√5 and Phi

[nothing]

You're multiplying the numerator and denominator of the definition of phi by pi.

1+√5/2 is a ratio that can be satisfied by not only 1/2, but π/2π.
There is no indication of a circle with 1/2, but there is with π/2π.

That is, you're multiplying phi by pi/pi; which is also not transcendental. It's 1.

There is a qualitative difference between 1+√5/2 and π+π√5/2π,
the latter serves as a rotating base whereas this is not implied in the former.
Again, mathematicians only see numbers in terms of quantity,
because of their incessant need to reduce everything to some arbitrary '1'.

The point of using π to generate the ratio is to understand
where the first natural '1' comes from in the first place: squaring Φ,
rather than arbitrarily starting with it.

Having a base of 2π is the same as having one golden spiral per π,
the circle within which is necessary to construct the pentagram Φ contains.

[wtf]

There is a qualitative difference between 1+√5/2 and π+π√5/2π,

You're missing parens. I assume you mean to claim that there is a difference between (1 + sqrt(5))/2 and (pi + pi sqrt(5)) / 2.

But this is no different than claiming 1/2 and 2/4 have a qualitative difference. What difference is that exactly? They're two expressions for the same thing.

[nothing]

There is a qualitative difference between 1+√5/2 and π+π√5/2π,

You're missing parens. I assume you mean to claim that there is a difference between (1 + sqrt(5))/2 and (pi + pi sqrt(5)) / 2.

But this is no different than claiming 1/2 and 2/4 have a qualitative difference. What difference is that exactly? They're two expressions for the same thing.

A base of 2π is a different context entirely: 2π is describing the geometry of a circle as two half-rotations.
By using 2π as a base, the '1' emerging from the square serves as an arbitrary scalable radius
(whose default is anyway '1') if/when coupled to the same π, thus becomes a function of π,
thus geometric relationships can be described if/when π itself is the base of the expression.

I personally do not care for any mathematics that is also not field geometry:
(astro-)physics uses π for a reason: it has something to do with the universe.
Numbers are arbitrary and mean nothing less whence they naturally arise:
'1' arises from Φ→Φ² thus let it be just that, and let it concern a base of π.

[Skepdick]

A base of 2π is a different context entirely: 2π is describing the geometry of a circle as two half-rotations.

By using 2π as a base, the '1' emerging from the square serves as an arbitrary scalable radius

It doesn't make any difference. You are still prescribing a quantitative unit.

Your unit is the circle.
An algebraist's unit is the number.

I personally do not care for any mathematics that is also not field geometry:

Field geometry and algebra are the same thing. If you have a fixed point (a unit) you can pivot/tunnel between domains. Do some homework on Brouwer's fixed point theorem.

If you approach both algebra and geometry from the perspective of computation. Everything reduces to a unit. Inversely - everything can be constructed/synthesised from a unit.

This follows directly from Kleene's recursion theorem, and Rogers's fixed-point theorem.

If F is a total computable function, it has a fixed point. Translation: ALL computable functions have a unit.

Your unit (axiom?) is π exists.

Furthermore, finite field geometry reduces to boolean algebra: Geometry of Synthesis.

I think wtf tried to paint the same picture using simpler/more intuitive concepts. Wrapping the number line around the unit-circle is the modulo operation, and when you start dealing with modulo operators you've done a full circle and arrived at the doorstep of generator functions.

[nothing]

It doesn't make any difference. You are still prescribing a quantitative unit.

Your unit is the circle.
An algebraist's unit is the number.

The circle need not equal '1', as '1' is generated by squaring Φ.
It is the only number of its kind that can do that.

Field geometry and algebra are the same thing. If you have a fixed point (a unit) you can pivot/tunnel between domains. Do some homework on Brouwer's fixed point theorem.

It is possible to do "algebra" that has nothing to do with our physical universe.
All it takes is one false premise, and the entire geometry fails to it.
When I say field geometry, I mean that agrees with the physical universe.

If you approach both algebra and geometry from the perspective of computation. Everything reduces to a unit. Inversely - everything can be constructed/synthesised from a unit.

