We know π is 4 (in counterspace).

Well then, what is Φ? (Legitimate question.)

user737 wrote: ↑Mon Jun 01, 2020 9:38 am

[1] The

**REAL**: Φ

^{0} = 1 and provides for the 'roots of +1'

*ONE* solution of

*apparent* polarity (±√1 becomes ±π in counterspace). The only way ("forward") is One.

The

**IMAGINARY** (3-fold -- Father, Mother, Child):

[1] Dimension

*i* as Φ

^{1} = Φ

^{1}
[2] Dimension

*j* as Φ

^{2} = Φ

^{1} + Φ

^{0}
[3] Dimension

*k* as Φ

^{3} = Φ

^{2} + Φ

^{1} AND/OR Φ

^{2} - Φ

^{1} = Φ

^{0}
...and form the 'roots of -1' as an infinite set of 3-vector solutions in "imaginary" (temporal) coordinate 3-space.

Once at Φ

^{2} we have a

*choice* or what we call the (potential) "

direction reversal".

(1) We can proceed "backward" towards Φ

^{3}, or

(2) We can proceed "forward" back towards Φ

^{0} (=1)*

***Gotcha!: "imaginary" is in counterspace and so we have crossed a unit boundary! "Forward" is "backward" and "backward" is "forward" as inward ↔ outward due the ***apparent* inversion when observing motion across *any* unit boundary.

Got myself! Came to me in the shower! (In the bathroom but I didn't first fall down and bump my head on the sink.)

Φ

^{3} is

*also* observed from across a unit boundary (same as Φ

^{2} and Φ

^{1}). Φ

^{0} IS the unit boundary.

This means Φ

^{2} - Φ

^{1} = Φ

^{0} = 1 → -1 due an

*apparent* inversion of polarity!

user737 wrote: ↑Wed Jan 29, 2020 2:46 pm

ckiit wrote: ↑Wed Jan 29, 2020 3:55 am

What are the odds of a single polynomial whose +/- root solutions are +1 and +phi^3, thus scale invariant?

Would that not be an equation which describes the physical universe?

+Φ

^{3} = Φ

^{2} + Φ = Φ

^{2} / (Φ-1) = (Φ+1) / (Φ-1) = -Φ × Φ

^{2}= -Φ

^{3}
(1/Φ)

^{2} + 1/Φ = +1

where Φ = (1 + √5)/2 -- the golden ratio

See that negative sign in front of Φ

^{3} off to the right? That's not a typo.

See how the order of operations is reversed from the first part of the sequence referencing Φ

^{2} and Φ

^{1}?

My mistake was in the

operation applied -- being the good, little linear thinker I am, exchanged sum for product as it was the only manner in which I

**believed** a difference could be captured. That was

**wrong**. I admit it. The difference is in the

*order of operations* (rotation is non-communitive):

*i*.

*j* =

*k* whereas

*j*.

*i* = -

*k* (the

*negative*).

Furthermore, scalar rotation inward and scalar rotation outward cannot be discriminated and so they are

both ±

*rotations*.

Let's correct this for the record:

±Φ^{3} = Φ^{2} + Φ^{1} = Φ^{2} / (Φ^{1} - Φ^{0}) = (Φ^{1} + Φ^{0}) / (Φ^{1} - Φ^{0}) = Φ^{2} - Φ^{1} = ±Φ^{3}

Just for fun: "product" of powers as

(a) ∑ of differences of 3 - 2 - 1 - 2 - 1 - 0 - 1 - 0 - 1 - 0 - 2 - 1 - 3 = -11 ⇒ flip/invert (with datum offset) ⇒ -(1/12)

(b) simple sum: 17

Background reading
Looking in

another *light*... let's make it

two lights (one seen, one unseen*) as a DUAL projection:

* Miles' "charge field"

One REAL forming the Cosmic Sector of 3D space (from our

*perspective* appears as inverse '1-x' region as equivalent “0D” space) as:

**Projective**: *no*

*parallelism* and *no* "points" -- NO geometry (4D as '2x2' matrix of motions i.e. quaternion) -- purely SCALAR: 2

^{0} = 1 scalar direction ("no dimension" of NRS) → Φ

^{0} = +1 ⇒ PROGRESSION as is

*conceptualized* as OUTWARD motion "away" from UNITY in both time and space (both linear and rotational).

Motion is defined as a simple

*ratio* of a scalar magnitudes (internal 2D structure) comprising a magnitude of spatial

*aspect* in proportion to a magnitude of temporal

*aspect* or vice versa.

