art theory 101~The LUFE Matrix Supplement: Examples & Proofs 

The LUFE Matrix Supplement: Examples and Proofs
The distillation of System International (SI) units into more fundamentally base units of SpaceTime (ST) dimensions
Short Title: The LUFE Matrix Supplement
Copyright© 1985, 1991, 2003, Reginald Brooks. All rights reserved
IntroductionLayout & RulesPage 7a
Contents
IntroductionLayout & Rules page 7a
Introduction
This supplement is designed to accompany the "The LUFE Matrix: The distillation of SI units into more fundamentally base units of space and time dimensions. The LUFE Matrix was developed and published by the author originally in 1985 and was subsequently presented in other papers culminating in the 199192 work from which the title and substance of this digitized version is based.
Because the LUFE Matrix addresses the most fundamental units of physical expression, it is concerned with all fields in physics, and many of those in pure mathematics.
Well over 200 examples and proofs are provided starting with the simple examples of classical Newtonian mechanics. As the reader gains understanding and proficiency in using the matrix, the examples, following in a very rough sense the historical development of physical thought, become ever so challenging. However, once through the section on electromagnetism, one has become fully proficient in applying the LUFE Matrix to the advanced fields of modern physics.
In a sense, the proof is in the pudding. the LUFE Matrix works in all fields of physics in which the physical expression can be reduced to spacetime (ST) dimensions. The examples presented here not only reveal the operational rules of the matrix and some of the suggested graphic representations of the matrix in action, they also prove their validity by being invariant in all forms of physical thought expressed in System International (SI) consistent mathematical form.
Thus mass, as S_{III}/T_{II} (Three dimensional units of space per two dimensional units of time) is valid as it is used in Newtonian mechanics, properties of matter, electromagnetism, special and general relativity, quantum mechanics (including dense matter, plasma, and low temperature physics), cosmology, particle physics, and quantum field theory.
The same for the other entities. The invariance of their ST designations throughout diverse fields of physics supports their validity. any mathematical operation in which the essential effect is to multiply or divide one or more entities (of SI derivation) can be restated on the LUFE Matrix...and in simpler, more fundamentally base terms.
The conceptual simplicity that the LUFE Matrix allows in this ultimate distillation of dimensional analysis far exceeds the minimal effort required to become proficient in its use.
Nature is multidimensional, maintaining an essentially perpendicular symmetry of space to time dimensions in forming the matter and fields to which we ascribe a reality. The LUFE Matrix is a new and original mathematical distillation of the fundamental units (and concepts) which account account for physical law. Use it wisely, peacefully and harmoniously at all times.
Although some of the introductory layout material presented here is repeated, the full account in The LUFE Matrix: IntroductionLayout should be reviewed for starters.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ sidebar 1: PREVIEW~~~~~~~~~~~~~~~~~~~~REVIEW can you say "Energy" ten different ways Quick & Simple Visual Overview Recommended: Before & After ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
The LUFE Matrix Layout
We start with the familiar x, yaxis of the Cartesian coordinate system in which four quadrants are formed, designated as: ST, S/T, 1/(ST), and T/S, clockwise. More on this in a bit.
The horizontal xaxis designates the pure spatial dimensions:
Space (S) as S_{I}, S_{II}, S_{III},…and so on.
The vertical yaxis designates the pure temporal dimensions:
Time (T) as T_{I}, T_{II}, T_{III},…and so on.
From the origin, the spatial axis is positive to the right, negative to the left. The temporal axis is positive up, negative down.
Some entities have purely spatial dimensions, e.g., displacement, length, radius, area, volume, wavelength, etc. These are located on the horizontal "space" axis.
Some entities have purely temporal dimensions such as elapsed time or frequency, the two of which are reciprocally related in that frequency is cycles per second. (The inverse of time, T =1/T=n = frequency.) These are located on the vertical "time" axis.
However, the vast majority of physical entities are expressed in the interactive ST dimensions in which a combination of horizontal spatial dimension(s) and vertical temporal dimension(s) gives the physical expression. This appears on the LUFE Matrix in one of the four quadrants, usually in the S/T lower right quadrant as most entities have a net dimensional expression of so many S dimensions per so many T dimensions. For example, the velocity of light, c equals so much displacement per unit of time = meters per second = space/time = S/T= wavelength (l ) times frequency (n ) = l n .
When an item is per something spatial, or per something temporal, that is effectively dividing that item by the spatial or temporal dimension(s). Meters per second translate as m/s, or positive space meters and negative time seconds on the LUFE Matrix. This would place velocity, or m/s in the lower right quadrant as S_{I}/T_{I }(it is equivalent to placing an item on the x, y coordinate system at x=+1, y=1). Another common way of expressing division of, or the denominator in a fraction, is with the negative exponential, i.e. m/s = m s^{1} and m/s^{2} = m s^{2}, and so on.
