Reciprocal Lineland -- a 1-dimensional RS Universe

bperet's picture
Submitted by bperet on Fri, 03/27/2009 - 20:23

I've always been a big fan of the "Flatland" concept used as an aide to understand dimensional interactions. So... I started thinking what would happen if we created an "Reciprocal System Flatland"... a 2-dimensional universe instead of a three-dimensional one? Might be a lot easier to conceive, because as a 3-dimensional observer, you could actually draw it on a sheet of paper.

Or better yet, as "A Square" found out when taken to "Lineland"... a 1-dimensional universe? Should make the relationships a bit more obvious, and may be a good method to refine them for later use in 2D and 3D.

Taking a look at the RS2 Lineland version of motion, from a material observer, "time" would be just a single, imaginary rotation (row) and "space" would be a single, "real" value (column):

\Large \begin{vmatrix} 1 & it \end{vmatrix} \times \begin{vmatrix} s \\ 1 \end{vmatrix} = \begin{vmatrix} 1+i0 & 0+it \end{vmatrix} \times \begin{vmatrix} s+i0 \\ 1 + i0 \end{vmatrix} = \begin{vmatrix} s + it \end{vmatrix}

A Cosmic sector observer would see the same thing, but with the aspects of space and time exchanged:

\Large \begin{vmatrix} 1 & is \end{vmatrix} \times \begin{vmatrix} t \\ 1 \end{vmatrix} = \begin{vmatrix} 1+i0 & 0+is \end{vmatrix} \times \begin{vmatrix} t+i0 \\ 1 + i0 \end{vmatrix} = \begin{vmatrix} t + is \end{vmatrix}

Natural Datum

The natural datum express here is 1+i0 = 1, same as Larson. In both RS and RS21 (RS2 Lineland), the natural datum serves as an "identity" operator; where I(x) = x. In the complex form, 1+i0 it represents both a linear and polar geometry, which are geometric reciprocals.

Since the complex form is already a "reciprocal" dichotomy, I would be interested to know if anyone can conceive of a "matrix" from which both the linear/real/space and polar/imaginary/time forms can be derived, containing all observer perspectives. Is there a more generic equation?

Rows and Columns

The selection of rows for time and columns for space is arbitrary. It can just as easily be done the other way around.

The "row" form is quaternion-like (1 iX jY kZ), with the scaling factor in the first element; the "column" form is homogeneous-coordinate-like, (X Y Z 1), with the scaling factor in the final element. I decided to use columns for "space" because that is the convention with computer transformation matrices. The only obvious problem is the Argand diagram usually plots the "real" on the X-axis, which would be a "row", so Argand plots would be rotated 90 degrees CCW.

I don't see a particular preference to one form or the other; if you do, please make a comment.

Forums:

bperet's picture

bperet

Fri, 03/27/2009 - 21:48

A received a suggestion from someone who knows Theodor Kaluza's matrix math well (circa 1921), and made the suggestion that I treat each element of a transformation matrix as a sub-matrix, rather than a complex quantity. Did a little research, and found that a complex number CAN be easily represented in a 2x2 matrix form:

\Large s + it = \begin{bmatrix} s & -t \\ t & s \end{bmatrix}

And the cosmic version:

\Large t + is = \begin{bmatrix} t & -s \\ s & t \end{bmatrix}

Which is rather interesting... as we all know, the "real" space of the material sector does not have negative coordinates--cannot have something -5 inches long. But apparently the Cosmic sector CAN have negative spatial coordinates--what Nick Thomas calls the realm of "counterspace".

bperet's picture

bperet

Fri, 03/27/2009 - 22:42

For a motion: s + it = \begin{bmatrix} s & -t \\ +t & s \end{bmatrix}

Additive inverse: -s -it = \begin{bmatrix} -s & +t \\ -t & -s \end{bmatrix}

Multiplicative inverse: \left (\frac{s}{s^2+t^2} \right ) + \left (\frac{-t}{s^2+t^2} \right )i = \begin{bmatrix} \frac{s}{s^2+t^2} & \frac{+t}{s^2+t^2} \\ \frac{-t}{s^2+t^2} & \frac{s}{s^2+t^2} \end{bmatrix}

Complex conjugate: s -it = \begin{bmatrix} s & +t \\ -t & -s \end{bmatrix}

The additive inverse is probably not applicable in a unity-identy datum, since s=0 is precluded. That means that there are two other realms where motion can manifest, other than as speed: the inverse and conjugate realms.

Starlight*

Sat, 03/28/2009 - 01:01

It's an interesting and exciting idea.
Funny you mentioned this. Last night as I was rustling through some papers in a box, I came across some drawings I drew a few years back. It could give you some ideas and even make an interesting addition to "Reciprocal System Flatland". There's even a song that goes along with it.
As far as mathematic equations, you'll have to figure that one out.