




from __future__ import annotations







def lowerCamelCase__  (     _lowercase					,		_lowercase = None					,		_lowercase = None ):

     '''simple docstring'''


     if start is None:
          UpperCAmelCase_     :				List[str]  =     0

     if end is None:
          UpperCAmelCase_     :				Dict  =     len(_lowercase ) - 1

     if start >= end:
          return

     UpperCAmelCase_     :				Optional[Any]  =     (start + end) // 2

     slowsort(_lowercase					,		_lowercase					,		_lowercase )
     slowsort(_lowercase					,		mid + 1					,		_lowercase )

     if sequence[end] < sequence[mid]:
          UpperCAmelCase_,       UpperCAmelCase_     :				List[str]  =     sequence[mid], sequence[end]

     slowsort(_lowercase					,		_lowercase					,		end - 1 )


if __name__ == "__main__":
     from doctest import testmod

     testmod()



from __future__ import annotations

import matplotlib.pyplot as plt  # type: ignore
import numpy

# initial triangle of Koch snowflake
lowerCamelCase		   =			numpy.array([0, 0])
lowerCamelCase		   =			numpy.array([0.5, 0.866_0254])
lowerCamelCase		   =			numpy.array([1, 0])
lowerCamelCase		   =			[VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]




def       a_ (       SCREAMING_SNAKE_CASE__   : list[numpy.ndarray]						,     SCREAMING_SNAKE_CASE__   : int	):
       '''simple docstring'''





       _lowerCamelCase			:						List[Any]            =initial_vectors
       for _ in range(SCREAMING_SNAKE_CASE__	):
              _lowerCamelCase			:						List[str]            =iteration_step(SCREAMING_SNAKE_CASE__	)
       return vectors




def       a_ (       SCREAMING_SNAKE_CASE__   : list[numpy.ndarray]	):
       '''simple docstring'''





       _lowerCamelCase			:						Dict            =[]
       for i, start_vector in enumerate(vectors[:-1]	):
              _lowerCamelCase			:						Tuple            =vectors[i + 1]
              new_vectors.append(SCREAMING_SNAKE_CASE__	)
              _lowerCamelCase			:						Tuple            =end_vector - start_vector
              new_vectors.append(start_vector + difference_vector / 3	)
              new_vectors.append(
                  start_vector + difference_vector / 3 + rotate(difference_vector / 3						,     60	)	)
              new_vectors.append(start_vector + difference_vector * 2 / 3	)
       new_vectors.append(vectors[-1]	)
       return new_vectors




def       a_ (       SCREAMING_SNAKE_CASE__   : numpy.ndarray						,     SCREAMING_SNAKE_CASE__   : float	):
       '''simple docstring'''





       _lowerCamelCase			:						Union[str, Any]            =numpy.radians(SCREAMING_SNAKE_CASE__	)
       _lowerCamelCase      ,	_lowerCamelCase			:						List[Any]            =numpy.cos(SCREAMING_SNAKE_CASE__	), numpy.sin(SCREAMING_SNAKE_CASE__	)
       _lowerCamelCase			:						str            =numpy.array(((c, -s), (s, c))	)
       return numpy.dot(SCREAMING_SNAKE_CASE__						,     SCREAMING_SNAKE_CASE__	)




def       a_ (       SCREAMING_SNAKE_CASE__   : list[numpy.ndarray]	):
       '''simple docstring'''





       _lowerCamelCase			:						Optional[Any]            =plt.gca()
       axes.set_aspect('equal'	)

       # matplotlib.pyplot.plot takes a list of all x-coordinates and a list of all
       # y-coordinates as inputs, which are constructed from the vector-list using
       # zip()
       _lowerCamelCase      ,	_lowerCamelCase			:						Tuple            =zip(*SCREAMING_SNAKE_CASE__	)
       plt.plot(SCREAMING_SNAKE_CASE__						,     SCREAMING_SNAKE_CASE__	)
       plt.show()


if __name__ == "__main__":
 import doctest

 doctest.testmod()

 lowerCamelCase		   =			iterate(INITIAL_VECTORS, 5)
 plot(processed_vectors)


