'''simple docstring'''






import random






def  UpperCamelCase__					(					_lowercase    :				int   ,    _lowercase    :				float   ,    _lowercase    :				bool = False     )							->	dict:
  __UpperCAmelCase:					dict		= {i: [] for i in range(_lowercase     )}

  # if probability is greater or equal than 1, then generate a complete graph
  if probability >= 1:
    return complete_graph(_lowercase     )
  # if probability is lower or equal than 0, then return a graph without edges
  if probability <= 0:
    return graph

  # for each couple of nodes, add an edge from u to v
  # if the number randomly generated is greater than probability probability
  for i in range(_lowercase     ):
    for j in range(i + 1   ,    _lowercase     ):
      if random.random() < probability:
        graph[i].append(_lowercase     )
        if not directed:
          # if the graph is undirected, add an edge in from j to i, either
          graph[j].append(_lowercase     )
  return graph






def  UpperCamelCase__					(					_lowercase    :				int     )							->	dict:
  return {
      i: [j for j in range(_lowercase     ) if i != j] for i in range(_lowercase     )
  }


if __name__ == "__main__":
    import doctest

    doctest.testmod()
'''simple docstring'''






from __future__ import annotations

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

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






def  UpperCamelCase__					(					_lowercase    :				list[numpy.ndarray]   ,    _lowercase    :				int     )							->	list[numpy.ndarray]:
  __UpperCAmelCase:					int		= initial_vectors
  for _ in range(_lowercase     ):
    __UpperCAmelCase:					Any		= iteration_step(_lowercase     )
  return vectors






def  UpperCamelCase__					(					_lowercase    :				list[numpy.ndarray]     )							->	list[numpy.ndarray]:
  __UpperCAmelCase:					Optional[int]		= []
  for i, start_vector in enumerate(vectors[:-1]     ):
    __UpperCAmelCase:					Union[str, Any]		= vectors[i + 1]
    new_vectors.append(_lowercase     )
    __UpperCAmelCase:					Optional[int]		= 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  UpperCamelCase__					(					_lowercase    :				numpy.ndarray   ,    _lowercase    :				float     )							->	numpy.ndarray:
  __UpperCAmelCase:					Tuple		= numpy.radians(_lowercase     )
  __UpperCAmelCase, __UpperCAmelCase:					Optional[Any]		= numpy.cos(_lowercase     ), numpy.sin(_lowercase     )
  __UpperCAmelCase:					Tuple		= numpy.array(((c, -s), (s, c))     )
  return numpy.dot(_lowercase   ,    _lowercase     )






def  UpperCamelCase__					(					_lowercase    :				list[numpy.ndarray]     )							->	None:
  __UpperCAmelCase:					Union[str, 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()
  __UpperCAmelCase, __UpperCAmelCase:					Dict		= zip(*_lowercase     )
  plt.plot(_lowercase   ,    _lowercase     )
  plt.show()


if __name__ == "__main__":
    import doctest

    doctest.testmod()

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