"Solving for Unity..." e=MC² → 16=Φπ² → 1=Φπ²/16

Discussion concerning the first major re-evaluation of Dewey B. Larson's Reciprocal System of theory, updated to include counterspace (Etheric spaces), projective geometry, and the non-local aspects of time/space.
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Φ as Equivalent Space is Mass

Post by user737 »

ckiit wrote: Tue Feb 04, 2020 10:40 am The following root form of Φ might give some insight as to how this symmetry operates internally:

Image
This screams equivalent space.

2π√5 + 6π is one five-rooted rotation (where one full rotation is 2π) plus 6π/2π = 3 more full rotations = 4 full rotations or total of 8π.

8π is 2 × 4π steradians where the steradian is the natural unit for solid rotation (2D rotation -- the photon as quaternion).

This is a dual quaternion and is indicative of i1 / j1 and i2 / j2 dimensions (double, double "magnetic" rotation) of motion.
user737 wrote: Mon Nov 25, 2019 7:46 am Turn IS the reciprocal of time (frequency) -- cycles per second instead of seconds per cycle -- which within the unit space boundary (time region) is an inverse speed (energy).

Those four dimensions (of equivalent space) would then be i1, j1, i2, and j2

kx is no longer an independent variable as it is defined as kx = ix.jx and furthermore ε = k and ε2 = 0 and ε ≠ 0
The remaining π in the numerator is indicative of the corresponding "electric" rotation (1D) where magnetic:electric in natural units is 2:1 just as 2π:π except here i.j = k and ε = k (Epsilon function links the pair of quaternions). This, to me, strongly implies π in this sense is a constant not a variable where π stands for ∞ in counterspace as a projection into space (i.e. max rotation before direction reverses and we re-trace our steps back to the start).

The root operation indicates a cast across a unit boundary (2D→1D). Quaternion is temporal motion (FTL as viewed from TSR) so this must be temporal → equivalent space.

The whole of the expression is normalized to 1 (2π)... clock space (“unit rotation”).

The normalization to 2π is legal as the numerator is of the right dimensionality: “square root” of time is equivalent space. If we were normalizing to provide for equivalent time (wholly separate expression not shown) the denominator would have to be 4π2.

This smells like mass (equivalent space) wherein temporal motion is normalized to a spatial volume (4D→3D), scaled by clock space, and projected into coordinate space.
bperet wrote: Fri Jul 19, 2013 9:02 am When you normalize time to "unit rotation," then you're taking a temporal volume and compressing it into this concept we call mass, which we can then use to measure its effect on space.
This preliminary finding certainly supports e=MC2 where 16=Φπ2

Now we're Cruisin'.
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NRS as Universal Transcendental Axes is 4x Invariant Cross Ratio from Unity

Post by user737 »

ckiit wrote: Sun Mar 22, 2020 3:33 pm Thus posited: symmetrically about the 4x4 matrix sits a single transcendental axes of '4' composed of 2x2 null binaries:
all+/-not denoted {alpha/omega} and causation+/-cessation denoted {beg/end}, each pair being a conjugate binary (+/-),
the transcendental nature of which owing to the axes being wholly space- and/or time-invariant,
thus must rest in/of and/or as '1' as well as being '4' (if indeed both transcendental and universal).
This must be true because no non-transcendental axes can ever be used to transcend a displaced body(s) beyond
the limitation/boundary of the axes itself. Because this axes transcends beyond space and time, it must follow that all
caused bodies contains this axes intrinsic to their own composition. In other words: all displaced bodies have local to them,
as part of their own constituency, a transcending axes immutably concerning unity at all times from all places.
mathtex2.jpg
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This is the mathematical basis for mass/gravity (i.e. a "2D rotational base"), from which all matter (geometric form) is derived as a series of recursive "shells" building outwards in counterspace (inwards in space): A/B and C/D (count of 4 ratios/motions) are each 2D rotations -- a pair of SOLID ROTATIONS (a pair of rotating photons -- double, double magnetic rotation) except here D is unity -- and form the axes of '4' in One. The goal is to be unified; take my hand, be my Brother. The only way is One.

