Question

$2\arctan(\phi^{-n})=\arctan\frac{p}{q}$ or $\arctan\frac{p\sqrt{5}}{q}$, where $\phi$ is the Golden Ratio. Is there a pattern in the $\frac{p}{q}$s?

score 461 · Answer 1 · 2019-11-14 17:25:02

461

Views

$2\arctan(\phi^{-n})=\arctan\frac{p}{q}$ or $\arctan\frac{p\sqrt{5}}{q}$, where $\phi$ is the Golden Ratio. Is there a pattern in the $\frac{p}{q}$s?

Published on 14 Nov 2019 - 17:25

score 5.4k · Answer 2 · 2019-11-18 12:48:18

5.4k

Views

Why does this process map every fraction to the golden ratio?

Published on 18 Nov 2019 - 12:48

score 243 · Answer 3 · 2019-11-28 12:31:55

243

Views

Proof Phi is Irrational by using another Irrational Number

Published on 28 Nov 2019 - 12:31

score 290 · Answer 4 · 2019-11-28 22:37:09

290

Views

Gosper's identity for the golden ratio: $\frac{2^{2/5}\sqrt{5} \, \Gamma(1/5)^4}{\Gamma(1/10)^2 \,\Gamma(3/10)^2} = \phi$

Published on 28 Nov 2019 - 22:37

score 360 · Answer 5 · 2019-12-02 15:03:03

360

Views

Golden ratio and floor function $\lfloor \phi ^2 n \rfloor - \lfloor \phi \lfloor \phi n \rfloor \rfloor = 1$ where $\phi = \frac{1+\sqrt{5}}{2}$

Published on 02 Dec 2019 - 15:03

score 161 · Answer 6 · 2019-12-05 20:07:45

161

Views

How can I prove analytically that the golden ratio is less then $\frac{\pi^2}{6}$.

Published on 05 Dec 2019 - 20:07

score 92 · Answer 7 · 2019-12-07 11:23:46

92

Views

Proving $ \lim_{n\to\infty} \dfrac{ \Phi^{n+1} - (1 - \Phi)^{n+1}}{\Phi^{n} - (1 - \Phi)^n} = \Phi $

Published on 07 Dec 2019 - 11:23

score 22 · Answer 8 · 2019-12-11 02:18:32

22

Views

When is $ a+b \phi >0$?

Published on 11 Dec 2019 - 2:18

score 203 · Answer 9 · 2019-12-22 08:14:23

203

Views

Roots of $x^2+bx+c=0$ with $|b|\leq 1$ and $|c|\leq 1$ have absolute value at most the Golden Ratio. Coincidence?

Published on 22 Dec 2019 - 8:14

score 653 · Answer 10 · 2019-12-27 15:50:43

653

Views

Show that a diagonal in a pentagon is the golden ratio

Published on 27 Dec 2019 - 15:50

score 176 · Answer 11 · 2020-01-13 01:34:46

176

Views

Solutions to $\prod_{n=1}^\infty \left ( 1+ \frac1{f(n)} \right ) = \varphi$

Published on 13 Jan 2020 - 1:34

score 374 · Answer 12 · 2020-01-18 18:37:14

374

Views

Deriving Parametric Equations For A Hyperbolic PHI Sine-Wave

Published on 18 Jan 2020 - 18:37

score 23 · Answer 13 · 2020-01-19 13:14:24

23

Views

Calculating n'th Fibonacci number through golden ratio

Published on 19 Jan 2020 - 13:14

score 896 · Answer 14 · 2020-01-20 12:59:09

896

Views

Why does the derivative equation of the unit semicircle equation intersect the semicircle at $x=-\frac{1}{\phi}$?

Published on 20 Jan 2020 - 12:59

score 404 · Answer 15 · 2020-01-20 20:49:57

404

Views

Parametric Equations for A Logarithmic Sine-wave With Alternately Offset Points of Hyperbolic Tangency

Published on 20 Jan 2020 - 20:49

List Question

$2\arctan(\phi^{-n})=\arctan\frac{p}{q}$ or $\arctan\frac{p\sqrt{5}}{q}$, where $\phi$ is the Golden Ratio. Is there a pattern in the $\frac{p}{q}$s?

Why does this process map every fraction to the golden ratio?

Proof Phi is Irrational by using another Irrational Number

Gosper's identity for the golden ratio: $\frac{2^{2/5}\sqrt{5} \, \Gamma(1/5)^4}{\Gamma(1/10)^2 \,\Gamma(3/10)^2} = \phi$

Golden ratio and floor function $\lfloor \phi ^2 n \rfloor - \lfloor \phi \lfloor \phi n \rfloor \rfloor = 1$ where $\phi = \frac{1+\sqrt{5}}{2}$

How can I prove analytically that the golden ratio is less then $\frac{\pi^2}{6}$.

Proving $ \lim_{n\to\infty} \dfrac{ \Phi^{n+1} - (1 - \Phi)^{n+1}}{\Phi^{n} - (1 - \Phi)^n} = \Phi $

When is $ a+b \phi >0$?

Roots of $x^2+bx+c=0$ with $|b|\leq 1$ and $|c|\leq 1$ have absolute value at most the Golden Ratio. Coincidence?

Show that a diagonal in a pentagon is the golden ratio

Solutions to $\prod_{n=1}^\infty \left ( 1+ \frac1{f(n)} \right ) = \varphi$

Deriving Parametric Equations For A Hyperbolic PHI Sine-Wave

Calculating n'th Fibonacci number through golden ratio

Why does the derivative equation of the unit semicircle equation intersect the semicircle at $x=-\frac{1}{\phi}$?

Parametric Equations for A Logarithmic Sine-wave With Alternately Offset Points of Hyperbolic Tangency

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