Back to Basics: How to Measure a Circle

Beyond the boundaries of established science an avalanche of exotic ideas compete for our attention. Experts tell us that these ideas should not be permitted to take up the time of working scientists, and for the most part they are surely correct. But what about the gems in the rubble pile? By what ground-rules might we bring extraordinary new possibilities to light?
d3x0r
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Re: Back to Basics: How to Measure a Circle

Sat Sep 19, 2020 6:07 pm
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).
But these additions are not rotations applied to rotations... they are the sum of impulses that result in an acceleration or velocity (or accumulate into a position)

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.
I would like to know more about these assignments, and why they think that that actually works... I can certainty generate the result of a rotation with 2 more rotations applied to it (and yes the order matters, A then B then C is not the same when 'applying' when 'adding' a+b+c IS actually the same).

There are a few places where additive rotations are better 1) it's an accurate SLERP operation that's cheaper...(these days) it's the pure rotation axis between one rotation frame and another... the vector between them is the rotation axis also.. the distance between them is how much around that axis the space is rotated.... and again, there is a place and time where X+Y+Z is the same no matter the order; however, due to physical limitations, the ability to USE that is limited.

Spatial position can be measured absolutely from one point to another, pick one on the floor, and one on the roof across the street, changes are high that most of those direct paths are not actually paths you can take to get to (x,y,z) target... it's the same with rotations;

If you're in a rocket with fixed point engines, the rotation of that body affects the directions it can apply thrust to change its axis of rotation; but all the thrusters at once, simply composite with addition.

Through experimentation and stubborn pursuit of 'what exactly ARE addable rotations' I can identify where they do work, but that's a very limited circumstance.... and unfortunately it's still a complex system of transformations to values that have to happen to rotate axis-angle around axis-angle .

(axis-angle is just angle-angle-angle separated ) like linear velocity can be speed*(x,y,z) or (speedX,speedY,speedZ) and the angles just add like normal linear x/y/z points (in some cases)

But absolutely, if you have two positions, you can immediately identify an axis of rotation and amount of rotation around that axis to get to that target point. (and compose a matrix to apply to get there even)
A physical system though has limitations, and a robot arm has 'yaw/pitch and roll' settings that are fix - and usually fixed as a result of each other... so a specific mounting order of 3 motors on a robot arm segment have to be applied separately - and that becomes a complex system of 3 equations that mutually affect each other.

---

And let me finish with reiterating

I would like to know more about these assignments, and why they think that that actually works... (how it's mapped) And I'm not certain WHY or the mechanism for it, but I suspect gravity is actually a spin axis; but rotates x/y/z at once - so it's more of a rotation around 'w'? but barring that I'd like to know what the SO(1) things actually imply, since I can directly visualize such things now.

A Gnostic Agnostic
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Re: Back to Basics: How to Measure a Circle

d3x0r wrote:
Tue Oct 06, 2020 11:42 am
by the above there's also no approximation when using Π/3 or any other Π based number, it's still a perfect ratio.
Correct, but this has nothing to do with the correct value of π as a discrete ratio.

The reciprocal of √Φ viz 1/√Φ = π/4. This is whence the "from Adam's own rib is derived Eve" analogy,
the real underlying relation being from √Φ's own reciprocal comes π/4. Adam and Eve aka. space and time
(line and curve) are reciprocally related. Mainstream science incl. EU does not know this (yet).

The above is why the Giza pyramid was constructed with a height of √Φ and 4x axial radii of unit length, hence 4/√Φ.
d3x0r wrote:
Tue Oct 06, 2020 11:42 am
4/√Φ (3.14460551103....) is NOT equivalent to 3.1415926
(or do some alegebra and move the /4)
Nobody claimed it was - the "approximated" π of 3.14159... is deficient at the thousandth decimal place.
The "approximation" methodology (of exhaustion) recursively misses an entire constituency of the circle.

If ones actually measures a circle without "approximating" it, it can be seen that the reciprocal of √2
viz. 1/√2 = √2/2 captures the square of the golden ratio such that the reciprocal yields the arc
associated with π/4. This is a natural property/consequence of ordinary mathematics.

Western science is not going anywhere until they learn how to measure a circle.
The correct value of π as 3.1446055... is in relation to the golden ratio (and light).

d3x0r
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Re: Back to Basics: How to Measure a Circle

A Gnostic Agnostic wrote:
Tue Oct 06, 2020 12:33 pm
Nobody claimed it was - the "approximated" π of 3.14159... is deficient at the thousandth decimal place.
The "approximation" methodology (of exhaustion) recursively misses an entire constituency of the circle.

