why is the golden ratio algebraic and the golden angle transcendental

Question

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.

Answer 1

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.

Answer 2

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.