Back to Basics: How to Measure a Circle

[A Gnostic Agnostic]

Yes, but 2 = 2*Pi/Pi, which is what you have in your expression.

I don't have 2*Pi/Pi in any expression of mine.
I do not even use such notation.
That is your expression.

I have:

Φ = (π+π√5)/2π
Φ² = (3π+π√5)/2π
both on a common base of 2π.

Just a yes/no question: do you acknowledge the addition
of a discrete 2π in the numerator of the squared expression?

Just a note: this addition of a discrete 2π in the numerator is the constituency of the +1 in/of Φ² = (Φ + 1).
The base of the expression(s) must always remain in 2π (full circle) format, as this serves as a scalar base of time (circular).

So which one is the most definitive proof? Let's analyze it.

They are all definitive - neither one more or less than the other.
The π by way of Φ video is the easiest (visual), it is only a couple of minutes:

https://www.youtube.com/watch?v=_j2tq1GsXbI

In my opinion, the "best" to "analyze" is the picture /w provided method
as the video can be used as visual reference if/when needed:

https://postimg.cc/9R1nKMK8

Open the picture and follow the method/annotation to follow
and point if/when/where you believe anything is "wrong".

I invite anyone/everyone else to do the same.

Method
1. Begin with a unit square about the origin (center of image).
(Recall: Φ² = (Φ + 1) and/or x² - x = 1 as geometric).
2. Extend two opposite sides by 0.5 units each towards each axial direction (±x, ±y).
(Note: this composes two 2x1 rectangles perpendicularly situated, each having a diagonal(s) √5).
3. Use the vertices of these rectangles to compose a circle whose diameter is (also) √5 (BC shown).
(Note: we so far have a √5 diameter circle with two √5-diagonal 2x1 rectangles inscribed perpendicular to one another).
4. Extend BC by 1 unit (A shown) such that AB = 1 and BC = √5 thus AC = (1+√5).
5. Divide AC in half as D such to satisfy (1+√5)/2 as AC/D.
(Note: point D represents the golden ratio 1.618... thus is of principle concern as it relates to the circle/square).
6. Focusing on D, rotate AC about the origin for one full rotation. Note the following of D upon doing so:
i. Point D incessantly coincides with a/the (entire circumference of) the circle 2r = 1 within the unit square.
ii. Point D "kisses" the associated unit square four times equidistantly (axially: ±0.5 on both x/y and/or real/imaginary planes).

Pi can thus be measured as a natural product of a square without the need for any approximation(s)
and can be expressed as a discrete ratio of the (square root of) the golden ratio. This ratio is 4/√Φ = π.

None of them are correct.

This is your belief, and it takes a "believer" to "believe" the opposite of what is true.
If you knew this to be true, you could reason why. We are all still waiting for that.

Pi is a number that relates the circumference of a circle to its diameter. There are no squares here. Anywhere.

What do you mean there are no squares anywhere?
The 2π we are measuring is the product of a square.
It naturally comes by way of a square.

Do you understand what it means to square something, as in Φ²?
If not... it speaks for itself.

Also:
Pi is not (only) a number, it is a ratio, and it doesn't only relate a circumference to a diameter.
That is what it is "defined" and/or "believed" to be, but the real underlying relation is 2r/c/4 = 8r/c.

A diameter is 2r. All diameters have a multiplicative '2' as their constituency. So a diameter d is actually 2r.
The same 2r may also compose a right angle which quarters the related circle.

Neither there are triangles, pentagons, hexagons, octagons or anything else you can inscribe a circle into.
So your argument about squares is invalid.

Perhaps there is another kind of 'invalid' at play here, as
nothing provided has invalidated any argument I have made
as I am still waiting for an actual argument. You can not
invalidate anything without making a actual argument, let alone
a valid one.

[paladin17]

I don't have 2*Pi/Pi in any expression of mine.
I do not even use such notation.
That is your expression.

I have:

Φ = (π+π√5)/2π
Φ² = (3π+π√5)/2π

(π+π√5)/2π = (1+√5)/2.
That is your expression. It doesn't depend on Pi. At all. It simply doesn't contain it.

The π by way of Φ video is the easiest (visual), it is only a couple of minutes:

https://www.youtube.com/watch?v=_j2tq1GsXbI

Unfortunately, the video contains no explanation at all.
It simply artificially introduces (at 01:46) Pi = 4/Sqrt(phi) without any proofs.

https://postimg.cc/9R1nKMK8

Open the picture and follow the method/annotation to follow
and point if/when/where you believe anything is "wrong".

The statements on the right ("Pi/4 can be expressed ...") and at the bottom ("Pi = 4/Sqrt(phi)") are wrong.
They don't follow from any considerations and are introduced artificially by the author.

Pi can thus be measured as a natural product of a square without the need for any approximation(s)
and can be expressed as a discrete ratio of the (square root of) the golden ratio. This ratio is 4/√Φ = π.

