

'''simple docstring'''





import math






def 					lowerCamelCase_     (							UpperCAmelCase_							:     int	)						->				List[Any]:




	'''simple docstring'''




	if 1 < number < 4:
		# 2 and 3 are primes
		return True
	elif number < 2 or number % 2 == 0 or number % 3 == 0:
		# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
		return False

	# All primes number are in format of 6k +/- 1
	for i in range(5							,       int(math.sqrt(__lowerCAmelCase	) + 1	)							,       6	):
		if number % i == 0 or number % (i + 2) == 0:
			return False
	return True






def 					lowerCamelCase_     (							UpperCAmelCase_							:     int = 1_0_0_0_1	)						->				Optional[int]:




	'''simple docstring'''




	try:
		_UpperCamelCase   :		Dict			=						int(__lowerCAmelCase	)
	except (TypeError, ValueError):
		raise TypeError('Parameter nth must be int or castable to int.'	) from None
	if nth <= 0:
		raise ValueError('Parameter nth must be greater than or equal to one.'	)
	_UpperCamelCase   :		Optional[Any]			=						[]
	_UpperCamelCase   :		Union[str, Any]			=						2
	while len(__lowerCAmelCase	) < nth:
		if is_prime(__lowerCAmelCase	):
			primes.append(__lowerCAmelCase	)
			num += 1
		else:
			num += 1
	return primes[len(__lowerCAmelCase	) - 1]


if __name__ == "__main__":
							print(f'{solution() = }')







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, 0.8_66_02_54])
lowerCAmelCase__								=		numpy.array([1, 0])
lowerCAmelCase__								=		[VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]






def 					lowerCamelCase_     (							UpperCAmelCase_							:     list[numpy.ndarray]							,       UpperCAmelCase_							:     int	)						->				list[numpy.ndarray]:




	'''simple docstring'''




	_UpperCamelCase   :		Tuple			=						initial_vectors
	for _ in range(UpperCAmelCase_	):
		_UpperCamelCase   :		str			=						iteration_step(UpperCAmelCase_	)
	return vectors






def 					lowerCamelCase_     (							UpperCAmelCase_							:     list[numpy.ndarray]	)						->				list[numpy.ndarray]:




	'''simple docstring'''




	_UpperCamelCase   :		int			=						[]
	for i, start_vector in enumerate(vectors[:-1]	):
		_UpperCamelCase   :		Union[str, Any]			=						vectors[i + 1]
		new_vectors.append(UpperCAmelCase_	)
		_UpperCamelCase   :		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							,       6_0	)	)
		new_vectors.append(start_vector + difference_vector * 2 / 3	)
	new_vectors.append(vectors[-1]	)
	return new_vectors






def 					lowerCamelCase_     (							UpperCAmelCase_							:     numpy.ndarray							,       UpperCAmelCase_							:     float	)						->				numpy.ndarray:




	'''simple docstring'''




	_UpperCamelCase   :		str			=						numpy.radians(UpperCAmelCase_	)
	_UpperCamelCase	,		_UpperCamelCase   :		Optional[Any]			=						numpy.cos(UpperCAmelCase_	), numpy.sin(UpperCAmelCase_	)
	_UpperCamelCase   :		Any			=						numpy.array(((c, -s), (s, c))	)
	return numpy.dot(UpperCAmelCase_							,       UpperCAmelCase_	)






def 					lowerCamelCase_     (							UpperCAmelCase_							:     list[numpy.ndarray]	)						->				None:




	'''simple docstring'''




	_UpperCamelCase   :		str			=						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(*UpperCAmelCase_	)
	plt.plot(UpperCAmelCase_							,       UpperCAmelCase_	)
	plt.show()


if __name__ == "__main__":
							import doctest

							doctest.testmod()

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




