Math Art with Fibonacci, Trigonometry, and Constants-AYNET

11月15日

Scientific Explanation of the Code
This Pine Script code is a dynamic visual representation that combines mathematical constants, trigonometric functions, and Fibonacci sequences to generate geometrical patterns on a TradingView chart. The code leverages Pine Script’s drawing functions (line.new) and real-time bar data to create evolving shapes. Below is a detailed scientific explanation of its components:

1. Inputs and User-Defined Parameters
num_points: Specifies the number of points used to generate the geometrical pattern. Higher values result in more complex and smoother shapes.
scale: A scaling factor to adjust the size of the shape.
rotation: A dynamic rotation factor that evolves the shape over time based on the bar index (bar_index).
shape_color: Defines the color of the drawn shapes.
2. Mathematical Constants
The script employs essential mathematical constants:

Phi (ϕ): Known as the golden ratio
(
1
+
5
)
/
2
(1+
5
​
)/2, which governs proportions in Fibonacci spirals and natural growth patterns.
Pi (π): Represents the ratio of a circle's circumference to its diameter, crucial for trigonometric calculations.
Euler’s Number (e): The base of natural logarithms, incorporated in exponential growth modeling.
3. Geometric and Trigonometric Calculations
Fibonacci-Based Radius: The radius for each point is determined using a Fibonacci-inspired formula:

𝑟
=
scale
×
𝜙
⋅
𝑖
num_points
r=scale×
num_points
ϕ⋅i
​

Here,
𝑖
i is the point index. This ensures the shape grows proportionally based on the golden ratio.

Angle Calculation: The angular position of each point is calculated as:

𝜃
=
𝑖
⋅
Δ
𝜃
+
rotation
⋅
bar_index
100
θ=i⋅Δθ+rotation⋅
100
bar_index
​

where
Δ
𝜃
=
2
𝜋
num_points
Δθ=
num_points
2π
​
. This generates evenly spaced points along a circle, with dynamic rotation.

Coordinates: Cartesian coordinates
(
𝑥
,
𝑦
)
(x,y) for each point are derived using:

𝑥
=
𝑟
⋅
cos
⁡
(
𝜃
)
,
𝑦
=
𝑟
⋅
sin
⁡
(
𝜃
)
x=r⋅cos(θ),y=r⋅sin(θ)
These coordinates describe a polar-to-Cartesian transformation.

4. Dynamic Line Drawing
Connecting Points: For each pair of consecutive points, a line is drawn using:

line.new
(
𝑥
1
,
𝑦
1
,
𝑥
2
,
𝑦
2
)
line.new(x
1
​
,y
1
​
,x
2
​
,y
2
​
)
The coordinates are adjusted by:

bar_index: Aligns the x-axis to the chart’s time-based bar index.
int() Conversion: Ensures x-coordinates are integers, as required by line.new.
Line Properties:

Color: Set by the user.
Width: Fixed at 1 for simplicity.
5. Real-Time Adaptation
The shapes evolve dynamically as new bars form:

Rotation Over Time: The rotation parameter modifies angles proportionally to bar_index, creating a rotating effect.
Bar Index Alignment: Shapes are positioned relative to the current bar on the chart, ensuring synchronization with market data.
6. Visualization and Applications
This script generates evolving geometrical shapes, which have both aesthetic and educational value. Potential applications include:

Mathematical Visualization: Demonstrating the interplay of Fibonacci sequences, trigonometry, and geometry.
Technical Analysis: Serving as a visual overlay for price movement patterns, highlighting cyclical or wave-like behavior.
Dynamic Art: Creating visually appealing and evolving patterns on financial charts.
Scientific Relevance
This code synthesizes principles from:

Mathematical Analysis: Incorporates constants and formulas central to calculus, trigonometry, and algebra.
Geometry: Visualizes patterns derived from polar coordinates and Fibonacci scaling.
Real-Time Systems: Adapts dynamically to market data, showcasing practical applications of mathematics in financial visualization.
If further optimization or additional functionality is required, let me know! 😊