I totally agree with your points 1-6, there's nothing wrong there. The wrong statements begin here.
Just as in the examples above, the [second] underlined statement doesn't follow from anywhere and is not proven.
Moreover, it cannot be proven (though you may try, of course).

The 2π we are measuring is the product of a square.
It naturally comes by way of a square.

No. It is a ratio between the circumference and diameter of a circle.
Sorry, no squares. No powers too.

[A Gnostic Agnostic]

This will be the last dialogue we have, as there are basic things
you do not seem to understand (and/or do not want to):

(π+π√5)/2π = (1+√5)/2.
That is your expression. It doesn't depend on Pi. At all. It simply doesn't contain it.

It is imperative to keep a base of 2π - you keep trying to reduce it back into arithmetic
when we are talking about a kinematic geometry. Leave the base as 2π and square it.

(π+π√5)/2π = Φ
(3π+π√5)/2π = Φ² = (Φ + 1)

The latter reflects an addition of 2π in the numerator emergent as a natural product of a square (on a base of the same 2π).
We do not reduce anything back into mundane arithmetic - we are talking about a scalar kinematic geometry that is intrinsically
normalize to '1' as the square arithmetically becomes one greater than itself. This is the universally unique quality of Φ, and
it happens to coincide with the addition of a discrete 2π.

Unfortunately, the video contains no explanation at all.
It simply artificially introduces (at 01:46) Pi = 4/Sqrt(phi) without any proofs.

If you don't understand how it follows, it is unfortunately a local barrier.
It was shown how/why each quarter of the circle is in relation to 1/√Φ
which is the part you seem to have trouble confronting/understanding.

The statements on the right ("Pi/4 can be expressed ...") and at the bottom ("Pi = 4/Sqrt(phi)") are wrong.
They don't follow from any considerations and are introduced artificially by the author.

Here we go again with the reflexive "wrong" with absolutely no substantive explanation.

For the others reading (he will not understand this): the √2/2 square correlating the 4 equidistant points
is the (only) square bound to the golden ratio. It follows that the reciprocal
of √Φ correlates 1/4 of the square to 1/4 of the circle, thus
1/√Φ = π/4 and from here on out, line and curve are immutably reciprocally related.

This poster will just cycle through denying this integral relation because
the answer doesn't agree with what he wants to be true (and/or false).

I totally agree with your points 1-6, there's nothing wrong there. The wrong statements begin here.
Just as in the examples above, the [second] underlined statement doesn't follow from anywhere and is not proven.
Moreover, it cannot be proven (though you may try, of course).

If 1-6 are correct, the deduction is correct. You can't say all the premises are correct
then deny the natural consequence - that is not how logic works.

For the others reading - this individual can/will only deny the result he doesn't like
because he doesn't like it. That is all that has and would happen should I not have chosen to disengage.

No. It is a ratio between the circumference and diameter of a circle.
Sorry, no squares. No powers too.

A diameter is 2r. A right angle is the same 2r.
2r/c/4 = 8r/c. This is a valid ratio.

A circumference/diameter relation needlessly discriminates against other possible relations,
such as the one above. It is true: present-day "approximated" π is ill-defined and ill-calculated.

For the others to close: the Giza pyramid was designed/constructed with a height of √Φ
and four axial radii of '1'. This implies precisely what I am indicating: each quarter is 1/√Φ.
The King's chamber is the same √5 integral to the circle viz. π² = 8√5-8.

However not even a Giza pyramid would convince some people of what is in plain sight, as
they will reject whatever they do not like and/or does not fit their presuppositions. This
all relates to the OP and highlights the need to incessantly challenge basic underlying assumptions.
This individual was unable to do this, and thus the impasse is duly reflect in/as the correspondence.

[antosarai]

That is all that has and would happen should I not have chosen to disengage.

...
Isn't it a pity?

But then perhaps you'd engage Mr. d3x0r and Additive Rotations - P=1/2at^2+vt+m for rotations?

Or isn't that kinematic geometry either

[d3x0r]

I've actually been there.. I get 4r and 2r^2 and even 2r^3 for metrics of circles and spheres in 1-norm space.

It doesn' t help sin/cos are still non-linear functions https://www.desmos.com/calculator/ucxjuvr0ry

Hmm this is ugly... will work on reformatting...

m(x,y) is a mod function (x/y - floor(x/y)) * y
w(x) is a wrap for 4... m(x,4)

s(x) = |(-(|w(x)-1|)+2|
S(x) = s(x/pi/2)

which makes a nice saw tooth wave for 'sin' in 1-norm... only it's not... there's a small deviation between sin/cos converted to 1-norm and a straight line...

https://www.reddit.com/user/dehker/comm ... inates_in/ (image)

while (x+z=45) with a linear iterator does nearly look smooth... there is a small nonlinearity between the middle and the ends (near the planes of intersect) where the middle is slightly more spread out than near the ends of the triangle.

