





from __future__ import annotations

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

# initial triangle of Koch snowflake
A       :   Union[str, Any]					= numpy.array([0, 0])
A       :   Optional[Any]					= numpy.array([0.5, 0.8_660_254])
A       :   Optional[Any]					= numpy.array([1, 0])
A       :   str					= [VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]







def   __lowerCamelCase   (				__a  :list[numpy.ndarray]       ,	__a  :int       )						->							list[numpy.ndarray]:






		"""simple docstring"""

		A__     =  initial_vectors
		for _ in range(__a       ):
				A__     =  iteration_step(__a       )
		return vectors







def   __lowerCamelCase   (				__a  :list[numpy.ndarray]       )						->							list[numpy.ndarray]:






		"""simple docstring"""

		A__     =  []
		for i, start_vector in enumerate(vectors[:-1]       ):
				A__     =  vectors[i + 1]
				new_vectors.append(__a       )
				A__     =  end_vector - start_vector
				new_vectors.append(start_vector + difference_vector / 3       )
				new_vectors.append(
				    start_vector + difference_vector / 3 + rotate(difference_vector / 3       ,	6_0       )       )
				new_vectors.append(start_vector + difference_vector * 2 / 3       )
		new_vectors.append(vectors[-1]       )
		return new_vectors







def   __lowerCamelCase   (				__a  :numpy.ndarray       ,	__a  :float       )						->							numpy.ndarray:






		"""simple docstring"""

		A__     =  numpy.radians(__a       )
		A__ ,     A__     =  numpy.cos(__a       ), numpy.sin(__a       )
		A__     =  numpy.array(((c, -s), (s, c))       )
		return numpy.dot(__a       ,	__a       )







def   __lowerCamelCase   (				__a  :list[numpy.ndarray]       )						->							None:






		"""simple docstring"""

		A__     =  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()
		A__ ,     A__     =  zip(*__a       )
		plt.plot(__a       ,	__a       )
		plt.show()


if __name__ == "__main__":
					import doctest

					doctest.testmod()

					A       :   List[str]					= iterate(INITIAL_VECTORS, 5)
					plot(processed_vectors)










def   __lowerCamelCase   (				__a  :int       )						->							Dict:






		"""simple docstring"""

		A__     =  len(__a       )
		A__     =  sum(__a       )

		A__     =  [[False for x in range(s + 1       )] for y in range(n + 1       )]

		for i in range(1       ,	n + 1       ):
				A__     =  True

		for i in range(1       ,	s + 1       ):
				A__     =  False

		for i in range(1       ,	n + 1       ):
				for j in range(1       ,	s + 1       ):
						A__     =  dp[i][j - 1]

						if arr[i - 1] <= j:
								A__     =  dp[i][j] or dp[i - 1][j - arr[i - 1]]

		for j in range(int(s / 2       )       ,	-1       ,	-1       ):
				if dp[n][j] is True:
						A__     =  s - 2 * j
						break

		return diff