That's it, that's just about as tough as it gets. We are going to refine this process and lay down some "rules of the matrix", but the idea is no more complicated than this. To avoid confusion of so much information, it is helpful to focus on the S/T quadrant realizing that the whole matrix is symmetric.
Notice that we have used the word dimension in three ways: (a) as spatial dimension, S; (b) as temporal dimension, T; and (c) as spacetime dimension, ST. From the origin, the number of spatial dimensions increases sequentially as S_{I}, S_{II}, S_{III},…and so on, each S unit represents one unit of spatial dimension.
And yes, while we can readily assign any one unit of S (S_{I}) as linear space, and two units of S (S_{II}) as area, and any three units of S (S_{III}) as volume, we must accept Nature's design in which four (S_{IV}) and five (S_{V}) or more spatial dimensions are required.
The same for the temporal dimension, each increases sequentially out form the origin as 1/T _{I}, 1/T_{II}, 1/T_{III},…and so on. Here again, one unit of time may be thought of as per second, and two units as per second second, or per second 2, and so on.
Each pure space or pure time dimension (i.e., S_{I}, S_{II}, S_{III},… and 1/T_{I}, 1/T_{II}, 1/T_{III},…etc.) is to be thought of as extending linearly to infinity (like a beam of light) and at a direction perpendicular to its axial location. Thus S_{I} extends vertically to infinity in both the positive and negative direction and 1/T_{I} extends horizontally to infinity in both the negative and positive direction.
It is on the S/T quadrant of the matrix that the pure spatial and pure temporal dimensions overlap…crossover…forming an area of ST, here as S/T, that defines dynamic, interactional ST dimension. Here is where most of the physical entities express themselves. In the ongoing example, the linear S_{I} dimension dynamically interacts with the linear 1/T_{I} dimension to give the S_{I}/T_{I} area which designates the velocity of light, c = S_{I}/T_{I} = meters/second = l n .
Although some of the introductory layout material presented here is repeated, the full account in The LUFE Matrix: IntroductionLayout should be reviewed for starters.
The LUFE Matrix Operational Rules
The LUFE Matrix Graphic Dynamics
Digital version for the computer: grays inactive, coloractive
Rules of engagement
In the following graphics, hover your cursor over the text and the graphic will change to reflect that text. (You must allow the computer to load the images for the first time, thereafter it will respond more quickly.)
(Please note: This area will be developed to specifically define how the working "examples & proofs" will be presented graphically once they have been digitized. Any number of simple graphic means to designate active and inactive areas, both net and operational, may be utilized. For nearly 20 years The LUFE Matrix has existed purely on paper with little more than outlines, parallel and squiggly lines, and encircled text and symbols...all in black and white. Whatever system you adopt to teach/learn from the matrix, it is most important to be consistent. The matrix is easy.)
Multiplication (addition) & Division (subtraction) Examples
~hover your cursor over the text on the right for images~ ~click image for full matrix blowup~ ~if needed, click and hold while you scroll~ 



NEXT: On to Page 7b Index: Examples and Proofs
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Page 5a DarkDarkLight: Dark Matter = Dark Energy
Page 5b The History of the Universe in Scalar Graphics
Page 5c The History of the Universe_update: The Big Void
Page 6a Geometry Layout
Page 6b Geometry Space Or Time Area (SOTA)
Page 6c Geometry SpaceTime Interactional Dimensions(STID)
Page 6d Distillation of SI units into ST dimensions
Page 6e Distillation of SI quantities into ST dimensions
Page 7 The LUFE Matrix Supplement: Examples and Proofs: IntroductionLayout & Rules
Index
 7/I/A1  7/I/A2  7/I/A3  7/I/A4  7/I/A5 
 7/I/B1  7/I/B2  7/I/B3  7/I/B4  7/I/B5 
 7/I/C1  7/I/C2  7/I/C3  7/I/C4  7/I/C5 
 7/II/A1  7/II/A2  7/II/A3  7/II/A4  7/II/A5 
 7/II/B1  7/II/B2  7/II/B3  7/II/B4  7/II/B5 
 7/II/C1  7/II/C2  7/II/C3 
 7/II/D1  7/II/D2  7/II/D3 
 7/II/E1  7/II/E2  
 7/III/A  7/III/B  7/III/C 
Page 7c The LUFE Matrix Supplement: References
Page 8a The LUFE Matrix: Infinite Dimensions
Page 9 The LUFE Matrix:E=mc^{2}
Page 10 Quantum Gravity ...by the book
Page 11Conservation of SpaceTime
Page 12LUFE: The Layman's Unified Field Expose`