaginary" numbers also have these properties because the operators are just axis designations and the magnitudes are of the real set. 2i + 3i = 3i + 2i; 2i * 3i = 3i * 2i. All it says is that you are doing a "real" operation on the "i" axis--could just as well be X, Y or Z, as well as i, j or k. You lose these properties when you exceed the single dimension that real numbers define, because we have no way to express an ordered set of planes or volumes, without reducing it to a single property such as area or volume--to stick it back on the 1D, real ordered set as a projection. Complex operations confuse people because they represent spin or twist, which cannot be represented by a displacement on a straight line--it requires a plane (Argand plane) to do it, which means you've lost the ordered set and therefore, the properties associated with the "real" line. This is the situation with straight lines, as well. A rectangle of (3,2) is not the same rectangle of (2,3), yet they are both "reals" and have the same area.
The trouble comes back to numbers and geometry again. There are geometric magnitudes for every number, but the converse is not true: there are geometric magnitudes for which there are no numbers. We can make symbols to represent such numbers, as we need to work with geometry algebraically, and that's a good thing, since modern science and technology would not exist without them, but the fact remains that there is this terrible incompatibility between discreet numbers and continuous geometry. Squaring the circle is impossible, which ultimately makes general relativity and quantum mechanics incompatible, theoretically. This is the biggest unsolved problem in theoretical physics.
The reason the algebraic properties are lost in the vector number system, as the dimensions of the system increase, is due to the fact that vector motion requires a starting and ending point in one "direction." There is no problem with 0d numbers, as the quantity of a collection of points (0d numbers) can be increased or decreased in one of two "directions," positive or negative "directions," to zero. However, if we extend these numbers to one dimension, we have to do so by means of rotation, and the starting and ending points we need to represent arc-lengths on a circle have to be arbitrarily chosen, because there's no way to say that one starting or ending point on the circle is greater or lessor than another.
Extending the numbers to two dimensions we not only lose the ordered property, but we also lose the commutative property because it's required to rotate the sphere of interest to point to the desired location on the surface, and this rotation now has multiple "directional" possibilities, making the outcome dependent upon the order of operations.
Of course, extending the numbers to three dimensions, really becomes problematic, because we've run out of dimensions at this point. There is no such thing as a fourth dimension, in Euclidean geometry, so it has to be invented and the operations required to manipulate these numbers is so order dependent that they lose the associative property.
Now, again these number systems correspond to vector motion; that is, change of position in one "direction" at a time. Rotation enables us to change the number of dimensions in which this motion is effective, but not without sacrificing something, in each higher dimension. This is not the case with number systems based on scalar motion, because it is motion in all possible "directions," of a given dimension, simultaneously. The binary expansion of the Tetraktys captures this dimensional truth and its relation to geometry through its correspondence with Larson's cube.
I know these scalar concepts are unfamiliar and may seem strange, but if you will start from the beginning and work your way through them, I think you will find that they are an exciting and useful way of developing Larson's new system of theory. At any rate, I do appreciate your engaging comments daniel. I hope you will continue.