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I'm referencing the golden angle from here https://en.wikipedia.org/wiki/Golden_angle, where it's defined as $ 2\pi(1-\frac{1}{\varphi})$.

All constructible numbers are algebraic numbers. A constructible number is constructed with lines and circles. So why is does the golden ratio (line) algebraic while the golden angle (circle, arc, angle) transcendental. I would assume at the very least the line and circles would agree with each other… or maybe it's the relative complement set $A \setminus C $, with constructible numbers $C$ and algebraic numbers $A$, where the line and circle start to diverge?

idk this isn't rigorous, it's more of an intuition or a metaphor. hopefully someone with rigour can help me out thank you.

EDIT: I have now heard all I want to hear about constructible vs algebraic numbers. Please comment on Brian Tung's linked post, Where is the sine function transcendental?, because I believe developing a stronger intuition in this area is better the former.

telepathyy
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    Your definitions are wrong. An algebraic number is the root of a polynomial with rational coefficients. You seem to be attempting to define "constructible number", which is a little more restrictive than what you posit: it is a length that can be constructed (from a base length) using only the operations of joining two points with a straight line, extending a line indefinitely in either direction, tracing a circle with a given center and going through a given point, and intersecting two constructions to determine a point. – Arturo Magidin Nov 04 '21 at 17:05
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    See algebraic number and compare with constructible number (in Wikipedia). $\sqrt[3]{2}$ is algebraic but not constructible. Every constructible number is algebraic. – Arturo Magidin Nov 04 '21 at 17:07
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    Wikipedia does say about the golden angle that "As its sine and cosine are transcendental numbers, the golden angle cannot be constructed using a straightedge and compass." That doesn't mean that this non-constructability characterises or defines transcendentalness. – Arthur Nov 04 '21 at 17:08
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    @ArturoMagidin Okay thank you I'll try to go back and reword. But my description was more of my attempt at an intuition to the question I raised above. – telepathyy Nov 04 '21 at 17:14
  • There was nothing rigorous about what I said, but could that help point someone in a direction to my answer? – telepathyy Nov 04 '21 at 17:15
  • The degree angle measurement of the golden angle is constructible (just like the golden ratio is). But its radian measurement is not, because it involves a multiple of $\pi$. In essence, you are trying to take an algebraic multiple of $\pi$, and that is never going to be algebraic, let alone constructible, for the same reason that you cannot square the circle. Your intuition fail is in thinking that this is about just "using circles and lines". You need more than that for constructibility. – Arturo Magidin Nov 04 '21 at 17:22
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    @ArturoMagidin There's a difference between angles that are constructible, which don't include trisections in general, and angles that have constructible measures in degrees, which is closed under division by $3$. I think the answer to OP's question is not "for the same reason you cannot square the circle" – jackson Nov 04 '21 at 17:26
  • OP: Side note—you probably mean $A \setminus C$ (i.e., the set of all elements of $A$ that are not in $C$), rather than $C \setminus A$, which is the empty set. But that's just an error of notation. – Brian Tung Nov 04 '21 at 17:29
  • @Jackson Thank you. He had me reading the squaring the circle wikepedia page for the last 5 minutes. (enjoyed it tho) – telepathyy Nov 04 '21 at 17:29
  • @Jackson: I refered to the "degree angle measurement", not the angle itself; I was thinking about a line segment whose length is the amount of degree in the angle, if that makes sense. Of course, whether an angle is constructible is independent of how you measure the angle; and you can also talk about constructing the angle itself as opposed to its degree angle measurement. The radian angle measurement (in the same sense) is a multiple of $\pi$, and so its constructibility amounts to "straightening" the arc of circle, which ends up running into the transcendence of $\pi$. – Arturo Magidin Nov 04 '21 at 17:29
  • OP: What do you mean by "golden angle"? I can't make out what refers to, exactly. – Brian Tung Nov 04 '21 at 17:29
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    @BrianTung: Wikipedia is your friend. – Arturo Magidin Nov 04 '21 at 17:30
  • @ArturoMagidin: Well, I wanted OP to add that in. But yes, thanks. :-) – Brian Tung Nov 04 '21 at 17:31
  • @telepathyy When talking about angles being transcendental you have to specify the units. A right angle, for example, is a whole number in degrees, but is transcendental in radians. – dxiv Nov 04 '21 at 17:31
  • @telepathyy: There is nothing "unlike" anything I said. Just because you were wrong doesn't mean you have to start dumping on the people who point out your errors. – Arturo Magidin Nov 04 '21 at 17:33

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The Lindemann-Weierstrass theorem leads to the result that sine and cosine are transcendental whenever their argument is algebraic (as the golden ratio is—in fact, the golden ratio is constructible), except when that argument is zero. I think that this should have relevance to your question, although I'm not sure I know what you mean by "golden angle."

ETA: Ha! I see that "golden angle" simply refers to the angle corresponding to the division of a circle's circumference into two parts related by the golden ratio. The shorter arc turns out, in radians, to be simply $(4-2\phi)\pi$, I believe. Since $4-2\phi$ is algebraic, and $\pi$ is transcendental, their product is transcendental. Sorry, I thought you meant something like the angle for which the sine or cosine or tangent was something related to the golden ratio.

In degrees, this value is $720-360\phi$, so it, like $\phi$ itself, is algebraic.

Some basic definitions:

  • An algebraic number is one that is the root of a non-zero polynomial with rational (or integer) coefficients. This includes complex numbers.
  • A constructible number is the length of a line segment that can be constructed with a finite sequence of compass-and-straightedge operations. That's a geometric interpretation; the algebraic interpretation is that it falls within a finite tower of quadratic extensions (which may be gobbledygook to you, conceivably), or alternatively that it is the result of combining a finite number of arithmetic operations plus square root as applied to rational (or integer) values. All constructible numbers are algebraic, but not all algebraic numbers are constructible; the latter set is a strict superset of the former. For example, as someone pointed out in the comments to the OP, $\sqrt[3]{2}$ is algebraic but not constructible. See the Wikipedia plot summary for doubling the cube.
  • A transcendental number is just one that is not algebraic. This too includes complex numbers.

See this question and its associated answer for more information. It even mentions the golden ratio, I believe.

Brian Tung
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  • You want to distinguish between constructing an angle, and constructing a line segment whose length has the same magnitude as the measurement of the angle in specific units. An angle is constructible if and only if you can construct two line segments that intersect and make the angle; this is equivalent to being able to construct the sine or cosine of the angle (since either will let you construct a right triangle with one angle equal to that given angle). The constructibility of the measurement depends on the units (right angle and $90$(${}^\circ$) constructible, $\frac{\pi}{2}$ not). – Arturo Magidin Nov 04 '21 at 17:37
  • @ArturoMagidin: Yeah, I see my geometric characterization of constructible number is pretty sloppy. Let me edit that. – Brian Tung Nov 04 '21 at 17:39
  • Thank you!!! thank you for your link, that really answered the core of what I was asking. – telepathyy Nov 04 '21 at 17:43
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Constructible numbers (which are all algebraic) are defined to be (ratios of) line lengths that can be constructed with ruler and compasses, not angles that can be constructed.

An easier example is the right angle $\pi/2$ that can easily be constructed using the bisection trick. If it were possible to 'unbend' a curve around that angle into a straight line of length $\pi/2$, then that would contradict algebraicity of constructible numbers. But that isn't possible with ruler and compasses.

John Gowers
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