I've been working with this other guy on IRC who instists he can find better series for sin/cos than taylor series ..... but really the methods to divide the circle into appropriate angles to get the right coordinates can be done with like a itertive binary division using iscosclese triangles - divide first by 45, then the 45 in half then that appropriate side in half etc... for 54 divisions you could get a good result for sin/cos up to the precision of a double... but that's a lot more work than 5or 6 iterations of taylor series.

[d3x0r]

https://www.youtube.com/watch?v=GGj399x ... 8718451C47 This playlist by this Insights into Mathematics guy...
An Invitation to Geometry | WildTrig: Intro to Rational Trigonometry 0 | N J Wildberger

https://www.youtube.com/watch?v=Nboydwa ... 7&index=88 And specifically this really puts together that turns, radians, degrees, or natural rotation units can all be used to describe an angle and they're all scalar multipliers; if you really don't like PI in the terms of working with circles, convert to something that doesn't have Pi; say the ancient babylonian system of degrees (Radians are actually a new-fangled thing that relates arc-length to angle... and then dealing with the curvature of a thing is easier.


The difference between a rotation and a curvature; a rotation is a pivot of a point around another point... a curvature applies a change relative to the other point's current position, going in a direction. One relies on basically 3 points the other is self-relative.

A constant K of curvature describes a circle with a radius of 1/K. ... There are no points in the universe, but there are lines - or things which have 0 curvature (is an infinite circle, and breaks the 1/K math) Whereas on the other side a circle with radius R has a curvature of 1/R... so a circle of 0 radius breaks the division in that direction.

Edit: I guess a full circle is actually 8 in natural rotation coordiantes 2+2+2+2 ... with a radius of 1... so really 8r or 4*diameter
I'll have to revisit the area and volume too

[paladin17]

(π+π√5)/2π = Φ
(3π+π√5)/2π = Φ² = (Φ + 1)

It is impossible to express phi through Pi (or vice versa) using these expressions, since the left side does not depend on Pi.

It was shown how/why each quarter of the circle is in relation to 1/√Φ

No, it wasn't. The expression that appeared there is a non sequitur.

If 1-6 are correct, the deduction is correct. You can't say all the premises are correct
then deny the natural consequence - that is not how logic works.

I can, if the "consequence" is not connected to the previous statements in any way whatsoever, which is what we have here.

A diameter is 2r. A right angle is the same 2r.
2r/c/4 = 8r/c. This is a valid ratio.

A diameter is a line segment. A right angle is an angle.
They are different, unrelated things.

A circumference/diameter relation needlessly discriminates against other possible relations

I agree with this. There are many ways of calculating or measuring Pi, none of which are connected in any way to phi.

[A Gnostic Agnostic]

That is all that has and would happen should I not have chosen to disengage.

...
Isn't it a pity?

But then perhaps you'd engage Mr. d3x0r and Additive Rotations - P=1/2at^2+vt+m for rotations?

Or isn't that kinematic geometry either

It is a pity because over 2000 years of human ignorance is contained in/as the "approximation" of π.
The magnitude of the error is, in my experience, wholly incomprehensible to most, thus we will have
people who do not pay the attention merited by the magnitude of the error. It is immense.

P=1/2at^2+vt+m may attempt to describe a kinematic situation(s) in a limited context, but is itself not kinematic.
A kinematic geometry must have a periodic (rotational) base (of 2π) that grounds/normalizes all motion indiscriminately.

The photon (and physical universe) can be represented as a quaternion(s) 1, i, j, k

q = a + bi + cj + dk

wherein ±π forms a dual conjugate quaternion(s) that give rise to recti-Euclidean and polar-Euclidean geometries (within which kinematics may be expressed).

I will address this in my reply to d3x0r.

Edit: I guess a full circle is actually 8 in natural rotation coordiantes 2+2+2+2 ... with a radius of 1... so really 8r or 4*diameter
I'll have to revisit the area and volume too

Yes, this is correct. I encourage the following thought experiment (for others):

1. Imagine two 2x1 rectangles perpendicularly (axially) situation about an origin.
2. Use the vertices to compose a circle with diameter √5.
3. Travel back-and-forth along each of the 4x √5 diagonals.

This is whence we get our symmetrical '8' that captures the circle, hence:

2r/c/4 = 8r/c or c = 8r
viz.
π = √(8√5-8) = 4/√Φ

There is no such thing as a perfect circle that is not in golden ratio proportion.
In fact, the quaternion can be used to describe the photon and/or physical universe:

MATRIX.jpg

https://i.postimg.cc/bJxF8KzY/MATRIX.jpg

wherein e^πi + e^-πi = 2cos(x) re: (2D→1D) motion
(this is with the correct value of π as 4/√Φ).

We already have the photon/physical universe modeled as a (dual) quaternion(s) and understand what light is and how it works. This is not the issue, the issue is humanity does not know how to measure a circle, and instead approximates it with straight lines that completely ignore the need to measure the √5 diameter circle (this circle normalizes all motion to the unit square and/or 2r = 1 circle).

