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Python/fractals/koch_snowflake.py at Write-for-current-Python · davgit/Python · GitHub
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fractals
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koch_snowflake.py
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koch_snowflake.py
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"""
Description
The Koch snowflake is a fractal curve and one of the earliest fractals to
have been described. The Koch snowflake can be built up iteratively, in a
sequence of stages. The first stage is an equilateral triangle, and each
successive stage is formed by adding outward bends to each side of the
previous stage, making smaller equilateral triangles.
This can be achieved through the following steps for each line:
1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step 1
as its base and points outward.
3. remove the line segment that is the base of the triangle from step 2.
(description adapted from https://en.wikipedia.org/wiki/Koch_snowflake )
(for a more detailed explanation and an implementation in the
Processing language, see https://natureofcode.com/book/chapter-8-fractals/
#84-the-koch-curve-and-the-arraylist-technique )
Requirements (pip):
- matplotlib
- numpy
"""
from
__future__
import
annotations
import
matplotlib
.
pyplot
as
plt
# type: ignore
import
numpy
# initial triangle of Koch snowflake
VECTOR_1
=
numpy
.
array
([
0
,
0
])
VECTOR_2
=
numpy
.
array
([
0.5
,
0.8660254
])
VECTOR_3
=
numpy
.
array
([
1
,
0
])
INITIAL_VECTORS
=
[
VECTOR_1
,
VECTOR_2
,
VECTOR_3
,
VECTOR_1
]
# uncomment for simple Koch curve instead of Koch snowflake
# INITIAL_VECTORS = [VECTOR_1, VECTOR_3]
def
iterate
(
initial_vectors
:
list
[
numpy
.
ndarray
],
steps
:
int
)
->
list
[
numpy
.
ndarray
]:
"""
Go through the number of iterations determined by the argument "steps".
Be careful with high values (above 5) since the time to calculate increases
exponentially.
>>> iterate([numpy.array([0, 0]), numpy.array([1, 0])], 1)
[array([0, 0]), array([0.33333333, 0. ]), array([0.5 ,
\
0.28867513]), array([0.66666667, 0. ]), array([1, 0])]
"""
vectors
=
initial_vectors
for
i
in
range
(
steps
):
vectors
=
iteration_step
(
vectors
)
return
vectors
def
iteration_step
(
vectors
:
list
[
numpy
.
ndarray
])
->
list
[
numpy
.
ndarray
]:
"""
Loops through each pair of adjacent vectors. Each line between two adjacent
vectors is divided into 4 segments by adding 3 additional vectors in-between
the original two vectors. The vector in the middle is constructed through a
60 degree rotation so it is bent outwards.
>>> iteration_step([numpy.array([0, 0]), numpy.array([1, 0])])
[array([0, 0]), array([0.33333333, 0. ]), array([0.5 ,
\
0.28867513]), array([0.66666667, 0. ]), array([1, 0])]
"""
new_vectors
=
[]
for
i
,
start_vector
in
enumerate
(
vectors
[:
-
1
]):
end_vector
=
vectors
[
i
+
1
]
new_vectors
.
append
(
start_vector
)
difference_vector
=
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
rotate
(
vector
:
numpy
.
ndarray
,
angle_in_degrees
:
float
)
->
numpy
.
ndarray
:
"""
Standard rotation of a 2D vector with a rotation matrix
(see https://en.wikipedia.org/wiki/Rotation_matrix )
>>> rotate(numpy.array([1, 0]), 60)
array([0.5 , 0.8660254])
>>> rotate(numpy.array([1, 0]), 90)
array([6.123234e-17, 1.000000e+00])
"""
theta
=
numpy
.
radians
(
angle_in_degrees
)
c
,
s
=
numpy
.
cos
(
theta
),
numpy
.
sin
(
theta
)
rotation_matrix
=
numpy
.
array
(((
c
,
-
s
), (
s
,
c
)))
return
numpy
.
dot
(
rotation_matrix
,
vector
)
def
plot
(
vectors
:
list
[
numpy
.
ndarray
])
->
None
:
"""
Utility function to plot the vectors using matplotlib.pyplot
No doctest was implemented since this function does not have a return value
"""
# avoid stretched display of graph
axes
=
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()
x_coordinates
,
y_coordinates
=
zip
(
*
vectors
)
plt
.
plot
(
x_coordinates
,
y_coordinates
)
plt
.
show
()
if
__name__
==
"__main__"
:
import
doctest
doctest
.
testmod
()
processed_vectors
=
iterate
(
INITIAL_VECTORS
,
5
)
plot
(
processed_vectors
)
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