Net push/pull (force), net "spin" (torque) is all that can be projected to lower stratum.

bperet wrote: ↑Wed Dec 04, 2019 11:09 am

user737 wrote: ↑Mon Nov 11, 2019 6:06 pm

EDIT: Bruce, please note det(A,B;C,D) needed for the invariant cross-ratio is missing D or if we wish, D = 1 + 0i. Is this not analogous to the cross-ratio at the affine level with one point located on the plane at infinity? Are we

*certain* the projective stratum is 15 DOF and not really 16 DOF? Thoughts welcome.

Yes, I could see that as analogous. I use 15 DOF because the 16th is not variable--it is the progression of the natural reference system, which is always there at unity. Not something that can provide another degree of freedom.

Direct question: direct hit, the significance of which has gone unrealized until now.

There are

**4** "points" (2D-2D); one (1) point is

*always* at infinity (by definition this

*creates* the Affine), and constitutes the

*natural datum* offset of '1' between the NRS (unity) and

*all other* projective stratums (zero-based). Wherein quaternion q = a + bi + cj + dk as a=1 or seen otherwise: A-B-C (Larson notation) where D = 1 + 0

*w*; the three remaining points dictate stratum.

This will help us to work through the relative datum offsets:

*** Observing from this scalar "direction" is 3D coordinate space + clock time ***

**All/One** as

**Three** (

**Four** as 3 of 3 plus 1) planes at P

_{∞} due

*perspective*: Outward is P

_{0}→Π

_{∞} ⇒ c-Projective stratum

**None** as

**Zero** (0 of 3 as

**Four** minus 1) points at Π

_{∞} ⇒ m-Affine stratum ⇔ c-Euclidean stratum ⇐

**Two** (2 of 3) planes at P

_{∞}
**One** (1 of 3) points at Π

_{∞} ⇒ m-Metric stratum ⇔ c-Metric stratum ⇐

**One** (1 of 3) planes at P

_{∞}
**Two** (2 of 3) points at Π

_{∞} ⇒ m-Euclidean stratum ⇔ c-Affine stratum ⇐

**None** as

**Zero** (0 of 3 as

**Four** minus 1) planes at P

_{∞}
**All/One** as

**Three** (

**Four** as 3 of 3 plus 1) points at Π

_{∞} due

*perspective*: Outward is Π

_{0}→P

_{∞} ⇒ m-Projective stratum

*** Observing from this scalar "direction" is 3D coordinate time + clock space ***

Within this real number is contained a three-fold "IMAGINARY" forming the Material Sector (from our

*perspective*) -- extending infinitely inward (in space) or OUTWARD IN TIME as:

ULTRA-HIGH SPEED REGION (3-x) as 2D space / 1D time --

**Affine***:

Motion as

*projected* with

*no* invariant points at infinity is

*orbital* (circumferential) ⇒ 2

^{1} = 2 scalar directions (1-vector) → Φ

^{1} = Φ

^{1} ⇒ m-Electric (c-Magnetic) i.e. +1→

*i* dimension as 1.

*i* =

*i* rotation is ELECTRIC.

The

**1-vector** (linearized) is a '1x2' matrix of complex elements (bi-vector) of binary polarity either '+' "positive" or '-' "negative" and, as no discernment is possible

*in equivalent space* ("with thine eyes") -- as rotational scalar "direction" cannot be directly observed, forms a 1-pole (

**2DOF**) system which reduces to a SHM (-1↔+1). Therefore ELECTRIC CHARGE (electric ionization level) is a 1-dimensional rotational vibration in the TR. Legacy science believe all ionic matter (i.e. charged matter) is "negatively" charged is due the failure to observe the proper datum of 1 whereas the shadow is a projection "about zero."

Turn m-Electric inside-out in (scalar) 1D is c-Magnetic (

*equivalent* 1D space). Integral 'either/or' perspective normalized to first-order clock space (2π) -- spin ½ (360° or 2π radians) normalized to unity time -- "½ spin" (in actuality, 1D angular) rotation from our

*perspective*.

********** 3D↔3D overlap forming 4D region conjugated across m-Electric ↔ c-Magnetic ⇔ m-Magnetic ↔ c-Electric **********

INTERMEDIATE SPEED REGION (2-x) as 1D space / 2D time --

**Metric** or Similar:

Motion as

*projected* with

*one* invariant point at infinity is

*growth measure* (hyperbolic) ⇒ 2

^{2} = 4 scalar directions (3-vector) → Φ

^{2} = Φ

^{1} + Φ

^{0} ⇒ m-Magnetic (c-Electric) i.e.

