The Golden Angle
Main Concept

The golden angle is closely related to the golden ratio, $\mathrm{\ϕ}$. The golden angle is $\mathrm{\varphi}$ rotations around a circle. The golden ratio, $\mathrm{\varphi}$, has the value $\mathbf{\varphi}\mathbf{}\mathbf{equals;}\frac{\mathbf{}\mathbf{1}\mathbf{}\mathbf{plus;}\mathbf{}\sqrt{\mathbf{5}}}{\mathbf{2}}\mathbf{equals;}\mathbf{}\mathbf{1.6180339887}\mathbf{..}\mathbf{period;}$. Two quantities are said to be in the golden ratio if the ratio of their sum to the larger quantity is equal to the ratio of the larger quantity to the smaller one. Algebraically, this is written as: $\frac{\mathit{a}\mathbf{\+}\mathit{b}}{\mathit{a}}\mathbf{}\mathbf{equals;}\mathbf{}\frac{\mathit{a}}{\mathit{b}}\mathbf{}\mathbf{equals;}\mathbf{}\mathbf{varphi;}$ where $\mathit{a}\mathbf{}\mathbf{gt;}\mathbf{}\mathit{b}$. We find that the ratios of successive terms of the Fibonacci sequence, namely, $\frac{2}{1}\,\frac{3}{2}comma;\frac{5}{3}comma;\frac{8}{5}comma;\frac{13}{8}comma;..period;$ provide the closest rational approximations of the golden ratio for a given size of numerator and denominator. In fact, they approach $\mathrm{\varphi}$ as we move further along in the sequence. In terms of degrees, $\mathrm{\varphi}$ rotations is equivalent to $\mathbf{222.4922359}\mathbf{..}\mathbf{\.}$ deg or $\mathbf{137.5077641}\mathbf{..}\mathbf{\.}$ deg. Both angles exhibit the same properties since $137.5077641..\.$ deg is $222.4922359..\.$ deg.

Interesting properties


One of the foremost properties of $\mathrm{\varphi}$ is that it is approximated worse, by the method of fractions of integers, than any other number, so it is known as the most irrational number. This becomes obvious when looking at the continued fraction for $\mathrm{\varphi}$, $\mathrm{\ϕ}equals;1plus;\frac{1}{1plus;\frac{1}{1plus;\frac{1}{1plus;\frac{1}{1..period;}}}}$. When approximating a number with a continued fraction, the continued fraction is truncated, and a fraction is used in the last step of the approximation. If this is a small number, such as $\frac{1}{200}$, then the approximation is considered precise. In ϕ's continued fraction, the final fraction is always "$\frac{1}{1}$", which is not small, and that's why ϕ is coined "the most irrational number". This fact makes the golden angle interesting. For instance, if a point were placed around a circle every $0.2$ rotations. After $5$ rotations, the point would have returned to its initial position since $0.2$ is a rational number that can be expressed as $\frac{1}{5}$. After $10$ rotations, the same is still true. Since a rational number of rotations around a circle produces a pattern that will repeat forever, a natural question would be to ask what happens when you rotate around a circle in the same way but place the points after an irractional number of rotations. Additionally, what would differentiate the graph of the most irrational number from that of other irrational numbers?




Use the interactive components in this worksheet to experiment with different values of f. What happens when you enter a rational number between [0,1]? What happens when you enter an irrational number? How about the golden angle/ratio?
Steps
1.

Use the slider to select the number of points displayed on the graph. Each new point will be placed around the central circle at the specified angle or rotation increment specified.

2.

Enter a real number in [0, 1] (if Enter a real number in [0, 1] is checked) or an angle in degrees in the input box.

Note: The real number in [0,1] is a number of rotations that will be displayed instead of an angle. If an integer number of rotations is entered, a straight line will be plotted since each new point is placed at one whole rotation from its previous point. If $0.6180339887..\.$ ($\mathrm{\ϕ}$1) is entered, the same thing will be displayed since the whole rotation component of the number will not change the angle at which the next point is placed.
If placing a point would result in its overlapping with another point, the point is placed as close as possible to the center, at the same angle, without overlapping with any points that are already placed, meaning that the radius at which the point is placed is increased until it will not overlap with any other points.
3.

After having entered an angle or number of rotations click Update Angle to register your changes.

What happens when you rotate around a circle in the same way but place the points after an irrational number of rotations? Additionally, what would differentiate the graph of the most irrational number from that of other irrational numbers?
Number of Points
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Optional: generate more than 100 points (may take a long time)
Number of Points:

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