In this case the unit is per rotation, with 2π being one rotation.
This need not even be given a number: just per rotation.

This follows directly from Kleene's recursion theorem, and Rogers's fixed-point theorem.

If F is a total computable function, it has a fixed point. Translation: ALL computable functions have a unit.

Your unit (axiom?) is π exists.

Thus one would indicate 2π as a rotational base, the coefficients determine the relationship.
It is like variables and coefficients switch roles: the coefficients are the variables, whereas
the variables have constants in place of them.

Furthermore, finite field geometry reduces to boolean algebra: Geometry of Synthesis.

I think wtf tried to paint the same picture using simpler/more intuitive concepts. Wrapping the number line around the unit-circle is the modulo operation, and when you start dealing with modulo operators you've done a full circle and arrived at the doorstep of generator functions.

I appreciate yours and theirs attempts to help but neither you nor wtf can understand the implicit relationship Φ has with π.
It is okay as I see that hardly anyone knows of these relationships, as because Φ exists π exists: they are inseparable.

I found that π as being 3.141... is not precise, though it is extremely close.
The geometric way to derive this relationship is by using a circle of diameter √5
and adding one unit length to it (1 + √5), then taking the midpoint (/2) and taking √Φ
as a base with 4 over it, the number of "quarter-circles" needed to generate the equivalent unit circle:


The flower on the right was made simply by rotating the golden spiral 180-degrees
and doubling the result each time, three times: 1→2→4→8, thus '9' acts as the center.

[Skepdick]

The circle need not equal '1', as '1' is generated by squaring Φ.
It is the only number of its kind that can do that.

You aren't hearing me (but do notice that you are using the word "generated" - and I pointed you to Generators in Number theory).

The English sentence "1 is generated by squaring Φ" translates to the Mathematical sentence: Φ*Φ = 1

IF Φ*Φ = 1
then √1 = √Φ*Φ
Therefore: 1 = Φ

But BY DEFINITION Φ = (1 + √5)/2

Nothing "needs equal 1". It's an axiom. You can equate anything to 1. That's what a "unit" is!

You can even equate: (1 + √5)/2 = 1
You totally can! But then you are re-defining the standard meanings of the Mathematical operators.
And you are totally allowed to do that (but you need to shift your perspective to higher order Lambda calculus/computational perspective and invent new operators)

Man is the measure of all things (said Protagoras)
YOU are the One (said Morpheus to Neo).
YOU decide what "equates to 1" (Said Skepdick)

The unit-circle is: x*x + y*y = 1
The unit-Φ is: Φ*Φ = 1
The unit-Human is: Human = 1

This trick (of equating things to 1) goes back as far as Eudoxus of Cnidus. It's the only way we know how to "prove things" - we only prove things in terms of other things we already accept.

It is possible to do "algebra" that has nothing to do with our physical universe.

So what?

It is also possible to do "geometry" that has nothing to do with our physical universe.
Mathematics gives you models. You decide how to apply them to reality.

All it takes is one false premise, and the entire geometry fails to it.

ALL premises are just that - premises. YOU decide what "equates to 1".
ALL premises are axioms. They are true on your best judgment to be able to make accurate assertions.

When I say field geometry, I mean that agrees with the physical universe.

What mechanism would I use to determine if your geometry agrees with the physical universe?

It's just a model. Can I compute/predict consequences with it?

In this case the unit is per rotation, with 2π being one rotation.
This need not even be given a number: just per rotation.

So you are doing: Divide by 2π and round down.

f(x) = floor(x / 2π)

You are constructing integers from circles!!!

Thus one would indicate 2π as a rotational base, the coefficients determine the relationship.

No! The coefficients determine a "complete rotation around the circle"

0.5 means .5 rotations.
1 means 1 rotation.
2 means 2 rotations.

It is like variables and coefficients switch roles: the coefficients are the variables, whereas
the variables have constants in place of them.

It's like the inverse of multiplication is division.
Multiplicators become denominators.

I appreciate yours and theirs attempts to help but neither you nor wtf can understand the implicit relationship Φ has with π.

And you are unable to explain it...

