√5 and Phi
√5 and Phi
(1 + √5) / 2 = 1.618...(Φ)
______________________
1 circle of 360 degrees as r = (1),
+√5 vertices of a pentagram (36x5=180),
/2 possible directions (180x2=360).
Using π instead:
(π + π√5) / 2π
yields the same 1.618...(Φ)
and if/when squaring as Φ²
naturally derives the '1'
thus: Φ² = Φ + 1.
π + π√5 / 2π = 1.618... (Φ)
(π + π√5 / 2π)² =
6π² + 2π²√5 / 4π² =
3π + π√5 / 2π = 2.618... Φ + 1
(π + π√5 / 2π = 1.618... (Φ))
_____________________________
2π, thus π → 3π is '1' full rotation as 2π
or the two "directions" of a pentagram:
By allowing π to produce Φ, the natural emergence of '1'
is the product of their golden arrangement squared, thus
Φ² = Φ + 1 and reflects the bidirectional symmetry
as it exists throughout all of nature, including the human body.
"Opening up" Φ by taking the root of it
yields +/ Φ such to yield the constituency of Φ:
____________________________________________
π thus relates to either +/ Φ with 2π capturing both.
1+√5
‾‾ 2 ‾‾ = 1.618...
is thus not as inductively rooted as:
π+π√5
‾ 2π ‾ = 1.618...
which naturally concerns the circle r = 1
if/when squared/reduced:
3π+π√5
‾ 2π ‾ = 2.618...
or simply Φ + 1, a difference of 2π.
Thus, understanding "1" as an qualitative expression of 2π ie. {one full rotation},
lends itself to there being an internal intrinsic symmetry which obeys the "golden rule".
In another manner of speaking: what Φ is to π, Φ² is to 3π, or the "addition" of one full rotation as 2π.
Plotting Φ produces a spiral relative to a fixed origin, thus
the arc of it can be used to measure π and found to be
in whole integer relation to Φ. This relationship is presently
ignored by establishment science owing to the same having
established false premises esp. as they concern the nature
(or relative lack thereof) of space and time: they are not
but aspects of motion, thus have no intrinsic geometry
until in relation to one another. The same is true for Φ and π
the former always geometrically satisfying x²x1=0 thus
π is only 'transcendental' outside of these parameters
erroneously otherwise taken to be always so.
______________________
1 circle of 360 degrees as r = (1),
+√5 vertices of a pentagram (36x5=180),
/2 possible directions (180x2=360).
Using π instead:
(π + π√5) / 2π
yields the same 1.618...(Φ)
and if/when squaring as Φ²
naturally derives the '1'
thus: Φ² = Φ + 1.
π + π√5 / 2π = 1.618... (Φ)
(π + π√5 / 2π)² =
6π² + 2π²√5 / 4π² =
3π + π√5 / 2π = 2.618... Φ + 1
(π + π√5 / 2π = 1.618... (Φ))
_____________________________
2π, thus π → 3π is '1' full rotation as 2π
or the two "directions" of a pentagram:
By allowing π to produce Φ, the natural emergence of '1'
is the product of their golden arrangement squared, thus
Φ² = Φ + 1 and reflects the bidirectional symmetry
as it exists throughout all of nature, including the human body.
"Opening up" Φ by taking the root of it
yields +/ Φ such to yield the constituency of Φ:
____________________________________________
π thus relates to either +/ Φ with 2π capturing both.
1+√5
‾‾ 2 ‾‾ = 1.618...
is thus not as inductively rooted as:
π+π√5
‾ 2π ‾ = 1.618...
which naturally concerns the circle r = 1
if/when squared/reduced:
3π+π√5
‾ 2π ‾ = 2.618...
or simply Φ + 1, a difference of 2π.
Thus, understanding "1" as an qualitative expression of 2π ie. {one full rotation},
lends itself to there being an internal intrinsic symmetry which obeys the "golden rule".
In another manner of speaking: what Φ is to π, Φ² is to 3π, or the "addition" of one full rotation as 2π.