If ones actually measures a circle without "approximating" it, it can be seen that the reciprocal of √2
viz. 1/√2 = √2/2 captures the square of the golden ratio such that the reciprocal yields the arc
associated with π/4. This is a natural property/consequence of ordinary mathematics.

Western science is not going anywhere until they learn how to measure a circle.
The correct value of π as 3.1446055... is in relation to the golden ratio (and light).
at 3 thousandths of a unit, such an error would be obviously wrong in many circumstances. While on the scale of an inch, it's not notable, in a circle the size of ... (looks for an example round Greek building, and fails) 1000 inches is some 83 feet, not even 1/3 of a football field, but then I couldn't find an equivalent circle of known measure that an equivalent size - at 1000 inches there'd be a 3 inch gap somewhere...

Honestly I've never needed PI, other than sin and cos functions are in units of PI; and I don't see that changing in hardware any time soon. It would be just as accurate if the scalar were 'OneOverSqrtPhi' or, in my preference 1.

Most machining of round things is just 'shave it down until it's round enough' or until 'this diameter matches this other diameter' and doesn't worry about the circumference anyway; so in math using the pi character is usually sufficient.

Having a larger value for the circumference vs radius one would end up with a spiral and not a circle.

https://youtu.be/gj-u8qXdmJA - using an inscribed and circumscribed polygon, and increasing the number of sides seems like a reasonable method to get to the ratio of radius to circumference... and it's not even very many required to get under 3.144 (/4)

I maybe missed it- but what is the reasoning that arrives that the value is also equivalent to 1/sqrt(phi) *4?

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Re: Back to Basics: How to Measure a Circle

d3x0r wrote:
Sat Oct 10, 2020 6:08 am
at 3 thousandths of a unit, such an error would be obviously wrong in many circumstances. While on the scale of an inch, it's not notable, in a circle the size of ... (looks for an example round Greek building, and fails) 1000 inches is some 83 feet, not even 1/3 of a football field, but then I couldn't find an equivalent circle of known measure that an equivalent size - at 1000 inches there'd be a 3 inch gap somewhere...
Such an error *is* obviously wrong, as the "missing" "dark" matter/energy is contained in/of (as) the incorrect value of π. If one measures a 1000mm diameter circle, one will find that 3141.6mm is too short. A real physical measurement of a real symmetrical circle of diameter 1000mm will measure no less than 3144.6mm if the measurement/instruments are sufficiently precise. The problem with the "approximated" π is it does not represent a real, physical circle, hence appears "transcendental". π can not be "transcendental" as all real circles minimally have a scalar radius of 1/2 such to scale with the golden ratio 1/2 + √5/2 whose reciprocal is discretely one less than itself (1/Φ = Φ-1) and whose square is discretely one greater than itself (Φ*Φ = Φ+1). This golden ratio is the only number in the universe with these unique properties, hence it must be contained in any real solution to unity, as it is: 1 = Φ(π/4)².
d3x0r wrote:
Sat Oct 10, 2020 6:08 am
Honestly I've never needed PI, other than sin and cos functions are in units of PI; and I don't see that changing in hardware any time soon. It would be just as accurate if the scalar were 'OneOverSqrtPhi' or, in my preference 1.
Well the physical universe needs π just as it does Φ. The sin and cos functions are based on an unreal π, so you are not missing out on anything there. What you may be missing out on is the natural reciprocal relation between line and curve (space and time). This follows as a natural consequence according to the real geometry that underlies π = 4/√Φ, the latter being a real root of f(x) = x⁴ - 16x² + 256. The roots of the latter function create a real/imaginary axis (symmetrical), the same the physical universe adheres to.
d3x0r wrote:
Sat Oct 10, 2020 6:08 am
Most machining of round things is just 'shave it down until it's round enough' or until 'this diameter matches this other diameter' and doesn't worry about the circumference anyway; so in math using the pi character is usually sufficient.
The circumference of a circle is fixed to the radius of the circle. The ratio c/d is fixed, and this ratio scales by way of the golden ratio. However note: a diameter is composed of 2r, thus c/2r discriminates against 2r/c/4=8r/c implying the 2r is a right angle instead of a flat diameter.
d3x0r wrote:
Sat Oct 10, 2020 6:08 am
Having a larger value for the circumference vs radius one would end up with a spiral and not a circle.
ie. it would not be a circle anymore because the scalar nature of the golden ratio would be broken.
d3x0r wrote:
Sat Oct 10, 2020 6:08 am
https://youtu.be/gj-u8qXdmJA - using an inscribed and circumscribed polygon, and increasing the number of sides seems like a reasonable method to get to the ratio of radius to circumference... and it's not even very many required to get under 3.144 (/4)
No, it will miss an entire constituency of the circle. The portion that is missed is recursively redistributed via the deficiency of the methodology. There is only one polygon that one must inscribe into an r = 1/2 circle, that is the square of the same area s = 1/√2. This square correlates the four axial points where both the circle and the square join/meet.
d3x0r wrote:
Sat Oct 10, 2020 6:08 am
I maybe missed it- but what is the reasoning that arrives that the value is also equivalent to 1/sqrt(phi) *4?
If you inscribe an r = 1/2 circle in a unit square s = 1, you will find the circle and the square meet at four symmetrically equidistant points. Both the circle and the square are composed of four right angles: the square via the four corners (out-in) and the circle via the axial radii (in-out). This is the geometric reason π must be in some relation to '4'. The circumference of the r = 1/2 circle is a consequence of the golden ratio, thus by taking the square it correlates these same four points.