The incorrect "approximated" value of π is the obstacle. If EU can pick up on this problem/solution, they will be able to show that Einstein's e = MC² invalidates Einstein's own theory. The universe is not gravity-centric, as:

π = 4/√Φ
π² = 16/Φ
16 = Φπ²

...is the real relation underlying e = MC², wherein Φ is a scalar 1D mass-electric constant and π² is a scalar 2D velocity-magnetic constant (gravity is an inward motion / acceleration). The real/imaginary numbers become linked/coupled to the physical universe by way of the equality above.

This union of real/imaginary happens at f(x) = x⁴ + 16x² -256 which is the "junction function" through which all such relations immutably transit. π is a root of this function (along with its own conjugate) and relates to imaginary counter-parts that are off-set by a factor of 16, hence 16 = Φπ².

[d3x0r]

P=1/2at^2+vt+m may attempt to describe a kinematic situation(s) in a limited context, but is itself not kinematic.
A kinematic geometry must have a periodic (rotational) base (of 2π) that grounds/normalizes all motion indiscriminately.

It is... when evaluated the angles get converted through sin/cos to periodic functions... and doe fold at 2pi.... but can also represent rotations within 2pi to 6pi... -2pi to 6pi... 6pi to 10pi,. .. and periouds of 4pi after that... so saying that 2pi is the limit is quite wrong.... the real rotation period of rotating a rotation around another rotation is 4pi, and not 2 or 3 pi.

Quaternions only represent 1/2 of the possible rotation coordinates... since the Theta angle put in is cos(theta/2) even when you get out is only +/- pi or +/- 180 degrees, and fails to represent any rotations with degrees beyond that. And also fails to represent any single rotation of 181 degrees... which is probably why spins re only in +/- 1/2 because that's the maximum extend of the rotation expressible with hamiltonians.

We don't know how light works - it relies on stretching space, which isn't a thing.


Don't really get where you're going with sqrt of 5... that looks like trying to force rotation coordinates back into 2-normal somehow... which is just going to be self defeating.

Even if pi is an approximation, as soon as you want 2/5'ths of an arc you get an infinite, non precisice value that's just as far off as PI in base 2 calculations....

Edit2: Quaternions also don't allow you to add rotations or compare them deferentially or relatively. So why even mention that as an altenrative when the system I presented already does work? Even Log(Quaternion) which is supposed to represent the angles of the rotation, doesn't, and doesn't provide additive rotations; if they did, I wouldn't have had to continue to work on them... but once I did figure out the error fractor between log-quaternions and actual additive rotations, it all became quite clear why a lot of other math ended up the way it did.

[d3x0r]

I apologize I'm stealing your thread a bit, and really if you want to talk about rotations and challenge the system I have; I've yet to be defeated by any challenges And SLERP as a smooth rotation function is fairly bizaare in natural rotation coodinates.

On topic - Circles
If you standardize a unit measure on the circumference/perimeter of a circle, that's a great, but then to construct that circle you need another distance, either in some other unit entirely (inches and centimeters for instance... but radial length vs arc length) so that it can remain in 'natural' units, but then the conversion will have a n/pi or n*pi scalar builtin ... kinda like that pesky 'G' constant. I'm not sure what you are arguing in favor of or against. Yes, we can only hope to approximate it, like Phi (golden ratio); e (as in the base of ln() ) ; sqrt(2); 1/3 in base 2....

[A Gnostic Agnostic]

It is... when evaluated the angles get converted through sin/cos to periodic functions... and doe fold at 2pi.... but can also represent rotations within 2pi to 6pi... -2pi to 6pi... 6pi to 10pi,. .. and periouds of 4pi after that... so saying that 2pi is the limit is quite wrong.... the real rotation period of rotating a rotation around another rotation is 4pi, and not 2 or 3 pi.

If it folds at 2π, then 2π precedes expressions requiring conversions.

2π is not the limit, it is the minimum which defines a beg/end circle/cycle.
I will elaborate shortly.

Quaternions only represent 1/2 of the possible rotation coordinates... since the Theta angle put in is cos(theta/2) even when you get out is only +/- pi or +/- 180 degrees, and fails to represent any rotations with degrees beyond that. And also fails to represent any single rotation of 181 degrees... which is probably why spins re only in +/- 1/2 because that's the maximum extend of the rotation expressible with hamiltonians.

This is not necessarily true. Quaternions may represent any/all possible rotation coordinates.
It is important to understand the geometry must be understood as projective following affine.
180-degrees is all that is needed to represent conjugation viz. ±180 degrees = 360.

The universe is a binary system: presence/absence (of) with 180 degrees of freedom for each.

We don't know how light works - it relies on stretching space, which isn't a thing.

I admit GR and/or EU do not know how light works. When I said "we" I was loosely referencing others.
There are other theories of the physical universe that know how light works (albeit suppressed).
The "stretching space" phenomena is an artifact due to the misunderstanding of space and time.
Space and time are not two separate things (or "things" at all). They are nothing but multiplicative
reciprocal aspects of motion. One may imagine a see-saw with space and time on either end
and reciprocity as the fulcrum, hence 1/√Φ = π/4 wherein from Φ (line) is derived π (curve).