It is okay as I see that hardly anyone knows of these relationships, as because Φ exists π exists: they are inseparable.

And that knowledge will be forever locked in your head until you can write it down for others to understand.

I found that π as being 3.141... is not precise, though it is extremely close.

There is an algorithm which can approximate it to n-th digit.

The geometric way to derive this relationship is by using a circle of diameter √5

√5= pi*r^2
5 = (pi*r^2)^2

And then?

and adding one unit length to it (1 + √5)

What does it mean to "add one unit length" to the diameter? "Unit length" of what?

[nothing]

You aren't hearing me (but do notice that you are using the word "generated" - and I pointed you to Generators in Number theory).

The English sentence "1 is generated by squaring Φ" translates to the Mathematical sentence: Φ*Φ = 1

IF Φ*Φ = 1
then √1 = √Φ*Φ
Therefore: 1 = Φ

But BY DEFINITION Φ = (1 + √5)/2

Nothing "needs equal 1". It's an axiom. You can equate anything to 1. That's what a "unit" is!

...are you intentionally being difficult? The OP clearly states squaring Φ derives (or "generates") itself +1.

You can even equate: (1 + √5)/2 = 1
You totally can! But then you are re-defining the standard meanings of the Mathematical operators.
And you are totally allowed to do that (but you need to shift your perspective to higher order Lambda calculus/computational perspective and invent new operators)

I don't mean to redefine anything - I simply allow '1' to be exactly where it comes from: the squaring of Φ.

Man is the measure of all things (said Protagoras)
YOU are the One (said Morpheus to Neo).
YOU decide what "equates to 1" (Said Skepdick)

These are all actually nonsense - man is certainly not the measure of all things - that is how solipsistic man is..

The unit-circle is: x*x + y*y = 1
The unit-Φ is: Φ*Φ = 1
The unit-Human is: Human = 1

sigh

This trick (of equating things to 1) goes back as far as Eudoxus of Cnidus. It's the only way we know how to "prove things" - we only prove things in terms of other things we already accept.

It is also done in versor mathematics to model electricity - it is convenient, but it needs to be remembered
that the '1' only applies to that local situation. I am only concerned what the universal '1' is... and it is
definitely not man. As I said, it comes from the squaring of Φ.

So what?

It is also possible to do "geometry" that has nothing to do with our physical universe.
Mathematics gives you models. You decide how to apply them to reality.

I already stated I don't care about geometries that do not apply to the physical universe.

ALL premises are just that - premises. YOU decide what "equates to 1".
ALL premises are axioms. They are true on your best judgment to be able to make accurate assertions.

No I do not... I let the universe decide. Premises are only local axioms that can be wrong.

What mechanism would I use to determine if your geometry agrees with the physical universe?

It's just a model. Can I compute/predict consequences with it?

The geometry of Φ and π generates the same geometry magnetism does,
you can use a Ferocell to see they are the same.

You can use it to model the universe in terms of motion, as
space and time are merely reciprocal aspects of motion
and that is all they are: numerator and denominator s/t.

So you are doing: Divide by 2π and round down.

f(x) = floor(x / 2π)

You are constructing integers from circles!!!

The other way around: circles from integers. The integers determine how the circles interact to produce what form.
You can plot an equation on any math site and apply cos-theta and just move up in .1 increments and see
how the forms spiral and split to make geometric forms. It is the same geometry "atoms" are comprised of,
which are just configurations of motion.

No! The coefficients determine a "complete rotation around the circle"

0.5 means .5 rotations.
1 means 1 rotation.
2 means 2 rotations.

It's like the inverse of multiplication is division.
Multiplicators become denominators.

No they don't: they indicate the relative magnitudes of each term.
For example every five half-rotations of a, b produces three equidistant points etc.
The geometry can only be understood if you see the base as constantly rotating.

And you are unable to explain it...

Unwilling, not unable. I don't have the patience esp. on these forums.

And that knowledge will be forever locked in your head until you can write it down for others to understand.

Or choose to... I'd not write something like that down, it would have to be an actual lecture/presentation
which I wouldn't mind doing but I am already doing this regarding another topic, though related, much more important.