The datum for {A, B, C, D} is the NRS -- unity... motion -- as these are scalar magnitudes (in actuality each 2D as there is a space aspect and a time aspect), the transcendental nature of which owing to the axes being wholly space- AND time-invariant. (Fixed it for us. ;) )

Before we start debating D is *not* a cross ratio, I beg to differ: (1/1) × (1/1) = (1/1)

A/B & C/D comprise the '2x2 null binaries' (+/-) as roots of no coordinate dimension to be evolved into coordinate dimension (raise power thru 'exponation' -- the integration process previously described for 1D→2D projection into equivalent space/time):

[1] all+/-not denoted {alpha/omega} as represented by inverses ALL (+1 or ∞ when observed from counterspace) -AND/OR- NOT (±π -- the metaphysical, ethereal "rotational base" -- nothing or '0' when observed from space), and

[2] causation+/-cessation denoted {beg/end} as represented by conjugate motion in TSR & motion in STR -AND/OR- conjugate motion in TR & motion in SR -- the totality of motion that seek the boundaries of {+1 ↔ -1} in space and {↔π↔π↔} in time... maximum of 2 displacement units in either "direction" (exclusive or: the first as speed, the second as energy)...
bperet wrote: Wed Jan 01, 2020 3:11 pm
user737 wrote: Wed Dec 04, 2019 12:09 pm Are we certain speed of progression = speed of light? If not, this is another DOF.
It would not be another DOF, as the existing 15 degrees cover the motion of the photon (as birotation).

The speed of the progression is fixed, which is the unity in the 16th position of a 4x4 matrix.
Take it away, Bruce:
bperet wrote: Sat Feb 28, 2009 12:42 pm Row vector is a temporal "absolute location", formed by the intersection of three planes.

Column vector is a spatial "absolute location", formed by the coordinates of a point.

A,B,C are the three, scalar dimensions of motion.

The "r" values are the rotations/shifts for the respective dimensions (sine component). The cosine component is multipled across the two adjacent, scalar dimensions. For example, rotation about Z includes a cosine component to A and B.

tx,ty,tz are translations. The "S" translations are the linear, spatial translation we are familiar with. The "T" translations are the temporal "translation", which we would view as a phase angle of the turn (unbounded angle).

Ct is Clock time. Notice that it is REAL, not imaginary (temporal), thus making clock time appear as though it was a "4th dimension" to space. It is actually a scale factor to space.

Cs is Clock space, an imaginary component that scales temporal vectors, and acts as a 4th dimension to time.
The inherent symmetry (of reciprocity) is immediately apparent and cannot be denied.
bperet wrote: Wed Feb 25, 2009 3:12 pm Where absolute locations in space (row) and time (column) are linked to the compound, scalar speeds (4x4 matrix). ALL numbers are complex quantities, not integers.
I wish there was a strikethrough font available on the board for just this reason: swap ‘row’ and ‘column’ in Bruce’s comment above as he later flipped convention (from whence this quote has come) in providing for the final matrix transform form seen at the top of this post.

Also, Bruce, you said ‘ALL’ (emphasis not mine) and I do not see that necessarily true (I am being very deliberate with my word choice here) for unity datum: 1 + 0w. This means there must be other transforms for those others combinations of speed regions besides what we use here for TR ↔ ST/TSR. Talk to me, Goose.

Location in space as represented by a Homogeneous coordinate (HC) column, with the 4th coordinate being clock time (scale factor for spatial location), and contain only the real (linear) aspect of the complex quantity:

| X + i0, Y + j0, Z + j0, 1=Ct |

Location in time as represented by a Quaternion row, with the 4th coordinate being clock space (scale factor for temporal location), and contain only the imaginary (rotational) aspect of the complex quantity:

| 1=Cs, 0 + iZ, 0 + jY, 0 + kX | *

*note: must read read row backwards (right-to-left) to transcribe proper ordering per existing convention