Don't really get where you're going with sqrt of 5... that looks like trying to force rotation coordinates back into 2-normal somehow... which is just going to be self defeating.

√5 is the key to any/all kinematics (motion) both physical and metaphysical.
It is required to relate rational and irrational numbers (as) incessantly normalized to unity.
It also constructs the pentagram (underlying the human body) which allows for the following:

Apex: local ± binary discretion (to be (+), or not to be (-): that is the ± discretion)
Arms: null ± binary operators viz. Α∞Ω (particular orientation/direction)
Legs: null ± binary roots viz. beg∞end (particular location)

From the apex down, there are two possible routes towards two possible roots:
Apex → R1 → Α∞Ω → R2
Apex → R2 → Α∞Ω → R1
wherein both may happen at the same time such to grant variable discretion (consciousness).

The three powers of Φ viz. Φ, Φ², Φ³ composes the full pentagram normalized to 1/unity
as the symmetry can be seen by looking at any golden ratio pentagram: each of the powers
construct a portion of the pentagram.

Edit2: Quaternions also don't allow you to add rotations or compare them deferentially or relatively. So why even mention that as an altenrative when the system I presented already does work? Even Log(Quaternion) which is supposed to represent the angles of the rotation, doesn't, and doesn't provide additive rotations; if they did, I wouldn't have had to continue to work on them... but once I did figure out the error fractor between log-quaternions and actual additive rotations, it all became quite clear why a lot of other math ended up the way it did.

I am not sure where that deduction is coming from, but it is certainly possible to add rotations (?).
Perhaps the problem is how you are treating time? (No fault of yours, Western science is clueless on time).

The 3D that are normally associated with space actually precedes all considerations of space and time entirely.
This means that whereas space has 3 dimensions, time has the same 3 dimensions despite our own being
bound to experience time as a linear progression.

"Adding" in time as a 4th dimension is a big mistake. There is no 4th dimension of time, only 3D:

s³/t = motion (3D physical)
t³/s = energy (3D metaphysical)
s³/t x t³/s = (st)²

s³/t³ = gravity
t³/s³ = mass
s³/t³ x t³/s³ = 1 = Φ(π/4)²

I apologize I'm stealing your thread a bit, and really if you want to talk about rotations and challenge the system I have; I've yet to be defeated by any challenges And SLERP as a smooth rotation function is fairly bizaare in natural rotation coodinates.

I'm not challenging your system, it is not in my interest to challenge anything except basic underlying assumptions that underlie Western science as a whole - in particular, how to measure a circle.

I otherwise appreciate any/all dialogue absent polemics.

On topic - Circles
If you standardize a unit measure on the circumference/perimeter of a circle, that's a great, but then to construct that circle you need another distance, either in some other unit entirely (inches and centimeters for instance... but radial length vs arc length) so that it can remain in 'natural' units, but then the conversion will have a n/pi or n*pi scalar builtin ... kinda like that pesky 'G' constant. I'm not sure what you are arguing in favor of or against. Yes, we can only hope to approximate it, like Phi (golden ratio); e (as in the base of ln() ) ; sqrt(2); 1/3 in base 2....

The unit measure is standardized to units of motion viz. s/t = 1 or the "speed" of light (photons).
The circle 2r = 1 (all circular motion) is a natural product of the square (root) of the golden ratio.
The "speed" of light is not (only) a speed, it is a scalar "rate".
The rate is scalar because it is based on the same scalar relation in/of the golden ratio.
One unit of motion has an energy constituency of 16 units of energy capturing
binary information re: alpha/omega/beg/end (photon) wherein 1 unit of energy is
incessantly bound to unity (thus only 15 of 16 are degrees of freedom).

G is not a constant, this is another false underlying assumption Western science has (there are a lot).

Radial length vs. arc length is already reconciled by the correct value of π viz. 1/√Φ = π/4.
This relation scales from the unit of motion (unity/light) onward.

The key is reciprocity. The link between space and counter-space is natural logarithmic
viz. a transform from recti-Euclidean to reciprocal polar-Euclidean which effectively
turns everything inside-out in 3D (see mass and gravity above).

[d3x0r]

This is not necessarily true. Quaternions may represent any/all possible rotation coordinates.
It is important to understand the geometry must be understood as projective following affine.
180-degrees is all that is needed to represent conjugation viz. ±180 degrees = 360.

all that is needed in what way? Since it's already established that they can't rotate say 367 degrees; or recover that it is at 367 degrees....

For an example a 5 segment arm with 5 motors that can go +/- 90 degrees, if they are all set at 60 degrees, the total is 300 degrees, which the quaternion only represents at -60.... so computing a minor change from the current position, ends up distributing -9 degrees and all the motors go from +60 to -9... all bad; This could be maintained in entire accuracy with simple additions.

and reciprocity as the fulcrum, hence 1/√Φ = π/4 wherein from Φ (line) is derived π (curve).

