Lineland Cosmogony: In the beginning...

bperet's picture
Submitted by bperet on Mon, 03/30/2009 - 15:42

Depending on theology, the initial state of the universe is either "void" or "one-ness", unity. From the void state, a single being usually emerges (such as Ymir in the Norse), again bringing the initial state to a unity. So, in the beginning, there was One-ness:

\Large \begin{vmatrix} \; 1 \; \end{vmatrix}

Which is a uniform motion prior to the dichotomy of space and time--a progression--but as yet, no direction. Most western theology has an opposition to this progression ("free will"), which Larson calls it a "direction reversal" where the initial condition is "outward" and the opposition is "inward." Eastern theologies, however, start with the concept of akasha, the aether, which is opposed by aksara brahma, which means "non-involuting"--in other words, the initial condition is "inward" and the opposition is "outward".

Therefore, the initial state of Unity is both inward (involuting) and outward (evoluting), and the selection depends upon the observer principle--where you put your "camera"--inside, looking out, or outside, looking in. This determines your base factors of which direction the progression is moving in, and defines the opposing direction.

This can be expressed by the distinction between rows and columns of a matrix, where the dichotomy is represented as a matrix transpose:

\Large \begin{vmatrix} 1 & 0 \end{vmatrix}^T = \begin{vmatrix} 1 \\ 0 \end{vmatrix} \; or \; \begin{vmatrix} 0 & 1 \end{vmatrix}^T = \begin{vmatrix} 0 \\ 1 \end{vmatrix}

Unity then being the inner product of the matrix and its transpose:

\Large \begin{vmatrix} 1 \end{vmatrix} \Rightarrow \begin{vmatrix} 1 & 0 \end{vmatrix} \cdot \begin{vmatrix} 1 \\ 0 \end{vmatrix} \; or \; \begin{vmatrix} 0 & 1 \end{vmatrix} \cdot \begin{vmatrix} 0 \\ 1 \end{vmatrix}

Note that the basal elements must consist of unity and zero... if both are entered as unity, the result is 2, not 1. If both are 0, the result is 0. This indicates that the dichtomy has BOTH a unit-datum (all) and a zero-datum (none).

The first row-column pair (left) is representative of the concept of unity expressed as infinity (all)... namely 1/0. The second row-column pair (right) is representative of the concept of zero (none)... namely 0/1.

This defines the "progression", as Larson calls it, in a one-dimensional lineland.

In order to "oppose" the progression, all that is needed is to do the "opposite"... namely, reverse the order of which things are done, by exchanging rows and columns, giving the outer product:

\Large \begin{vmatrix} 1 \\ 0 \end{vmatrix} \otimes \begin{vmatrix} 1 & 0 \end{vmatrix} = \begin{vmatrix} 1 & 0\\ 0 & 0 \end{vmatrix} \; or \; \begin{vmatrix} 0 \\ 1 \end{vmatrix} \otimes \begin{vmatrix} 0 & 1 \end{vmatrix} = \begin{vmatrix} 0 & 0\\ 0 & 1 \end{vmatrix}

Opposition to the progression increased the dimensionality of the matrix to TWO. And also note that the resultant matrix is NOT a uniform motion... but the sum is the identity matrix:

\Large \begin{vmatrix} 1 & 0\\ 0 & 0 \end{vmatrix} + \begin{vmatrix} 0 & 0\\ 0 & 1 \end{vmatrix} = \begin{vmatrix} 1 & 0\\ 0 & 1 \end{vmatrix}

Therefore, opposition against progression (aka "free will") results in a manifest dichotomy of dichotomies, of increased complexity (or density). One dichotomy being the infinity-zero relationship, or evolutive (yang) and involutive (yin) principles, and the other being the 1-0 datums of which they are expressed, resulting in a quadration--FOUR aspects of ONE dimension.

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bperet's picture

bperet

Tue, 03/31/2009 - 13:57

Based on the current modalities of symbolism, the involutive concept is "yin" and the evolutive is yang. For a general matrix application, that would create these definitions:

Evolutive (yang) = \begin{vmatrix} 0 & 1 \end{vmatrix} \; or \; \begin{vmatrix} 0 \\ 1 \end{vmatrix}

Involutive (yin) = \begin{vmatrix} 1 & 0 \end{vmatrix} \; or \; \begin{vmatrix} 1 \\ 0 \end{vmatrix}

Which are based on standard, mathematical conventions of "real" (yang) and "imaginary" (yin) number systems.

The determining factor is where the unity operator is placed. In real coordinates, represented as a homogeneous coordinate vector [ x y z 1 ] (3D) or [ x 1 ] (1D), the unitary operator (scale factor) is the last entry in the vector. In imaginary coordinates, represented by quaternions [ 1 iX jY kZ ], the unitary operator is the first entry in the vector.