4x4matrix.jpg
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The 4x4 matrix includes a single transcendental axis (diagonal -- top left to bottom right) which comprises A, B, C, D (where D = 1 + 0w) as shown. A, B, C are the 3 scalar dimensions of motion and are each measured as an invariant (unchanging) cross ratio from unity:
bperet wrote: Wed Feb 25, 2009 3:12 pm The 4x4 matrix contains the various speeds of the three, scalar dimensions along the diagonal, with transforms for rotation (turn) and shift (translation) multiplied in, as complex quantities. One interesting result is that spatial location is altered by temporal location, and vice versa, and that no single scalar dimension is directly represented in the system, as Larson claimed, but it is the net motion of all three dimensions that is represented. (Not a problem most of the time, since two of the dimensions are usually at unity--identity--and have no effect).
ns/nt x (1/1) = nt/ns IS the cross ratio (with respect to UNITY) and forms the basis for what is a dimension:
user737 wrote: Mon Apr 06, 2020 11:33 am No dimension (no cross ratio!): point ↔ radius
D at 1 + 0w (complex) provides for UNITY datum (1, ±π) within the larger framework of 4×4=16 elements and serves at least two purposes: (1) necessary to transform Projective (parallelism not conserved) → Affine (parallelism conserved) where the ideal plane at infinity is introduced to create the concept of parallelism, and (2) establishes the universal datum reference as outward motion at the "speed of light" (+1).

Projective invariants include incidence and cross ratio only; Affine introduces an additional invariant property, parallelism as all parallel lines now meet at an ideal "point" at infinity. Differing sets of parallel lines meet at differing "points" on the 'plane at infinity' and the collection of "points" at the plane at infinity are labeled the ideal 'line at infinity'. We call this perspective.

The projection of the real aspect (Φ-based) of the motion → equivalent space, and can be directly observed/measured as anything with a net spatial displacement (some particles per "point" particle perspective, all atoms).

The projection of the imaginary aspect2-based) of the motion → equivalent time, and can be indirectly observed/measured as a 1D "force" vector -- a scalar push/pull + 1D scalar bi-rotation (spin = torque), a complex expression. All "forces" are temporal motion (motion in time) and includes the wave perspective of some particles and all "force fields".

Seems not only do we not observe the NRS, nor do we observe space or time. We manifest equivalent space and equivalent time.
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Re: "Solving for Unity..." e=MC² → 16=Φπ² → 1=Φπ²/16

Post by Djchrismac »

You're really starting to see the code behind the matrix now, top work!

I feel this image is worthwhile adding again for reference:
Regional Dimensions.png
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user737 wrote: Thu May 28, 2020 10:31 pm The remaining π in the numerator is indicative of the corresponding "electric" rotation (1D) where magnetic:electric in natural units is 2:1 just as 2π:π except here i.j = k and ε = k (Epsilon function links the pair of quaternions). This, to me, strongly implies π in this sense is a constant not a variable where π stands for ∞ in counterspace as a projection into space (i.e. max rotation before direction reverses and we re-trace our steps back to the start).

The root operation indicates a cast across a unit boundary (2D→1D). Quaternion is temporal motion so this must be temporal → equivalent space.

The whole of the expression is normalized to 1 (2π)... clock space (“unit rotation”).

The normalization to 2π is legal as the numerator is of the right dimensionality: “square root” of time is equivalent space. If we were normalizing to provide for equivalent time the denominator would have to be 4π2.

This smells like mass (equivalent space) wherein temporal motion is normalized to a spatial volume scaled by clock space and projected into coordinate space.
What came to mind reading this was DNA code telling a shell when to stop growing. Is DNA projecting the code by creating a blueprint in 3D time that gets projected as a structure in 3D space?
user737 wrote: Fri May 29, 2020 12:13 am D at 1 + 0w (complex) provides for UNITY datum (1, ±π) within the larger framework of 4×4=16 elements and serves at least two purposes: (1) necessary to transform Projective → Affine stratum where the ideal plane at infinity is introduced to create the concept of parallelism, and (2) establishes the universal datum reference as outward motion at the "speed of light" (+1).
All matter is retarded motion after all.
user737 wrote: Fri May 29, 2020 12:13 am Projective invariants include incidence and cross ratio only; Affine introduces an additional invariant property, parallelism as all parallel lines now meet at an ideal "point" at infinity. Differing sets of parallel lines meet at differing "points" on the 'plane at infinity' and the collection of "points" at the plane at infinity are labeled the ideal 'line at infinity'. We call this perspective.
Like this? :D
lightspeed.jpg
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user737 wrote: Fri May 29, 2020 12:13 am The projection of the real aspect (Φ-based) of the motion → equivalent space, and can be directly observed/measured as anything with a net spatial displacement (some particles per "point" particle perspective, all atoms).