3.1415926blah is a lot closer to accurate than 3.14460551103....
And by that you're replacing one approximation with another; thought you'd at least have converted to a 1-norm arclength and have integral circumference.

I'm curious how you can apply 1/√Φ in any way to get an meaningful circularity

The 3D that are normally associated with space actually precedes all considerations of space and time entirely.
This means that whereas space has 3 dimensions, time has the same 3 dimensions despite our own being
bound to experience time as a linear progression.

And you again forgot rotations... which are a dimensions of objects in space regardless of linear position or velocity. Every Game/physics engine has a position and an orientation, velocity and angular velocity, accerelation of of course angular accelerations. While it is possible to describe the rotations using sets of coordinates for bivectors and matrices; but can more simply just use X/Y/Z coordinates that are the angles of rotation. Any vector of rotation describes an axse

"Adding" in time as a 4th dimension is a big mistake. There is no 4th dimension of time, only 3D:

s³/t = motion (3D physical)
t³/s = energy (3D metaphysical)
s³/t x t³/s = (st)²

s³/t³ = gravity
t³/s³ = mass
s³/t³ x t³/s³ = 1 = Φ(π/4)²

You say there is no time... and then you define a bunch of terms with time. If time isn't a dimension, what is it? And what makes it not exactly like any other dimension with a measure?

Dimensions can also be say... PVT (pressure,volume,temperature) ... or LC, EI (inductance,capacitance, volts, amps) and combinations thereof to get power, work, etc... but all of those are emergent properties from the base rotation and linear motion dimensions.

G is not a constant, this is another false underlying assumption Western science has (there are a lot).

Radial length vs. arc length is already reconciled by the correct value of π viz. 1/√Φ = π/4.

The key is reciprocity. The link between space and counter-space is natural logarithmic
viz. a transform from recti-Euclidean to reciprocal polar-Euclidean which effectively
turns everything inside-out in 3D (see mass and gravity above).

There really isn't counter-space... other than (to programmers) a space containing scratch variables and counters; which none of which have any units or metrics.

Pi is as good of a bit pattern as any; I'm not so bothered by having to multiply turns times 2pi... have to anyway when a force is applied at a radius of 1, for 1m/s then the circle only gets a arc-length/angular speed of 1/2pi m/s ...
(and I don't see how replacing 3.1415 with 3.1446 is going to make that any more correct...)

---
I have to make a bunch of changes to undo some of the calculations i made for additive rotations.

So buried in some abstract math concept there actually is something that says rotations just add; after I spent all that time trying to prove that they actually do... it's definatly needlessly complicated... https://en.wikipedia.org/wiki/Lie_product_formula basically A+B is the same as applying A and B at the same time, or, by iterating small steps of A and B continuously to get the result... So for some axis*angle + another axis*angle is just a third which is the effective rotation as if both of the individual rotations happened simultaneously. (makes adding multiple torque sources quite easy) But anyway I was converting everything to a 1-norm and doing the math, because the numbers looked better; but turns out it's really just a toString() sort of phenomenon to make the values human-readable as opposed to math-readable (typically the phrase is machine-readable). But a rotation of (90,0,0) and (0,90,0) to (90,90,0) looks a lot better as (90,90,0) than (127.6,127.6,0) IMO (which is what it would be to get to show as 90,90, or in total flipped over 180 degrees from the start).

[paladin17]

So buried in some abstract math concept there actually is something that says rotations just add; after I spent all that time trying to prove that they actually do... it's definatly needlessly complicated... https://en.wikipedia.org/wiki/Lie_product_formula basically A+B is the same as applying A and B at the same time, or, by iterating small steps of A and B continuously to get the result... So for some axis*angle + another axis*angle is just a third which is the effective rotation as if both of the individual rotations happened simultaneously. (makes adding multiple torque sources quite easy) But anyway I was converting everything to a 1-norm and doing the math, because the numbers looked better; but turns out it's really just a toString() sort of phenomenon to make the values human-readable as opposed to math-readable (typically the phrase is machine-readable). But a rotation of (90,0,0) and (0,90,0) to (90,90,0) looks a lot better as (90,90,0) than (127.6,127.6,0) IMO (which is what it would be to get to show as 90,90, or in total flipped over 180 degrees from the start).

You have to keep in mind though that 3D rotations (e.g. Lie group SO(3) - rotations around 3 orthonormal vectors) are non-Abelian. I.e. the order of the application of rotations matter. Rotate around X to x degrees and then rotate around Y to y degrees is not the same as in the reverse (in the general case anyways).
This actually is very important in quantum field theory, where the fundamental interactions are represented by abstract rotations in the abstract space (see principle of local gauge invariance) and some of these rotational groups are Abelian (e.g. U(1) which represents electromagnetism) while others aren't (e.g. SU(3) which represents the strong nuclear interaction), which leads to radically different properties of the resulting theories.

[A Gnostic Agnostic]

all that is needed in what way? Since it's already established that they can't rotate say 367 degrees; or recover that it is at 367 degrees....

