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 space-time (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.

The horizontal x-axis designates the pure spatial dimensions:

Space (S) as S_I, S_II, S_III,…and so on.

The vertical y-axis 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/s² = 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 space-time 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: Introduction-Layout should be reviewed for starters.

1. The area on the matrix defines the physical entity and its units of expression. Interactional ST dimensions have areas within a quadrant, while purely spatial or temporal dimensions have linear "spaces" along their respective axis. The identity of a physical entity, be it a quantity or the units to describe/measure it, in terms of net amounts of so much spatial dimension and/or temporal dimension is constant and does not change regardless of its operation location on the matrix.



2. All net areas and spaces are counted from the origin, 0, where the x-(space) and y-(time) axial coordinates cross.



3. Pure spatial (like length) or temporal (like time, frequency or temperature) dimensions never appear by themselves in a quadrant. They are confined to their static, linear space on the axis. These are referred to as Space Or Time Areas (SOTA). Pure spatial dimensions always go to infinity in both the up and down directions for each spatial dimension. Pure temporal dimensions always go to infinity in both the left and right direction for each temporal dimension. Each is like a bipolar laser.



4. It is only in perpendicular combination(s) that space and time dimensions dynamically interact to form ST interactional dimensions (STID).



5. Once you are on the matrix proper, that is on the STID area, then the multiplication (by addition) or division (by subtraction) of any other dimensions, be they ST interactional dimensions or pure spatial or temporal dimensions, is begun from that area already defined on the matrix at that point (not from the origin, 0), using its usual ST designation. For example, acceleration (S_I/T_II) is velocity (S_I/T_I) per unit of time (T_I), so if velocity is first located on the matrix at the S _I/T_I ST interactional area of the S/T quadrant, then dividing this per unit of time (T_I), which is the same as multiplying it by 1/T_I, requires in this case that we add one unit of pure temporal dimension (1/T_I) to the existing S< SUB>I/T_I STID area to give acceleration (S_I/T_I · 1/T_I = S_I/T_II).
   
   Tip: Get on the matrix proper first with the larger STID areas, then combine the smaller STID units and/or pure SOTA units. Remember to keep the distinction between SOTA and STID areas clear in mind during all operations. Each is composed of pure dimensions that run to infinity, but only STID areas have both space and time dimensions. Once we get into the mathematical equations, all of which can be simplified and solved on the matrix, we will then begin to add and/or subtract various SOTA and STID areas to others already on the matrix. This entails building out (for multiplication we add areas) or in (for division we subtract areas) as the equations are solved. SOTAs are added or subtracted to the grid row or column, to which their bipolar, laser-light-like influence extends, thus they act next to the sides of an existing area. On the other hand, STID areas are added or subtracted diagonally to other STID areas. This is only natural as the STID areas have their dual, perpendicular, bipolar, laser-light-like influence going both horizontally and vertically.



6. Only physical entities that have net dimensional quantities appear on the matrix or are involved in any of the operations of the matrix. Dimensionless units include any pure numbers, integers, fractions, geometric ratios, radians, trigonomic, logarithmic, or other functions, and the like. In short, any units such that when their SI units are converted to ST units and there remains no net ST units then such entities are dimensionless units. Examples include Newton's gravitational constant, Coulomb's constant, the permittivity of free space, the dielectric constant and the fine structure constant.



7. The LUFE Matrix readily displays the dynamics of mathematical operations involving the multiplication or division of physical entities. Addition or subtraction does not effect the matrix, nor does the order in which mathematical operation take place. It is only the net, remaining area (or linear space) that counts.



8. Multiplication in the matrix is akin to exponential multiplication…the product of two or more dimensional quantities is found by the addition of their dimensional designations. Two examples: S_I · S_IV = S_V, and, S_I/T_I · S_IV/T_III = S_V/T_IV.



9. Division is similar to exponential division…the quotient of two or more dimensional quantities is found by the subtraction of their dimensional designations. Two examples: S_V/T_IV ¸ 1/T_I = S_V/T_III, and (S_III/T_II) ² ¸ (S_I)² = S_IV/T_IV.



10. The LUFE Matrix is equally valid for the MKS, CGS and other less universal systems of units as the idea is to reduce and distill these systems to one in which there are only two fundamental base units, space and time.

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.)

General-SOTA (Space OR Time Areas)

·Green-Horizontal-Time goes to infinity both ways

·Blue-Vertical-Space goes to infinity both ways

General-STID (Space-Time Interactional Dimensions)

·Deep RGB & O Color Blend-Boxes- go to infinity all 4 ways

Division

·Squiggly Lines=Division (subtraction)-lines go to infinity

·Yellow-box Outlines=net area subtracted out

Multiplication

·Parallel Lines=Multiplication (subtraction)-lines go to infinity

·Grayscale-box Outlines=specific ID area for that quantity added

Net Areas (after operations)

·Bold Black Symbol=Net Area ID

·Bold Black-box Outline=Net Area after operations completed

·Deep RGB & O Color Blend-Boxes-, or,

·Green-Horizontal-Time box, or

·Blue-Vertical-Space box, shows Net Area*

·PurpleDot=Diagonal Corner-of Net Area*

Special Cases, i.e. Examples

·Various other Colored Dots=Diagonal Corners of specific ID areas & operations

·Large, complex layouts will have simplified graphics

~hover your cursor over the highlighted text to show its matrix location~

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~

~best viewed 800x600, ~toggle F11 for full image Graphic Examples                       # description ST unit 1 space axis - 2 time axis added - 3 axes in color - 4 grid boxes added - 5 text added to boxes - 6 working color template - 7 1-space dimension SI 8 per 1-time D 1/TI 9 1-space/1-time = c=velocity of light SI/TI 10 2-space/2-time = c2=velocity of light2 SII/TII 11 Multiplication (addition) & Squares SV/TIV 12 Division (subtraction) & Square Root SI/TI In #11 we start with mass, m, and diagonally add the velocity of light, c, twice to give the energy, E. In #12 the reverse, we start with the energy, E, and diagonally subtract out the velocity of light, c, twice. The order does not matter, only the Net Area remaining.

NEXT: On to Page 7b- Index: Examples and Proofs

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Page 8- The LUFE Matrix: Infinite Dimensions