The projection of the imaginary aspect (π2-based) of the motion → equivalent time, and can be indirectly observed/measured as a 1D "force" vector -- a scalar push/pull + 1D scalar bi-rotation (spin = torque), a complex expression. All "forces" are temporal motion (motion in time) and includes the wave perspective of some particles and all "force fields".

Seems not only do we not observe the NRS, nor do we observe space or time. We manifest equivalent space and equivalent time.
What the Bleep do we know?

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kNOw

Post by user737 »

Djchrismac wrote: Fri May 29, 2020 5:18 pm Image
This needs to be graphically upgraded to 3D. Will get working on that.
Djchrismac wrote: Fri May 29, 2020 5:18 pm You're really starting to see the code behind the matrix now, top work!


The Youtube embed is groovy.
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Unit Dual Quaternion as 1=Φπ²/16

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!RESERVED!
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Direction Reversal / Upside-Down

Post by ckiit »

user737 wrote:Sat May 30, 2020 9:01 pm Couldn't wait... had to share now!

1 = Φπ2/16 is the is product of a normed QUATERNION (3D time) and its RECIPROCAL (3D space) as one in UNITY !!!
Can this be seen in/as a teeter-totter wherein these two quaternions are either ends re: a fixed unity fulcrum?
If so, would this not mean that any two (reciprocally related) quaternions are bound to this same fulcrum?
user737 wrote:Sun May 31, 2020 1:32 pm YES! The answer is YES!

THE FULCRUM IS UNITY DATUM -- 1/1

I REPEAT THE FULCRUM IS LIGHT

A CROSS RATIO IS A SEE-SAW WHERE UNITY IS THE FULCRUM AND THE TWO OBSERVERS ARE SEATED OBSERVING ONE-ANOTHER INVERTED ACROSS THE UNIT BOUNDARY (THE FULCRUM). WHEN ONE GOES UP, THE OTHER GOES DOWN...

THE TWO ASPECTS OF MOTIONS ARE PROJECTED FROM A COMMON CENTER.

(I am excited!)

kris
What I see is the ±1 and ±i (2x2) inner-square of the Q8 here:
q8.jpg
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is "functionally" related to:
Number Axesbef.jpg
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wherein f(x) is a/the terminal function (fulcrum)
through which any/all magnitudes transit (motion)
thus clarifying that the 'ordinary' number system
is underlying both: real, rational terminating integers and
imaginary, irrational non-terminating ratios (line and curve).
x=4 and x=4i incessantly returns the raw integer(s) ± 256 wherein '4' is
therefor *transcendental (according to the presence/absence of i acting on it)
and 'i' is a mere "direction reversal" which takes the same magnitude 256
and reverses it as -256, hence 4/√Φ also "couples" conjugation in general:
______________________________________________________
*π is "transcendental" in no part (partial or whole) to the
(erroneously approximated) legacy π number 3.14159...
but rather wholly and explicitly to the "real" integer '4' as it rests upon
a bed of MOTION: √Φ (√5 is a/the key to motion to the same degree(s)
it is the key circle to measure such to retrieve the precise π of 4/√Φ).
This places emphasis on both: the (scientific) imperative to recognize
and acknowledge we live in a universe of motion, and (thus)
the central finding of Dewey B. Larson.
Direction Reversal - Upside Down.jpg
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wherein 2 / (1+√5) or "upside-down" Φ yields Φ - 1 (0.618...).

I see the 25/2 as 2(2π + 2π + π)/2π which captures
each full rotational direction (there are 2, hence the base, and 4π+π power) plus
one "in-between" (captured by the remaining lone π) which may go "one way or the other"
according to the observer (factor) and/or root of the impetus (conscious and/or not).

