User:AndrewMurray1.618

$$\varphi$$ = 1.61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788 06752 08766 89250 17116 96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364 86444 92410 44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007 65217 05751 79788 34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 15317 14101 17046 66599 14669 79873 17613 56006 70874 80710 13179 52368 94275 21948 43530 56783 00228 78569 97829 77834 78458 78228 91109 76250 03026 96156 17002 50464 33824 37764 86102 83831 26833 03724 29267 52631 16533 92473 16711 12115 88186 38513 31620 38400 52221 65791 28667 52946 54906 81131 71599 34323 59734 94985 09040 94762 13222 98101 72610 70596 11645 62990 98162 90555 20852 47903 52406 02017 27997 47175 34277 75927 78625 61943 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$$\varphi = \sqrt[3](\ 2+\sqrt{5})$$

$$\varphi - 1 = \tfrac{1}{\varphi}$$.

$$\tfrac{1}{\sqrt[3](\ 2+\sqrt{5}} = \sqrt[3](\ 2+\sqrt{5}) - 4$$

$$\varphi$$2 = $$\varphi + 1$$

$$\varphi = \frac{1+\sqrt{5}}{2}$$


 * $$\varphi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}\,.$$


 * $$-\frac{\varphi}{2}=\sin666^\circ=\cos(6\cdot 6 \cdot 6^\circ).{}$$


 * $$\varphi = 1+2\sin(\pi/10) = 1 + 2\sin 18^\circ$$
 * $$\varphi = {1 \over 2}\csc(\pi/10) = {1 \over 2}\csc 18^\circ$$
 * $$\varphi = 2\cos(\pi/5)=2\cos 36^\circ.\,$$


 * $$\sum_{n=1}^{\infty}|F(n)\varphi-F(n+1)|

= \varphi\,.$$


 * $$\varphi^{n+1} = \varphi^n + \varphi^{n-1}\,.$$



\begin{align} 3\varphi^3 - 5\varphi^2 + 4 & = 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 \\ & = 3[(\varphi + 1) + \varphi] - 5(\varphi + 1) + 4 \\ & = \varphi + 2 \approx 3.618. \end{align} $$

The golden ratio's decimal expansion can be calculated directly from the expression
 * $$\varphi = {1+\sqrt{5} \over 2},$$

√5 ≈ 2.2360679774997896964.


 * $$x_{n+1} = \frac{(x_n + 5/x_n)}{2}$$

xn and xn−1.


 * $$x_{n+1} = \frac{x_n^2 + 1}{2x_n - 1},$$

x − 1 − 1/x = 0,
 * $$x_{n+1} = \frac{x_n^2 + 2x_n}{x_n^2 + 1}.$$

F25001 and F25000,


 * $$\Phi = \varphi -1\,.$$


 * $$r_u = \frac{a}{2} \sqrt{\varphi \sqrt{5}} = \frac{a}{4} \sqrt{10 +2\sqrt{5}} \approx 0.9510565163 \cdot a $$


 * $$r_i = \frac{\varphi^2 a}{2 \sqrt{3}} = \frac{1}{12} \sqrt{3} \left(3+ \sqrt{5} \right) a \approx 0.7557613141\cdot a $$


 * $$ r_m = \frac{a \varphi}{2} = \frac{1}{4} \left(1+\sqrt{5}\right) a \approx 0.80901699\cdot a $$

$$ \varphi $$ (also called $$\tau$$) is the golden ratio.


 * (0, ±1, ±φ)
 * (±1, ±φ, 0)
 * (±φ, 0, ±1)

φ = (1+√5)/2

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to &phi;, the golden ratio. Specifically, a golden spiral gets wider (or further from its origin) by a factor of &phi; for every quarter turn it makes.

The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of b.


 * $$r = ae^{b\theta}\,$$

or


 * $$\theta = \frac{1}{b} \ln(r/a),$$

with e being the base of natural logarithms, a being an arbitrary positive real constant, and b such that when &theta; is a right angle (a quarter turn in either direction):


 * $$e^{b\theta_\mathrm{right}}\, = \varphi$$

Therefore, b is given by


 * $$b = {\ln{\varphi} \over \theta_\mathrm{right}}$$

The numerical value of b depends on whether the right angle is measured as 90 degrees or as &pi;/2 radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of b (that is, b can also be the negative of this value):


 * $$|b| = {\ln{\varphi} \over 90} = 0.0053468\,$$ for &theta; in degrees;


 * $$|b| = {\ln{\varphi} \over \pi/2} = 0.306349\,$$ for &theta; in radians.


 * $$r = ac^{\theta}\,$$

where the constant c is given by:


 * $$c = e^b\,$$

which for the golden spiral gives c values of:


 * $$c = \varphi ^ \frac{1}{90} = 1.0053611$$

and


 * $$c = \varphi ^ \frac{2}{\pi} = 1.358456$$