Apollonian Gasket Pattern using Python

 

import matplotlib.pyplot as plt

import numpy as np

def draw_circle(ax, x, y, r):

    circle = plt.Circle((x, y), r, edgecolor='black', facecolor='none')

    ax.add_patch(circle)

def apollonian(ax, x, y, r, depth):

    if depth == 0:

        return

    draw_circle(ax, x, y, r)

    new_radius = r / 2

    apollonian(ax, x - r, y, new_radius, depth-1)

    apollonian(ax, x + r, y, new_radius, depth-1)

    apollonian(ax, x, y + r, new_radius, depth-1)

fig, ax = plt.subplots(figsize=(6, 6))

ax.set_aspect('equal', 'box')

ax.set_xlim(-1.5, 1.5)

ax.set_ylim(-1.5, 1.5)

initial_radius = 1

apollonian(ax, 0, 0, initial_radius, 5) 

ax.set_title('Apollonian Gasket')

plt.show()

#source code --> clcoding.com

Code Explanation:

1.Importing Necessary Libraries

import matplotlib.pyplot as plt

import numpy as np

matplotlib.pyplot: This is the main library for plotting 2D and 3D figures in Python. We're using it to create and display the plot.

numpy: This is a powerful numerical library in Python used for handling arrays and performing mathematical operations. It is useful for generating coordinate data.

 2. Drawing a Single Circle

def draw_circle(ax, x, y, r):

    circle = plt.Circle((x, y), r, edgecolor='black', facecolor='none')

    ax.add_patch(circle)

Purpose: This function draws a single circle on the plot.

x, y: Coordinates of the circle's center.

r: Radius of the circle.

plt.Circle(): This creates a circle object at the given position with the specified radius. The edgecolor='black' ensures that the circle's edge is outlined in black, and facecolor='none' ensures that the circle has no fill color (transparent).

 3. Recursive Function to Create the Apollonian Gasket

def apollonian(ax, x, y, r, depth):

    if depth == 0:

        return

    # Draw the current circle

    draw_circle(ax, x, y, r)

   

    # Calculate the radius of the three new tangent circles

    new_radius = r / 2

   

    # Recursively draw the three smaller tangent circles

    apollonian(ax, x - r, y, new_radius, depth-1)

    apollonian(ax, x + r, y, new_radius, depth-1)

    apollonian(ax, x, y + r, new_radius, depth-1)

Purpose: This function is the core of generating the Apollonian Gasket. It starts by drawing a circle and then recursively creates three smaller tangent circles inside the original circle.

Key Steps:

Base case (Stopping condition): If the depth reaches 0, the recursion stops. This prevents the function from going into infinite recursion.

Drawing the Circle: The current circle is drawn using draw_circle().

Calculating New Radius: Each recursive step reduces the radius of the new circles to half of the current radius (new_radius = r / 2). This makes the fractal progressively smaller.

Recursive Calls: The function then calls itself three times:

x - r, y: Creates a circle to the left of the current circle.

x + r, y: Creates a circle to the right of the current circle.

x, y + r: Creates a circle above the current circle.

 4. Setting Up the Plot

fig, ax = plt.subplots(figsize=(8, 8))

ax.set_aspect('equal', 'box')

ax.set_xlim(-1.5, 1.5)

ax.set_ylim(-1.5, 1.5)

Purpose: Sets up the plotting area.

plt.subplots(figsize=(8, 8)): Creates a new plot with a figure size of 8x8 inches.

ax.set_aspect('equal', 'box'): Ensures the aspect ratio of the plot is equal, so circles are not distorted. It makes sure the X and Y axes have the same scale.

ax.set_xlim(-1.5, 1.5) and ax.set_ylim(-1.5, 1.5): These functions define the range of the X and Y axes. It ensures the circles are centered and visible within the plot.

 5. Calling the Recursive Function

initial_radius = 1

apollonian(ax, 0, 0, initial_radius, 5)  # Depth=5 for deeper recursion

Purpose: This line starts the recursion.

The initial_radius = 1 defines the radius of the first circle (the largest one).

apollonian(ax, 0, 0, initial_radius, 5): This call starts the fractal drawing from the center of the plot (0, 0) with an initial radius of 1, and a recursion depth of 5. The depth determines how many iterations of circles will be drawn. Higher depth values make the fractal more intricate.

 6. Displaying the Plot

ax.set_title('Apollonian Gasket')

plt.show()

ax.set_title('Apollonian Gasket'): Adds a title to the plot.

plt.show(): This line displays the plot on the screen.