A 367 degree rotation is the same result as a 7 degree rotation.
The difference would be merely the velocity (v = s/t) such to arrive at.
(If I spin a radial arm 367 degrees, then do the same only 7 degrees,
the outcomes are equivalent, though the means are not).

For an example a 5 segment arm with 5 motors that can go +/- 90 degrees, if they are all set at 60 degrees, the total is 300 degrees, which the quaternion only represents at -60.... so computing a minor change from the current position, ends up distributing -9 degrees and all the motors go from +60 to -9... all bad; This could be maintained in entire accuracy with simple additions.

I don't know what a 5 segment arm with 5 motors whose range of freedom is (restricted to) +/- 90 degrees
all (somehow) situated about the same 60 degree orientation... is meant to describe as (if) something "real".

To be real, one would have to begin with a quaternion of a photon.
This is why the "speed of light" (s/t = 1) should serves as a unit datum.
The axis for this photon is contained in/as the roots of the function
to which π as 4/√Φ is one-of-four roots, that f being f(x) = x⁴ + 16x² - 256:

±√9.88854381999... (beg/end location)
±i√25.88854381999... (Α/Ω direction)

These four roots compose a real/imaginary axis that couples rational/irrational
number relations (all of them), as Φ and π are the "scalar" relations associated:

Φ = (1+√5)/2 viz. couples irrational/rational
π = 4/√Φ viz. couples rational/irrational

π = 4/√Φ = √(8√5-8)
π² = 16/Φ = (8√5-8)
16 = Φπ²
(e = MC²)
1 = Φπ²/16
= Φ(π/4)²
...ad infinitum ...

Do you see the integer difference of '16' in the four roots?
This is how rational/irrational are in relation to one another.

Einstein's e = MC² is composed of a mass-electric scalar constant M
(whose properties resolutely reflect the same scalar nature of the golden proportion)
and a velocity-magnetic scalar constant C whose product is energy.

How this relates to how to measure a circle: these four roots reflect the four
equidistant points that relate a unit square (space and time are discrete units)
to the associated 2r = 1 circle. Further: measuring the circle properly without
approximation shows that line and curve (implies: space and time relations)
are reciprocally related. Reciprocity will be the key to future physics
because the physical universe is based on a bed of (incessant) reciprocity.

EU has a chance to push this forward, but so far, no dice.
Again: the barrier is in the "approximation" of π rather than
leaving it as a discrete ratio of the (root of the) golden proportion.

The golden proportion gives rise to the perfect circle.
In other words: if you treat Φ as Adam and π as Eve,
from Adam comes Eve, thus from Φ comes π.

The re-coupling of these will spark a progression to a golden age.
This can not happen until reciprocity is consciously acknowledged
as principally underlying the laws which govern the physical universe,
as evidenced upon properly measuring a circle without "approximation".

3.1415926blah is a lot closer to accurate than 3.14460551103....

False - 3.14159... is an "approximation" which misses an entire constituency of the circle.
In doing so, it is impossible to sync/normalize to the scalar rate associated with Φ.

I know humanity has had this "approximated" value of π for ~2000+ years, thus is "believed" to be correct,
but this is the same problem I am highlighting: if there is no challenging of basic underlying assumptions,
there is no progress and human beings continue to circle/cycle through periods of prolonged ignorance/suffering.

And by that you're replacing one approximation with another; thought you'd at least have converted to a 1-norm arclength and have integral circumference.

4/√Φ
8/√ˉ2(1+√5)ˉ
-2√ˉ2(1+√5)ˉ + √ˉ10(1+√5)
√(8√5-8)
etc.

are not approximations. These are all discrete ratios/relations of the same underlying relation
whose numerical output is 3.1446055... wherein 3.141.... is an "approximation" of the former, and
not the other way around (there is a lot of upside-down in Western academia).

Arc lengths are naturally quarters:
1/√Φ = π/4 viz.
1/√Φ + 1/√Φ + 1/√Φ + 1/√Φ =
π/4 + π/4 + π/4 + π/4 =
4/√Φ (and/or all above equivalent expression)

Let √(8√5-8) be C in/of e = MC²:

e = M(8√5-8)
e = 8M√5 - 8M
M = e(√5+1) / 32

Mass becomes Φ if/when e = 16.

I'm curious how you can apply 1/√Φ in any way to get an meaningful circularity

...1/√Φ is equivalent to π/4.

And you again forgot rotations... which are a dimensions of objects in space regardless of linear position or velocity. Every Game/physics engine has a position and an orientation, velocity and angular velocity, accerelation of of course angular accelerations. While it is possible to describe the rotations using sets of coordinates for bivectors and matrices; but can more simply just use X/Y/Z coordinates that are the angles of rotation. Any vector of rotation describes an axse

No, I did not forget rotations (?) - that is what the photon does, it rotates
and does not move relative to itself and/or other photons.

