A Python script that creates n-flakes

Question

I have written a simple Python script that generates n-flakes using matplotlib.

Wikipedia article on N-flake.

I wrote two functions, one function generates pentaflake, the other generates hexaflake, I adapted my code from some turtle example code, I understand the mathematics, but I am not sure about the performance.

Pentaflake, five iterations:

enter image description here

Hexaflake, six iterations:

enter image description here

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Code

import math
import matplotlib.pyplot as plt
import random
from functools import lru_cache
from matplotlib.collections import PolyCollection
from PIL import Image
@lru_cache(maxsize=None)
def cos(d): return math.cos(math.radians(d))
@lru_cache(maxsize=None)
def sin(d): return math.sin(math.radians(d))
@lru_cache(maxsize=None)
def tan(d): return math.tan(math.radians(d))
def pentagon(x, y, r, direction):
 return [(x + r * cos(direction+i*72), y + r * sin(direction+i*72)) for i in range(5)]
def pentaflake(n):
 pentagons = []
 ratio = (1+2*cos(36))
 def worker(x, y, r, direction, n):
 if not n:
 pentagons.append(pentagon(x,y,r,direction))
 return
 r2 = r/ratio
 d = 2*r2*cos(36)
 for _ in range(5):
 x2,y2 = x+d*cos(direction),y+d*sin(direction)
 worker(x2,y2,r2,direction,n-1)
 direction += 72
 worker(x,y,r2,direction+180,n-1)
 
 worker(0, 0, 2, 90, n)
 return pentagons
def hexagon(x, y, r):
 return [(x + r * cos(90+i*60), y + r * sin(90+i*60)) for i in range(6)]
def hexaflake(n):
 hexagons = []
 def worker(x, y, r, n):
 if n==0:
 hexagons.append(hexagon(x,y,r))
 return
 r2 = r/3
 for i in range(6):
 x2,y2 = x+r*cos(90+i*60)*2/3,y+r*sin(90+i*60)*2/3
 worker(x2,y2,r2,n-1)
 worker(x,y,r2,n-1)
 
 worker(0, 0, 2, n)
 return hexagons
def plot_nflake(iterations, width=1920, height=1080, lw=0, color='#4d0dff', alpha=1, show=True, method=pentaflake):
 assert method in (pentaflake, hexaflake)
 polygons = method(iterations)
 fig = plt.figure(figsize=(width/100, height/100), dpi=100, facecolor='black')
 ax = fig.add_subplot(111)
 ax.set_axis_off()
 if not color:
 color = '#'+random.randbytes(3).hex()
 ax.add_collection(PolyCollection(polygons, facecolors=color, lw=lw, alpha=alpha))
 plt.axis('scaled')
 fig.subplots_adjust(left=0, bottom=0, right=1, top=1, wspace=0, hspace=0)
 fig.canvas.draw()
 image = Image.frombytes(
 'RGB', fig.canvas.get_width_height(), fig.canvas.tostring_rgb())
 if not show:
 plt.close(fig)
 else:
 plt.show()
 return image

How do I improve the performance of this script?

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Probably irrelevant, but I made these two images by editing the example pictures:

enter image description here

enter image description here

• Reinderien: Have you made an attempt at vectorising pentagon and hexagon? That would be a good place to start.