*i*→

*j* dimension as

*i*→

*i*.

*j* =

*k* rotation is MAGNETIC.

The

**3-vector** (linearized) is a '2x2' matrix of complex elements (dual bi-vectors) formed as combinations '++', '+-', '-+', and '--' as rotational scalar "direction" cannot be discerned in space, this 2-pole system degenerates into

**3DOF** in observation (-1↔-0/+0↔+1) as '-0' and '+0' are indistinguishable and

*order matters* i.e. '+-' is not the same as '-+' although '++' IS the same as '--' from our vantage point. Correspondingly, MAGNETIC CHARGE (magnetic ionization level) is a 2-dimensional rotational vibration in the TR.

Turn m-Magnetic inside-out in (scalar) 2D is c-Electric (

*equivalent* 2D space). Double integral perspective in coordinate space/time normalized to second-order clock space (4π

^{2})** -- "spin 1" (720° or 4π steradians) normalized to unity time --

*single* (in actuality, 2D solid) rotation from our

*perspective*.

LOW-SPEED REGION (1-x) as 3D time --

**Euclidean**:

Motion as

*projected* with

*two* invariant points at infinity is

*step measure* as recti-Euclidean (recti-linear) in conjunction with

*reciprocal* TR as polar-Euclidean (back to growth measure) ⇒ 2

^{3} = 8 scalar directions (7-vector) → Φ

^{3} = Φ

^{2} + Φ

^{1} and equivalent linear "polarization" as rotational conjugate Φ

^{2} - Φ

^{1} = Φ

^{0} ⇒ Mass as Gravity i.e.

*k*→-1 dimension as

*i*.

*j*.

*k* (= -1) rotation is MASS (

*equivalent* 3D space).

The

**7-vector** (linearized) is a '4x4' matrix of complex elements (two of dual, dual bi-vectors) formed as combinations '++', '+-', '-+', and '--' with '++', '+-', '-+', and '--' ⇒ two pairs of the sets of '++++', '+++-', '++-+', '++--', '+-++', '+-+-', '+--+', '+---', '-+++', '-++-', '-+-+', '-+-+', '--++', '--+-', '---+, and '----' form a dual 4-pole system. Out of all of these combinations there is only

one member of the set that could be considered redundant with the exception of the whole second set (order of operations matter): '++++' and '----' are equivalent as, again, rotational scalar "direction" cannot be directly observed. Our 16DOF becomes

**15DOF** with the remaining magnitude capturing UNITY (1 + 0

*w*). If this had a geometric shape it would be the star tetrahedron. The CENTER of the two tetrahedrons are ONE in unity (in coordinate space) -- the whole is one in actuality.

Turn mass inside-out in (scalar) 3D -- as is the job of the second of the

*dual* quaternion -- and we have GRAVITY (3D speed). Mass and gravity are a conjugate reflection in one. Gravity (-1) is opposite scalar "direction" than Progression (+1). Whereas (2π×4)π

^{2} → 8π

^{3} is a (3D) hyper rotation projected about an arbitrary coordinate zero-datum center, we must utilize infinite factorials in performance of a recursive sum (double, as in

*difference*, of double integral) back to the

*local equivalent* unity-datum (natural harmonic of unity) in both "directions" in determination of

*equivalent* 3D space --

*dual* (in actuality, hyper or

*dual* solid) rotation from our

*perspective* -- 1440° or 8π

^{3} hyper-radians. The net sheer is what we measure as temperature.

GRAVITATIONAL CHARGE becomes isotopic mass and is the atomic capture of

*charged* electron neutrinos (υ

_{e}*) -- a 2D "magnetic" base of zero net displacement plus photon as linear vibration (1D electric charge).

Looking at the units: 2D energy (inverse speed) is t

^{2}/s

^{2} (weight or momentum) per speed (s/t) -- speed (light) IS charge ⇒ t

^{2}/s

^{2} / s/t = t

^{3}/s

^{3} (mass).

Compounding the motion that is the atom creates additional inward motion (mass and gravity) until such time as it reaches a maximum (5: 5: 0). The charge is

*required* to provide for the capture as isotopic mass otherwise the net zero displacement of the electron neutrino (moving at the "speed of light" due the one free dimension) moves through all net displacements, spatial or temporal, as like a current of no resistance. Light binds all.