It actually has a birotation which incessantly allows a "choice-between-two"
as divided into yang-and-yin-like ± wholly dependent on the "charge" of the body
interacting with.

In this way, light is the conduit through which positive and negative charges
(equivalent: any dichotomy such as rational/irrational, real/imaginary etc.) transit,
the charge(s) itself being a property of the subject/object, not of the photon/light.

All a photon can do is (bi)-rotate, it can not accelerate/decelerate
unless apparently as a consequence of being introduced into a medium,
such as water slowing down light such to speed up again upon exit.
The underlying "rate" is fixed, hence it speeds back up again.

720° or 4π steradians is normalized to unity time: one single rotation from our perspective.
Note: rotations are non-commutative because order of operations MATTERS.

You say there is no time...

No, I do not say this anywhere. I said time is not a 4th dimension.

and then you define a bunch of terms with time. If time isn't a dimension, what is it? And what makes it not exactly like any other dimension with a measure?

Time and space are not two separate things - they are merely reciprocal aspects of motion.
They have no further significance than a numerator and a denominator:

s/t = v (speed)
t/s = e (energy)
___________________
space and time are reciprocally related viz.
s/t x t/s = 1 wherein '1' is normalized to unity.

s³/t relates 3D of space on a base of normalized (to light/unity) time viz. s³/1.
t³/s relates 3D of time on a base of normalized (to light unity) space viz. t³/1.

2º = 1
2¹ = 2
2² = 4
2³ = 16 = Φπ²

Time is not a dimension - it has the same dimensions as space does.
Novae are explosions in 3D space, whereas things like
quasars are explosions in 3D time. There is no such thing as
"infinite mass" and/or "singularities" - there is a discrete threshold.

Dimensions can also be say... PVT (pressure,volume,temperature) ... or LC, EI (inductance,capacitance, volts, amps) and combinations thereof to get power, work, etc... but all of those are emergent properties from the base rotation and linear motion dimensions.

The only real constituency of the physical universe is motion.
All particles/molecules/atoms are particular configurations
of motion(s) of increasing complexity.

All particles can be described by some discrete displacement(s) from unity datum 1/1 = light.
If s/t = 1, it is a photon being described.
If s/t ≠ 1, it is "something else" being described as a displacement(s) from (in relation to) light.

There really isn't counter-space... other than (to programmers) a space containing scratch variables and counters; which none of which have any units or metrics.

All motion in time is counter-spatial. The underlying geometry is the inverse of space wherein
less space means more energy, hence recti- and polar-Euclidean such to describe the relation.

Implosion technology (rather than explosion technology) will be the future of energy generation.

Pi is as good of a bit pattern as any; I'm not so bothered by having to multiply turns times 2pi... have to anyway when a force is applied at a radius of 1, for 1m/s then the circle only gets a arc-length/angular speed of 1/2pi m/s ...
(and I don't see how replacing 3.1415 with 3.1446 is going to make that any more correct...)

A circle with a radius of 1 is not a normalized circle - you need a circle with a diameter of 1 viz. 2r = 1 or r = 1/2.
It is also necessary to understand that π is not a circumference/diameter relation (another false assumption),
all diameters are resolutely a product of a doubling (2) of the real radius, thus a diameter is fixed as 2r.

Because 2r is what is needed to compose a right-angle and corresponding π/4,
2r/c/4 = 8r/c is the correct way to understand the underlying relation, and not
a "circumference to diameter" as "defined" by presumptuous mathematicians.

The golden ratio is practically in/of/as light (itself), thus
if not reflected in/as π, there is no scalar synchronization.
This is why the "approximated" π of 3.141... is "transcendental":
it has nothing to do with reality. The correct 4/√Φ is not "transcendental"
as π is describing a REAL radius in relation to four really equidistant axial points.
Such a relation can not be "transcendental" unless one has the wrong answer
due to wrong conception/methodology (this being the case for the "approximation" of π).

The correction to π will invariably spark tremendous scientific reformation, however
as we know, there are men on this planet that want all control and all power over all people.
The solving of e = MC² can be (has been) used to solve the biggest problem(s) on this planet
to no degree(s) of error(s).

[d3x0r]

3.1415926blah is a lot closer to accurate than 3.14460551103....

False - 3.14159... is an "approximation" which misses an entire constituency of the circle.
In doing so, it is impossible to sync/normalize to the scalar rate associated with Φ.

4/√Φ
8/√ˉ2(1+√5)ˉ
-2√ˉ2(1+√5)ˉ + √ˉ10(1+√5)
√(8√5-8)
etc.

are not approximations. These are all discrete ratios/relations of the same underlying relation
whose numerical output is 3.1446055... wherein 3.141.... is an "approximation" of the former, and
not the other way around (there is a lot of upside-down in Western academia).

...1/√Φ is equivalent to π/4.

[/quote]

by the above there's also no approximation when using Π/3 or any other Π based number, it's still a perfect ratio.

4/√Φ (3.14460551103....) is NOT equivalent to 3.1415926
(or do some alegebra and move the /4)