There is an algorithm which can approximate it to n-th digit.

√5= pi*r^2
5 = (pi*r^2)^2

And then?

Why would one want to do this?

What does it mean to "add one unit length" to the diameter? "Unit length" of what?

Why not just look at the graph? The diameter is indicated √5 and one is explicitly added/labeled.

√5 = diagonal between corners of two unit squares inside circle
+1 = adding one unit length to the √5 diagonal, thus (1 + √5)
/2 = mid-point on the (1 + √5) extended diagonal

[Skepdick]

...are you intentionally being difficult? The OP clearly states squaring Φ derives (or "generates") itself +1.

"Derives" or "generates" implies a function. Surely you have functional notation at your disposal? Express it.

Are you saying f(Φ*Φ) = Φ + 1

What are you saying? Formulate it first, then we can see if we can construct it!

I don't mean to redefine anything - I simply allow '1' to be exactly where it comes from: the squaring of Φ.

OK, but what you are saying is ambiguous to me.

"Squaring of Φ" means Φ*Φ to me (a.k.a Φ^2). So "allow '1' to be exactly where it comes from: the squaring of Φ."

parses in my head as: 1 = Φ*Φ

Whereas this sentence "squaring Φ derives (or "generates") itself +1."

parses in my head as: f(Φ*Φ) = Φ + 1

Which one do you mean?

These are all actually nonsense - man is certainly not the measure of all things - that is how solipsistic man is..

The only argument for realism is the fear of subjectivity. --Jean-Yves Girard

It is also done in versor mathematics to model electricity - it is convenient, but it needs to be remembered
that the '1' only applies to that local situation. I am only concerned what the universal '1' is... and it is
definitely not man. As I said, it comes from the squaring of Φ.

The Universal '1' is The Universe itself. ALL of it.

Everything else is the parts you have reduced it to.

I already stated I don't care about geometries that do not apply to the physical universe.

And I already pointed out to you that "applicability" is moot. How do you determine whether your geometry applies to "the physical universe" or not?

Have you considered the possibility that geometry os only a construct of the human mind for the purposes of understanding the universe?

No I do not... I let the universe decide. Premises are only local axioms that can be wrong.

Oh, you do.... how? How do yo ask the universe the question.

Hey Universe, does my geometry apply to you?
And then how does the Universe answer you? Do you get an e-mail? SMS?

How does the universe talk to you?

The geometry of Φ and π generates the same geometry magnetism does,
you can use a Ferocell to see they are the same.

The geometry for magnetism is constructed by humans. From human constructs.

You can use it to model the universe in terms of motion, as
space and time are merely reciprocal aspects of motion
and that is all they are: numerator and denominator s/t.

Space and time are resources from the lens of linear logic.
Resources required to compute functions.

The other way around: circles from integers.

It's not the other way around. Geometrically π is the ratio of a circle's circumference to its diameter. If you don't have circles - you don't have π.

But never mind that! You literally just told me that you care about geometries, but now you are doing algebra. Huh?

Are you constructing algebra from geometry; or geometry from algebra?

No they don't: they indicate the relative magnitudes of each term.

In respect to what?

For example every five half-rotations of a, b produces three equidistant points etc.
The geometry can only be understood if you see the base as constantly rotating.

3 equidistant points where? On the same circle? In a co-domain?

Unwilling, not unable. I don't have the patience esp. on these forums.

Well then.

Why would one want to do this?

Because you said π is an approximation. So is √5.

Why not just look at the graph? The diameter is indicated √5 and one is explicitly added/labeled.

it doesn't matter what the diameter is. Let it be 1. or x. It's arbitrary.

[nothing]

Are you saying f(Φ*Φ) = Φ + 1

What are you saying? Formulate it first, then we can see if we can construct it!

You follow this by answering your own question as it is stated/related in the OP:

Whereas this sentence "squaring Φ derives (or "generates") itself +1.

Φ*Φ = Φ + 1

The only argument for realism is the fear of subjectivity. --Jean-Yves Girard

This is backwards. It should read "the only argument for subjectivity is the fear of realism."
People are afraid of what they do not want to be real/unreal, such as a "believer" who fears their "belief" is false.
This is the case of "belief"-based ideologies, which is where fear comes from: fearing the unreality of ones own "belief".