Since I plan to develop this as a computer model, conforming to the existing standards for the construction of real coordiantes and rotational systems is helpful, since I can them make use of standard, shared libraries, and not have to re-invent the wheel. In the virtual reality realm, real coordinates are normally represented by a column vector, so that would make the material aspects: [ 0 1 ]T for "space" and the matching row, [ 1 0 ], "time".

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bperet

Wed, 04/08/2009 - 00:12

I noticed that I've created a somewhat philosophical system here... the [1] is the Pythagorean monad, the opposition in to rows and columns, the dyad, and the resolution of those into a transformation matrix, harmonia.

The transform, harmonia, is what Larson terms "scalar motion"--an abstract change in ONE dimension, in this particular case of LineLand. What is transformed (changed) is the information in the rows and columns. Since the transform, itself, is a scalar relationship, the rows and columns are also scalar speeds. Of course, we don't comprehend a native universe of motion, we need to create a coordinate realm for lineland--something that can be measured.

This is where it gets a bit tricky, because in order to create a coordinate system, we must compound a number of assumptions made by human perception, that we normally take for granted.

First among these assumptions is the concept of direction. For this, we need to define an infinity--something to point. There are two possibilities: the evolutive concept, where parallel lines meet at a plane at infinity, and the involutive concept (vanishing point), where parallel lines meet at a point at infinity. Problem is--in a 1-dimensional LineLand, NO PLANES! The point becomes the dual of itself, so the ONLY choice is a point at infinity, which can be in either natural datum. An evolutive, unit datum for "real", and an involutive, zero (angular) datum for "imaginary" (rotational operations).

This defines a geometric relationship, but... there is another type, a sequencing relationship, which arises with the concept of "direction". Direction is an ordered sequence of "low to high" or "high to low", in either length or angle. Another directional sequence that arises is the temporal counterpart, the common "arrow of time" concept, which can go forward or backward.

I'm creating a computer model parallel to this research, and am now a bit stuck, as I don't know how to classify a "sequence" in an object-oriented language like C++, which I am using. Sequences refer to Larson's concept of "clock time" and "clock space", which are scalars, and are assumed to be in numerical sequence. In dealing with a complex quantity, the linear concept of a sequence does not work (which is why complex numbers are not commutative). So I am trying to find the assumption that builds the concept of sequence in one or more dimensions. Thoughts?

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Horace

Thu, 04/09/2009 - 10:05

"I don't know how to classify a "sequence" in an object-oriented language like C++"

How about treating consecutive memory addresses as "the sequence".

C++ is not so far gone as e.g. C#, as not to understand memory addresses and pointers.

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bperet

Thu, 04/09/2009 - 18:05

In order to define the concept of direction, the plane at infinity must be defined. It can be represented by a row or column vector which can be included in the transformation of motion:

\begin{vmatrix} 0 & 1 \end{vmatrix} or \begin{vmatrix} 0 \\ 1 \end{vmatrix}

At this point, some conventions need to be established, as the "either-or" approach will get too confusing. In computer modeling, the convention is to represent points used to construct objects as column vectors, which defines the plane at infinity as a row vector, [ 0 0 0 1 ], the dual of zero. In a 1-dimensional realm, the point is self-dual, so we have a point at zero and a point at infinity. Combining the zero, infinity and a "scalar magnitude" into the matrix transform yields the result:

\begin{vmatrix} A & 0 \\ \infty & 1 \end{vmatrix}

Where:

  • A is the magnitude (scalar).
  • 0/1 is the point at the origin (column).
  • ∞, 1 is the plane at infinity (row).

With a direction defined by zero and infinity, points can be located on a line and transformed along that line. A location of a point is defined by how far from the origin it is, and the transform can be used to move a point along that line:

\begin{vmatrix} A & t \\ 0 & 1 \end{vmatrix} \times \begin{vmatrix} x \\ 1 \end{vmatrix} = \begin{vmatrix} Ax+t \\ 1 \end{vmatrix}

The location is scaled by A then translated by 't' along the lineland coordinate system. In a more generalized form:

\begin{vmatrix} A & t \\ I & u \end{vmatrix} \times \begin{vmatrix} x \\ w \end{vmatrix} = \begin{vmatrix} Ax+tw\\ Ix+uw \end{vmatrix}

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bperet

Thu, 04/09/2009 - 18:29

How about treating consecutive memory addresses as "the sequence".

I've looked at enumerations and collations and the sorting process itself. Basically, a sequence is a list that uses a function to control the placement of elements.

Since we are defining a Universe where the start point is unity, what must be "assumed" in order to create a function--relation between objects--that can be used to define a sequencing system like "clock time"?

We take things like clock time for granted, but in order to work out a proper model, I need to find how the transformation works that creates such functional sequences. Or should things like numerical magnitude just be a given, and sequences are somehow translated back to a numeric value for ordering?