Here is an image showing the underlying geometric relation:
https://i.postimg.cc/15gS29s1/Circle-Pi7.jpg

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Re: Back to Basics: How to Measure a Circle

d3x0r wrote:
Tue Oct 06, 2020 11:45 am
Sat Sep 19, 2020 6:07 pm
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).
But these additions are not rotations applied to rotations... they are the sum of impulses that result in an acceleration or velocity (or accumulate into a position)
I don't quite get what you mean by "rotations applied to rotations". You mean the situation when the second rotation happens around a new Y axis (as if it was "locked" to the rotating body)?
d3x0r wrote:
Tue Oct 06, 2020 11:45 am
Sat Sep 19, 2020 6:07 pm
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.
I would like to know more about these assignments, and why they think that that actually works... I can certainty generate the result of a rotation with 2 more rotations applied to it (and yes the order matters, A then B then C is not the same when 'applying' when 'adding' a+b+c IS actually the same).
Which assignments do you mean? SU nomenclature? Or the [gauge] field theory in general? There is quite a lot of literature on both topics.
Long story short: somebody has found certain symmetries in electromagnetic field wave function, so they expressed these symmetries in the language of the group theory, and then were able to find similar symmetries for other two interactions (so in the end we have U(1) group "generating" electromagnetism, SU(2) "generating" weak nuclear interaction and SU(3) "generating" strong nuclear interaction - all through this local gauge invariance formalism).
d3x0r wrote:
Tue Oct 06, 2020 11:45 am
Spatial position can be measured absolutely from one point to another, pick one on the floor, and one on the roof across the street, changes are high that most of those direct paths are not actually paths you can take to get to (x,y,z) target... it's the same with rotations;
Yes, of course. Any rotation around 3 axes, strictly speaking, can be expressed as a single rotation around a single axis (which, of course, except trivial cases would not coincide with any of the 3) - see Euler's rotation theorem. And curiously enough the same idea was used by F. I. Fedorov in the realm of particle physics (see his book "Lorentz group" and related works of other people, such as this one).
d3x0r wrote:
Tue Oct 06, 2020 11:45 am
But absolutely, if you have two positions, you can immediately identify an axis of rotation and amount of rotation around that axis to get to that target point. (and compose a matrix to apply to get there even)
A physical system though has limitations, and a robot arm has 'yaw/pitch and roll' settings that are fix - and usually fixed as a result of each other... so a specific mounting order of 3 motors on a robot arm segment have to be applied separately - and that becomes a complex system of 3 equations that mutually affect each other.
I can see that. So we're talking about physical systems that have certain limitations. If I understand you correctly, you're trying to work with what may be formally called nonholonomic constraints on rotations.

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Re: Back to Basics: How to Measure a Circle

Short 8-page paper:

'On How to (Properly) Measure a Circle
(Without the Need/Inclining for Approximation)'

https://vixra.org/abs/2010.0100

i. Methodology to measure a circle without the need/inclining for "approximation" (using the golden ratio).
ii. Outstanding Riemann Hypothesis Millennium Problem: the barrier of a deficient π of 3.14159... ("blunder of millennia").
iii. Rational Precedent to e = MC² as 16 = Φπ² implying corollary rational unity as 1 = Φ(π/4)² wherein Φ and π are one (via.)
v. Reciprocity, the latter resolutely being the nature of the relation between space and time:

1 = Φ(π/4)²
s/t = v, motion (measured in/as a speed or velocity)
t/s = e, energy (measured in discrete units using ordinary mathematics)
s/t x t/s = 1
∴ all universal motion has a corresponding t/s energy constituency proportional to the reciprocal of its own s/t velocity.

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