Plotting Φ produces a spiral relative to a fixed origin, thus
the arc of it can be used to measure π and found to be
in whole integer relation to Φ. This relationship is presently
ignored by establishment science owing to the same having
established false premises esp. as they concern the nature
(or relative lack thereof) of space and time: they are not
but aspects of motion, thus have no intrinsic geometry
until in relation to one another. The same is true for Φ and π
the former always geometrically satisfying x²x1=0 thus
π is only 'transcendental' outside of these parameters
erroneously otherwise taken to be always so.
Last edited by nothing on Wed Mar 18, 2020 10:26 pm, edited 16 times in total.

 Posts: 2997
 Joined: Wed Feb 10, 2010 2:04 pm
Re: √5 and Phi
Jimmy Walker was almost there...
Phi no mite!
Imp
Phi no mite!
Imp
Re: √5 and Phi
However Φ is a real solution to a quadratic equation.
By allowing use of π to generate both Φ and '1' naturally,
the circle derived/described by Φ² = Φ + 1
(as simply r = 1) is naturally both real and transcendental:
a real circle whose constituency is naturally imbued with Φ
concerning every 2π (ie. r = 1), therefor a coupling of 1:2π
such that golden proportion is the default.
Each of the two coils is comprised of two
conjugate pentagrams which all "tap" the circle
at r = 1.618... (Φ) and extends to r = 2.618... (Φ²).
These two coils coincide with the bidirection symmetry
both: of the pentagram, and of the 2π upon which Φ rests.
Last edited by nothing on Fri Feb 28, 2020 5:46 pm, edited 1 time in total.
Re: √5 and Phi
What I am reading into your sentence (by way of implication) is that there is no way to construct a transcendental number from algebraic ones?
Our history aside, I would appreciate a true/false (boolean) response. (only because it's easier than piling on yet more homework).
Re: √5 and Phi
I don't know what you mean by "construct" in this context. If x is transcendental then x  x = 0 is algebraic. So you have to be careful to say exactly what you mean.Skepdick wrote: ↑Sun Feb 09, 2020 8:24 pmWhat I am reading into your sentence (by way of implication) is that there is no way to construct a transcendental number from algebraic ones?
Our history aside, I would appreciate a true/false (boolean) response. (only because it's easier than piling on yet more homework).
OP claimed to have demonstrated the equation Φ² = Φ + 2π. This can not possibly be true. It would imply that Φ²  Φ = 2π. You'd have an algebraic number on the left and a transcendental on the right, being equal. But algebraic and transcendental are mutually exclusive classes of real numbers.
By the way how do I know that Φ²  Φ is algebraic? It's because Φ is algebraic, being the root of a polynomial having integer coefficients, as the OP himself demonstrated. And we know that the product and difference of algebraic numbers is algebraic.
We know that pi is transcendental because Lindemann proved it in 1882.
https://en.wikipedia.org/wiki/Ferdinand ... ence_proof
Re: √5 and Phi
Lets try.
Can I produce some algorithm/computable function which (given infinite time) and using ONLY the set of ALL Algebraic Numbers (if such a set exists?  maybe I am just making it up) as input to this function.
Can such a contraption output a transcendental number?
Although I THINK, you have answered my question by pointing out that x  x = 0. You've given me a fixed point, and I THINK a fixed point between two domains (algebraic and transcendental) is a sufficient criterion for construction.
Yeah, my ask is separate from the OP.wtf wrote: ↑Sun Feb 09, 2020 9:20 pm OP claimed to have demonstrated the equation Φ² = Φ + 2π. This can not possibly be true. It would imply that Φ²  Φ = 2π. You'd have an algebraic number on the left and a transcendental on the right, being equal. But algebraic and transcendental are mutually exclusive classes of real numbers.
Re: √5 and Phi
Sure. The sequence of rational (hence algebraic) numbers 3, 3.1, 3.14, 3.141, ... has the limit pi. So an infinite sequence of algebraics can have a transcendental limit. One you allow infinitary processes it's easy to sneak up on transcendentals using only algebraics.Skepdick wrote: ↑Sun Feb 09, 2020 9:24 pmLets try.