In other words: from the apex of any pentagram (√5) there exists
two possible "directions" hence 4π captures both configurations:
two Roots (R) and two Operators (O) as R1, R2 and O1, O2:
R1 → O1/2 → O1/2 → R2 and
R2 → O1/2 → O1/2 → R1
which may (or may not!) be analogous to:
R+1 → O± → O± → R-1 and
R-1 → O± → O± → R+1
wherein O1/2 ( O± ) refers to one-of-two in any given binary, thus
the root order determines the ± "orientation" which is obviously
magnitude-invariant: it is just an upside-down or right-side-up situation.
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Re: Direction Reversal / Upside-Down (Think3D!)

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ckiit wrote: Sun May 31, 2020 2:53 pm If so, would this not mean that any two (reciprocally related) quaternions are bound to this same fulcrum?
Essentially, yes! The reciprocally-scaled (norm'ed) conjugate quaternion is reciprocally related to the same fulcrum (UNITY) → change of dimensionality!

The scalar part (a) of quaternion q = a + bi + cj + dk must necessarily be UNITY -- unit clock time or unit clock space; allowing for UNITY "3D" normalization such that all coordinate dimensions are normalized to the same scale provides for the Euclidean projection. One stratum up at metric/similar allows for differing scalar magnitudes per (coordinate) dimension wherein nonuniform scaling remains a valid concept given the assumptions in place.

Key to further understanding.
user737 wrote: Fri May 29, 2020 5:09 pm 23 = 8 as < 1, i, j, k > × < 1, -i, -j, -k > (dual quaternion) as 8D equivalent space → which bring us back to 20 = 1 -- which is self-referential with respect to UNITY in-and-of Phi (Φ): mass or its reciprocal aspect inside-out in 3 dimensions: gravity. A quaternion and its conjugate quaternion are reciprocals, scalarly-separated (turn) by a first-power to second-power, co-normed scalar ratio.
unit_quad.png
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ckiit wrote: Sun May 31, 2020 2:53 pm ...plus one "in-between" (captured by the remaining lone π) which may go "one way or the other"...
This is the link between the two quaternions -- each "magnetic" double rotating system. This is charge. Charge is a captured photon (+1 as conjugate ±π in counterspace). There must be a vibrational aspect (½ effective equivalent temporal displacement, ½ effective equivalent spatial displacement) to bind the two motions together as one at a single "location" in coordinate space (or coordinate time).

"½ effective equivalent" means x½ = √x wherein xeq ∝ √x. When comparing absolute primary magnitudes it will be necessary to place in ratio the "square root" of their equivalent proportions. This is a loss of dimensionality data and constitutes a loss of resolution.

A resonance point IS a unity vibration, as ½ of the vibration is ineffective (being in the same "direction" as the progression) and as 1×1=1, this additional scalar consideration cannot be additive -- the point is marked as ||q1|| / ||q2|| = 1 or ±π in counterspace -- equivalent 2π as "phase lock" in 3D coordinate space as we are unable to consciously distinguish between rotationally scalarly inward and rotationally scalarly outward (the bi-rotation).

Just as motion can be reduced to an equivalent "electric" (1D)* scalar ratio (ratio of ratios or cross-ratio), there is also a corresponding loss of resolution at the next higher power: 3D, as this is but a projective decomposition (P4 → P3) to 3D coordinate space (+ clock time). Mass is also but a shadow.

* "Electric Universe" main thesis is everything is electric. These believers "see" but the shadow, and only in 1D at that!

Do not let thine eyes deceive thee. Know of a higher POWER.
ckiit wrote: Sun May 31, 2020 2:53 pm ...real, rational terminating integers and
imaginary, irrational non-terminating ratios (line and curve)...
I appreciate how this is communicated. Equivalent space (or equivalent time) is the domain of the discrete (terminating), ordinal coordinate systems, whereas in this context the "imaginary" comprises the NRS of continuous (non-terminating) scalar motion (i.e. ratios). A divisional algebra is required to provide for a discrete projection.
ckiit wrote: Sun May 31, 2020 2:53 pm ...thus the root order determines the ± "orientation" which is obviously...
Yes, of course: i.j = k however j.i = -k -- order of operation matters as rotation is decidedly non-commutative. Pick up a Rubik's cube. Rotate in the x-plane then rotate in the y-plane. Reset. This time rotate first in the y-plane followed by rotation in the x-plane (same two planes, reverse order). Did you get the same result? (No, no you didn't.)