I am not afraid of challenging mainstream views if/because I know where they are faulty, and why.
That is what knowledge serves towards: knowing all *not* to simply "believe".
However knowledge is not the be-all end-all: consciousness prevails over it.

Only a sound consciousness can resolutely and inductively attain what the true nature of reality is / is not.

The Universal '1' is The Universe itself. ALL of it.

Everything else is the parts you have reduced it to.

You're upside-down again: any particular body in/of of the physical universe
is some displacement(s) from '1' being unity over itself.
In order for it to be physical, it needs a displacement(s).
Therefor light as c = 1 is light that is not "displaced" from anything
but all is displaced from it. This is true both physically and metaphysically,
as any universal "geometry" must satisfy both.

viewtopic.php?f=12&t=28457

And I already pointed out to you that "applicability" is moot. How do you determine whether your geometry applies to "the physical universe" or not?

Have you considered the possibility that geometry os only a construct of the human mind for the purposes of understanding the universe?

To the first: the geometry derived by only π and Φ captures the same geometry as magnetism
Top: image generated in paint.net using only a simple golden spiral in relation to its own π.
Bottom: Ferocell image showing the magnetic "lines of force" generated by a magnet.



Nature is extremely simple: one geometric Φ, one transcendental π.
Calculating the real value of π requires knowing it comes from Φ.

Mainstream academia does not know π is a feature of Φ thus can be found directly, and thus is only "approximated" by them (3.141...).
It is still transcendental but the relationship can be known to the same precision (if not now more) than the Egyptians (π = 4/√Φ = 3.144...)
who built the Giza plateau / pyramids. It is the exact same geometry: knowledge of how π and Φ share in one another as curve-and-line.

Oh, you do.... how? How do yo ask the universe the question.

Hey Universe, does my geometry apply to you?
And then how does the Universe answer you? Do you get an e-mail? SMS?

How does the universe talk to you?

It is called conscience - to ask questions (ie. who/what/where/why/when/how, if to, if not to etc.) and altruistically seek the answer.
The quality of the conscience will always reflect the quality of the question(s) it can generate/address.
If you ask questions, and have a true desire to know, the "universe" may relatively align itself to the inquiry over time.
If you don't ask questions, and have no desire to know, the universe lets you do whatever, because that is how "choice" works.

The geometry for magnetism is constructed by humans. From human constructs.

Present-day mathematics attempting to describe this geometry is certainly constructed by humans.
However Φ and π is not constructed by humans. The latter is approximated by humans, but that is why
they can never get astronomical calculations precise yet: they have the wrong value of π knowing not
it is naturally coupled to Φ. This humanity is very devolved by comparison to earlier civilizations.

Technology is nice, but not when it causes suffering/death. Not a problem with the technology, but with humans using it.

Space and time are resources from the lens of linear logic.
Resources required to compute functions.

The universe is not linear; neither is space, neither is time.
Even Einstein's GR "field equations" are highly non-linear.
The only "functions" space and time serve are s/t, numerator/denominator.

That is all, no more no less.

It's not the other way around. Geometrically π is the ratio of a circle's circumference to its diameter. If you don't have circles - you don't have π.

But never mind that! You literally just told me that you care about geometries, but now you are doing algebra. Huh?

Are you constructing algebra from geometry; or geometry from algebra?

Φ generates the circle whose diameter is √5, hence (1+√5)/2.
The +1 adds one unit length to the to √5, and the mid-point
coincides with a unit circle which fits perfectly in a unit square.

Because four circumferences are needed to make the full circle,
π becomes a function of 4/√Φ which is 3.144... this is the true π.

The approximations based on billions of triangles of 3.141 is demonstrative
of humanity's ignorance of what π even is, let alone where it really comes from.

Therefor any algebra that misses this relation is invariably severed from the reality.
The approximation is still there, and it is accurate enough to go unnoticed
but still ultimately incorrect. The bigger problem is humanity suffering to understand
the relationship between π and Φ because it relates to the fabric of the universe:
curve and line,
likeness and image,
woman and man,
female and male
etc.