Can I produce some algorithm/computable function which (given infinite time) and using ONLY the set of ALL Algebraic Numbers (if such a set exists?  maybe I am just making it up) as input to this function.
Can such a contraption output a transcendental number?
The set of all algebraic numbers exists and is important. It's the set of algebraic numbers. They're a field, meaning you can add, subtract, multiply, and divide (except by 0) without leaving the set.
https://en.wikipedia.org/wiki/Algebraic ... ic_numbers
Re: √5 and Phi
Thanks! I learned something.wtf wrote: ↑Sun Feb 09, 2020 9:27 pmSure. The sequence of rational (hence algebraic) numbers 3, 3.1, 3.14, 3.141, ... has the limit pi. So an infinite sequence of algebraics can have a transcendental limit. One you allow infinitary processes it's easy to sneak up on transcendentals using only algebraics.Skepdick wrote: ↑Sun Feb 09, 2020 9:24 pmLets try.
Can I produce some algorithm/computable function which (given infinite time) and using ONLY the set of ALL Algebraic Numbers (if such a set exists?  maybe I am just making it up) as input to this function.
Can such a contraption output a transcendental number?
Last prod. Is the infinitary process sufficient for getting from algebraic > transcendental, or is the fixed point (x  x = 0) also necessary?
Re: √5 and Phi
The algebraic numbers are closed under addition and multiplication (and their inverses). But it's certainly possible for a finite linear combination of transcendental numbers to be algebraic, as the pi  pi = 0 examples shows.
Offhand I can't think of any other classes of examples. But as a striking example of the limits of our knowledge, it is unknown whether e + pi is rational or not. There's no fundamental reason it couldn't be, and nobody's ever proved it one way or the other.
https://math.stackexchange.com/question ... nalnumber
Last edited by wtf on Sun Feb 09, 2020 9:36 pm, edited 1 time in total.
Re: √5 and Phi
OK. This is harder to parse without some homework.wtf wrote: ↑Sun Feb 09, 2020 9:34 pm The algebraic numbers are closed under addition and multiplication (and their inverses). But it's certainly possible for a finite linear combination of transcendental numbers to be algebraic, as the pi  pi = 0 examples shows. Offhand I can't think of any other classes of examples.
I'll go take care of that. Thanks!
Re: √5 and Phi
If x is algebraic and y is algebraic then x + y, xy, x  y, and x/y (if y isn't 0) are algebraic. These facts aren't immediately obvious but can be proved.Skepdick wrote: ↑Sun Feb 09, 2020 9:35 pmOK. This is harder to parse without some homework.wtf wrote: ↑Sun Feb 09, 2020 9:34 pm The algebraic numbers are closed under addition and multiplication (and their inverses). But it's certainly possible for a finite linear combination of transcendental numbers to be algebraic, as the pi  pi = 0 examples shows. Offhand I can't think of any other classes of examples.
I'll go take care of that. Thanks!
Re: √5 and Phi
Yeah, that's intuitive to me and I would accept it without proof.wtf wrote: ↑Sun Feb 09, 2020 9:37 pmIf x is algebraic and y is algebraic then x + y, xy, x  y, and x/y (if y isn't 0) are algebraic. These facts aren't immediately obvious but can be proved.Skepdick wrote: ↑Sun Feb 09, 2020 9:35 pmOK. This is harder to parse without some homework.wtf wrote: ↑Sun Feb 09, 2020 9:34 pm The algebraic numbers are closed under addition and multiplication (and their inverses). But it's certainly possible for a finite linear combination of transcendental numbers to be algebraic, as the pi  pi = 0 examples shows. Offhand I can't think of any other classes of examples.
I'll go take care of that. Thanks!
Where I am tripping up is mapping your language to mine so I can grok it e.g "closed under addition...." sounds like what I call total, halting functions etc.