The change in "direction" is manifest as a change in polarity (positive ↔ negative) on the real number line.
ckiit wrote: Sun May 31, 2020 2:53 pm ...magnitude-invariant: it is just an upside-down or right-side-up situation.
NRS is magnitude-invariant -- "its magnitudes are absolute" -- and the cast into 3D coordinate space/time (conjugates) are just upside-down and inside-out in 3 (scalar) dimensions of motion as seen as discrete units of motion when normalized to a clock (clock time or clock space -- also conjugate expressions of the progression).

Equivalently-scaled conjugate quaternions would each argue the other is entirely upside-down, inside-out, etc. in 3D (not 4D as the real scalars are UNITY in both instances).
ckiit wrote: Sun May 31, 2020 2:53 pm I see the 25/2 as 2(2π + 2π + π)/2π which captures
each full rotational direction (there are 2, hence the base, and 4π+π power) plus
π is a "magnetic" rotation and so double π is two, double rotating systems -- two rotating photons -- the rotational base -- the quaternion (4D). There are then two of these -- the atom. Again, I suspect the remaining π captures the binding vibrational charge (secondary motion in counterspace).

As previously noted, the 2π in the denominator is indicative of normalization to clock space or unit rotation.

I do not presently have a good grasp as to why two raised to the power and so may be completely wrong on this point.

q8v2.jpg
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This is quite interesting. I realize now I have still been visualizing a dual quaternion as two overlapping spheres in space when they are but scalar motions... NO geometry. I have allowed my crutch to become my reality. Bad. As these are scalar motions they will be nested in counterspace.

These are the 8 quantum levels of the atomic system... 4 discrete (i.e. quantum) inverse speeds (i.e. relative base energy datums) from the first quaternion and 4 more discrete energy levels from the second quaternion further nested within extended to counterspace infinity at the center 'point at infinity.'

The second quaternion rotation is an intrinsically-linked albeit separate "imaginary" rotation within the context of the containing quaternion. This makes the atom a 6D structure or an equivalent 7D for purposes of interaction in space.
bperet wrote: Fri Mar 02, 2012 9:48 am Based on my recent update using Mathis' information, I would conclude that projected motion is 4-dimensional (the underlying, scalar motion is still 3D) and that tetrahedral geometry is a better representation of this structure. Unlike the cube and the sphere, the tetrahedron is self-dual; the geometry is the same even if you yank it inside-out, so it's applicability to either space/time (Material) or time/space (Cosmic) is uniform. The illusion of cube and sphere would be an artifact of our consciousness normalizing the tetrahedral system (outsides = linear, insides = polar).
Only 8 years behind Bruce. We were both having mid-morning tea:
user737 wrote: Wed May 20, 2020 9:02 am The process of creation of the unit circle ("circling the square") is in itself the act of normalization to clock time!

Conversely, the process of creation of the unit square ("squaring the circle") is in itself the act of normalization to clock space.
Look at all those 2's and 0's. Digits confirm.

Whereas:
bperet wrote: Wed Jul 25, 2012 9:54 pm The projection of a scalar speed onto a coordinate system always ends up with BOTH the linear and rotational shadows at the same time.
And:
user737 wrote: Wed May 27, 2020 4:00 pm Neither Φ nor π are solely primary motion. They are both combinations of primary and secondary motions.
This whole board is just one big game of connect-a-dot mixed with logical thinking. It's not in linear, book format so it's almost like a Choose Your Own Adventure novel (closed set of rotational operators that always get you to "the end"). How absolutely befitting coming from a man like Bruce (yin thinker).

dual_tetra.jpg
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Djchrismac wrote: Fri May 29, 2020 5:18 pm Like this? :D
Image

Source: Dual Quaternion model -- two interpenetrated tetrahedrons, the star tetrahedron

These appear to be suitable dimensions for initial consideration:

Image
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