All of this symmetry is duly reflected in geometry: π and Φ.

In respect to what?

Itself, in relation to any given rotating base over time.
The entire function ultimately concerns c = 1 as a displacement(s) therefrom.

Because space and time are multiplicative reciprocals, s³/t implies s/t³.

Time also has 3 dimensions, however I am not going to get into that as
General Relativity has just retarded humanity too much to not see this.

In the same way Φ relates to the addition of '1' to itself as a function of its own square,
space and time obey an inverse square law which is directly related, though mainstream physics
either does not know the relationship or does not tell people, as it essentially destroys
a lot of the General Relativity / Quantum / Religions businesses on the planet.

3 equidistant points where? On the same circle? In a co-domain?

On the perimeter of the circle whose diameter is √5.
Because Φ implies π (and vice versa) all is in relation to the same.

Because you said π is an approximation. So is √5.

It need not be approximated, it can be left as an irrational.
Approximating irrationals is like trying to turn a curve into a line.

4/√Φ is not an approximation of π,
it is exactly π if/when left in this form
relative to all else concerning it.

it doesn't matter what the diameter is. Let it be 1. or x. It's arbitrary.

This is where the confusion is.
The diameter of the circle must be √5 because '1' is in relation to it.
By taking the diameter of √5, adding the '1' such to have a line (1+√5),
finding the half-way point of this line (equivalent: /2)
reveals the relevant unit circle inside a unit square
contained within the circle whose diameter is √5.

[Skepdick]

Φ*Φ = Φ + 1

The above sentence is FALSE.

https://www.wolframalpha.com/input/?i=% ... E%A6+%2B+1

Are you sure you don't mean: f(Φ*Φ) = Φ + 1 ?

https://www.wolframalpha.com/input/?i=f ... E%A6+%2B+1

[wtf]

Φ*Φ = Φ + 1

The above sentence is FALSE.

Jeez Louise man, that's its defining property. Of course it's true. Congrats on finding a Wolfram bug but that's all you did. Take phi = (1 + sqrt(5)) / 2 and work it out by hand. Here, I'll do it for you.

phi^2 = ((1 + sqrt(5))/2)^2 = (1 + 5 + 2 sqrt(5)) / 4

= (6 + 2 sqrt(5)) / 4 = (3 + sqrt(5)) / 2 and

phi + 1 = (1 + sqrt(5) / 2) + 1 = (1 + sqrt(5)) / 2 + 2/2 = (3 + sqrt(5)) / 2

so the left and right sides are equal. It's all explained right here.

https://en.wikipedia.org/wiki/Golden_ratio

What's most likely going on is that Wolfram is distinguishing between lower-case phi and upper case phi. That's also explained in the Wiki article.

[wtf]

dbl post

[wtf]

triple post. sorry. I always hit quote when I should hit edit.

[Skepdick]

Jeez Louise man, that's its defining property. Of course it's true.

The "defining property" is to re-define phi?

And just a few days ago you were giving me shit on not using "standard definitions"...

Congrats on finding a Wolfram bug but that's all you did. Take phi = (1 + sqrt(5)) / 2

Perhaps the representation is getting in your way?

Φ*Φ = Φ + 1

is really

lhs(Φ) = Φ*Φ
rhs(Φ) = Φ + 1
lhs(Φ) = rhs(Φ)

https://repl.it/repls/DentalPrettyStrategy

Code: Select all

from math import sqrt

lhs = lambda phi: phi*phi
rhs = lambda phi: phi + 1

phi = float(( 1 + sqrt(5)/2))

assert lhs(phi) == rhs(phi)

Obviously that's true...

which would've been the same as saying x^2 - x = 1

The solution to which is Φ

[wtf]

Obviously that's true...

Can it be that you not only failed ninth grade high school algebra, but also don't know that this is NOT how you test for floating point equality? Can it possibly be true that you were unable to follow the simple chain of equalities I wrote down? And that you actually have no idea how to test floats for equality? Or read a Wiki page? You're really an idiot.