Mandelbrot Set

Define the escape time function
Define the proximity function using NumPy
Convert pixel coordinates to NumPy arrays
Calculate the distance from the origin using the Euclidean norm
Define the maximum proximity (distance) for color mapping
Define the gradient color function
Map the distance to a value between 0 and 1
Convert the distance to a color gradient using HSV color space
Set the maximum number of iterations
Iterate over the complex plane

Moredelbrot.py

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+import turtle
+import colorsys
+import numpy as np
+
+# Set the screen's background color
+turtle.bgcolor("black")
+# Set the turtle's pen size to 1 pixel
+turtle.pensize(1)
+# Set the turtle's speed to the maximum
+turtle.speed(0)
+# Set the screen size
+turtle.screensize(canvwidth=7680, canvheight=4800, bg='black')
+
+# Define the escape time function
+def escape_time(c, max_iterations):
+    z = 0
+    for i in range(max_iterations):
+        z = z*z + c
+        if abs(z) > 2:
+            return i
+    return max_iterations
+
+# Define the proximity function using NumPy
+def proximity(x, y):
+    # Convert pixel coordinates to NumPy arrays
+    x_arr = np.array(x)
+    y_arr = np.array(y)
+
+    # Calculate the distance from the origin using the Euclidean norm
+    distance = np.linalg.norm(np.array([x_arr, y_arr]), axis=0)
+
+    return distance
+
+# Define the maximum proximity (distance) for color mapping
+max_proximity = (400 ** 2 + 400 ** 2) ** 0.5
+
+# Define the gradient color function
+def gradient_color(distance):
+    # Map the distance to a value between 0 and 1
+    normalized_distance = distance / max_proximity
+    # Convert the distance to a color gradient using HSV color space
+    hue = normalized_distance  # Use the distance directly as hue for simplicity
+    saturation = 1.0
+    value = 1.0
+    r, g, b = colorsys.hsv_to_rgb(hue, saturation, value)
+    return r, g, b
+
+# Set the maximum number of iterations
+max_iterations = 256
+
+# Iterate over the complex plane
+for x in range(-400, 400):
+    for y in range(-400, 400):
+        # Convert pixel coordinates to complex number
+        c = complex(x/200, y/200)
+        # Calculate the escape time for the current point
+        i = escape_time(c, max_iterations)
+        if i == max_iterations:
+            # Get the proximity (distance) to the origin
+            dist = proximity(x, y)
+            # Get the gradient color based on proximity
+            r, g, b = gradient_color(dist)
+            # Set the pen color
+            turtle.pencolor(r, g, b)
+            # Move the turtle to the current point and draw a dot
+            turtle.penup()
+            turtle.goto(x, y)
+            turtle.dot()
+
+# Hide the turtle and keep the screen open
+turtle.hideturtle()
+turtle.exitonclick()