sierpinski-animations

The Essence of the Project

This project embodies a synthesis of mathematics and aesthetics. It encourages active exploration of fractal geometry through adjustable variables. Experimentation is welcomed.

Installation

pip install matplotlib numpy

Visual samples from the repository:

• 3DPyramidBook:

• RareSquare:

RareSquare Explanation

This script animates the transformation of the Sierpinski triangle into a dynamic fractal thing. Drawing upon recursion and rotational geometry, underpinned by principles of dynamical systems, the script showcases a temporal evolution of the fractal form.

• Recursive Algorithm: Utilizes the plot_sierpinski_fold function for recursive midpoint computation and vertex-based rotations, incrementally crafting the fractal depth.

• Geometric Rotations: Employs the rotate function to animate the fractal transformation, applying linear algebra to manipulate rotation angles within a two-dimensional plane.

• Animation Mechanics: The update_fold function drives the animation, managing frame-by-frame progression and utilizing angular interpolation to simulate the fractal's folding from 120 degrees down to zero.

• Visualization: Leverages Matplotlib's FuncAnimation for the graphical rendering, orchestrating the fractal's evolution into a tree-like structure through a seamless visual loop.

• Rendering: Executes the plt.show() command to deliver the fractal animation to the screen, revealing the progressive stages of the Sierpinski triangle's metamorphosis into a three-dimensional fractal geometry.

The interplay of mathematical elegance and computational graphics offers viewers a dynamic exploration of fractal transformations.

MeltingMountain:

MeltingMountain with +2 on the fractal complexity:

MeltingMountain Explanation

This Python script intertwines recursion and linear algebra to create the Sierpinski Triangle fractal, a self-replicating geometric pattern derived from a simple equilateral triangle. It showcases the emergence of complexity from simple rules.

• Recursion and Iteration: The generar_sierpinski function recursively generates the fractal structure, employing midpoint calculations to subdivide triangles, demonstrating a core concept of fractals in dynamical systems.

• Affine Transformations: It applies affine transformations to maintain the collinearity and proportional distances of points, highlighting the geometric invariance inherent in fractal transformations.

• Color Palette Creation: The generar_colores_pastel function generates a pastel color palette, utilizing vector operations to create visually appealing color schemes for the fractal.

• Rotational Transformations: Rotations are achieved through the abrir_fractal function, which uses matrix operations to pivot points around an axis, marrying linear algebra with trigonometry.

• Vector Calculus in Motion: Vector operations, including addition and scalar multiplication, facilitate the opening motion of the fractal, serving as a practical application of vector calculus in geometry.

• Computational Optimization: Optimized for performance, the script leverages NumPy for its efficient handling of arrays, enhancing computational operations for vertex manipulation and transformation.

This script not only produces a visual fractal phenomenon but also serves as an applied example of mathematical concepts in computational geometry and graphics, offering a dynamic view into the unfolding of fractal patterns.