Middle two (Φ

^{1} & Φ

^{2}) are

*conjugates*, bookends (±Φ

^{3} & ±1) are an equivalently-polarized pair of

*inverse conjugates*. Dualized combination contains a

*nested* *local* (biased)

*apparent* reciprocal conjugate pair as Φ

^{2} ± Φ

^{1}. Very chic.

* did you know?: Affine is also defined as a relative by marriage ⇒

*related through marriage* (of Φ and π). Your children will most likely be similar (Metric "blending" as the combination of two into one). Isn't it amazing to watch your child's appearance change back-and-forth between you and your spouse's facial feature set as they

*grow*? The Affine/Metric/Euclidean stratums are

*static* “timeslices” in order as opposed to the harmonious

*dynamic* of the Projective stratum.

** this implies that π is a

*unit* and π

^{2} a

*unit measure* -- much like area (say, m

^{2}) is length (m) × width (m). These are radians (π) and steradians (π

^{2}). What is π

^{3}?

*Hyper*-radians? These would

all be necessary for

*equivalent* unity-scaling to a common clock. Played out in

*linear* time clock, the

*step measure* aspect of time is to scale

unbounded turn as

bounded rotational cycles (in time):

**1**0 CALL FORM (BIRTH)

**2**0 CALL ATTRACT

**3**0 CALL GROW

**4**0 CALL DISBURSE (DEATH)

GOTO 10

From a "point" of equivalent "0D" comes a Radius of equivalent "0D" (as 1D Line), then a Planar/Angular 1D rotation (as 2D Plane), then a Solid 2D rotation (as 3D Volume), then a Hyper 3D rotation (as 4D Hypervolume of equivalent "0D" from one

*perspective*). This then collapses back into a 3D Volume (Solid 2D rotation) into a 2D Plane (Planar/Angular 1D rotation), then a 1D Line (as Radius of equivalent "0D") becomes a "point" of equivalent "0D" (from the "other"

*perspective*).

The first half is a process as-observed

*with respect to* time; the second half are polarizing (poles)

*in time*. The offset datum

*is* equivalent CLOCK TIME. The whole may also be viewed in the opposite (i.e. scalar "direction")

*perspective* in CLOCK SPACE in what would be described as "backwards" time to the Material-biased observer.

Lines 10 & 40 are inverse conjugates and are

*perceived* as polarizing motions at a "point" (no dimension)

*in time*; Lines 20 & 30 are perceived as on-going

*processes* possessing dimension --

*inverse speeds* projected as equivalent

durations with respect to clock time -- which overlap

*in time* as conjugate motions.

This makes sense as we must attract others' knowledge to grow ourselves, and our growth in knowledge begets the attraction of others. Start (birth) and stop (death) are but points of reference and contain within no real significance. They are but markers which signal the cessation of the current cycle and the genesis of the next. There "was" no beginning and there "will be" no end. The time is

**now**.

We can flip the WHOLE thing over (space ↔ time) to create the same structure from Progression to 1D/2D/3D

*equivalent* time, and countertime (SR), and the time-space (TS/STR) to from the "unseen" half of the

**Universe of Motion** -- motion in 3D time -- what Larson refers to as Cosmic, the land of what legacy science refers to as anti-matter (really,

*inverse*-matter). One of one is one. This series is based in π! How exciting and waiting to be discovered!

The dimensions ARE the

*turns* between the planes

*in time* as

*projected* as 1D/2D/3D equivalent space. Again, we don't

*see* space. We see

*time* (as equivalent space) as points "in" space; we are collectively observing

**one** compound motion

*of time*. Now realize

in order to experience the 'flow

*of time*' we must create an

ordered set

*of time*... we are watching a movie and where YOU focus YOUR energy (your conscious observation

*and* resulting action) is where YOU affect change as a whole.

user737 wrote: ↑Sun May 31, 2020 2:56 pm

ckiit wrote: ↑Sun May 31, 2020 2:53 pm

I see the 2

^{5/2} 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**.

Lol. This is the discrete value for

*e* (=2.71828...): Quantum

*e* = 2 ... and I bet dollars to donuts that the current irrational value is

**wrong**. This would introduce additional complexification to the matter noted

here.

Time to improve

the chart.

user737 wrote: ↑Mon May 25, 2020 10:19 am

There

*must* be a natural basis for

*e* and it

*must* be expressed as a recursive function of '1' as are both Φ (Phi) and π (Pi).

We must make

**whole** all so-called '

constants' (

*pseudo*-discrete entities), such as they are presented, as they are neither dynamic nor rational, as they should be (i.e. they are but approximations of Truth in their current form), and instead recognize the UNITY in ALL.