I am always super interested in both science and art, science and art are two different ways to describe the objective reality, they are two sides of the same coin, in many areas they overlap, and mathematics is the most prominent one of such areas.
So I used the last three days to write a bunch of functions that generate artistic geometrical shapes using mathematics and visualize the shapes using matplotlib.
Previously I had absolutely zero experience with matplotlib, in the past I have used GeoGebra 5 to draw shapes, but doing it manually is really slow and prone to error, and I don't use brushes because I want my images to be precise, so I have decided to draw pictures with code and here I am.
In this gigantic script I have combined:
Dice Average Formula
The following equation is used to calculate the average value of a set of dice.
AV = ( (M + 1 )/ 2 ) * N
Where AV is the average dice value M is the max value of all dice N is the total number of die
Full Saturation Spectrum
Explanations by me: the RGB color model uses one byte for each channel to represent colors, one byte is eight bits, 2 to the power of 8 is 256, so each channel can have a value from 0 to 255, and the total number of colors possible in this model is 16777216 (2 to the power of 24).
I don't know about much about color theory, but in my observation, fully saturated colors have only one or two bytes that are set, and at least one of the bytes is fully set. So the total number of fully saturated colors is 6 * 255 = 1530.
And of course, programming, I have written over 400 lines of code, spent more than 12 hours on this project, expressed math in code, just to create, something with absolutely no practical value whatsoever—ART.
Code
from functools import lru_cache
from matplotlib.patches import Arc, Circle, Rectangle
from itertools import combinations
import matplotlib.pyplot as plt
import random
import math
import re
FIBONACCI = [2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233]
DIRECTIONS = [('right', 'up'), ('right', 'down'), ('left', 'down'), ('left', 'up')]
ANGLES = [(90, 180), (0, 90), (-90, 0), (180, 270)]
CCW_ANGLES = [(0, 90), (90, 180), (180, 270), (270, 360)]
cycles = [
(4, 2, 1),
(-2, -1),
(-14, -7, -20, -10, -5),
(
-50, -25, -74,
-37, -110, -55,
-164, -82, -41,
-122, -61, -182,
-91, -272, -136,
-68, -34, -17
)
]
loops = {term: len(cycle) for cycle in cycles for term in cycle}
stops = {term: cycle[-1] for cycle in cycles for term in cycle}
@lru_cache(maxsize=None)
def fibonacci(n):
s = []
a, b = 0, 1
for i in range(n):
s.append(a)
a, b = b, a + b
return s
@lru_cache(maxsize=None)
def spectrum_position(n):
if not isinstance(n, int):
raise TypeError('`n` should be an integer')
if n < 0:
raise ValueError('`n` must be non-negative')
n %= 1530
if 0 <= n < 255:
return f'#ff{n:02x}00'
elif 255 <= n < 510:
return f'#{510-n:02x}ff00'
elif 510 <= n < 765:
return f'#00ff{n-510:02x}'
elif 765 <= n < 1020:
return f'#00{1020-n:02x}ff'
elif 1020 <= n < 1275:
return f'#{n-1020:02x}00ff'
elif 1275 <= n < 1530:
return f'#ff00{1530-n:02x}'
@lru_cache(maxsize=None)
def factorize(n):
assert isinstance(n, int)
assert n >= 2
factors = []
while not n % 2:
factors.append(2)
n /= 2
for i in range(3, int(n**.5+1), 2):
while not n % i:
factors.append(i)
n /= i
if n > 1:
factors.append(int(n))
return factors
@lru_cache(maxsize=None)
def dice_average(n):
factors = factorize(n)
if len(factors) == 1:
return (1, n * 2 - 1)
else:
maxf = max(factors)
factors.remove(maxf)
return (math.prod(factors), maxf * 2 - 1)
@lru_cache(maxsize=None)
def hailstone(n):
if not isinstance(n, int):
raise TypeError('n should be an integer')
if n == 0:
raise ValueError('n should not be 0')
sequence = [n]
loop_iter = 0
looping = False
while True:
if not looping and n in loops:
loop_terms = loops[n]
looping = True
if looping:
loop_iter += 1
if loop_iter == loop_terms:
break
if n % 2:
n = 3 * n + 1
else:
n /= 2
sequence.append(int(n))
return sequence
def sin(x): return math.sin(math.radians(x))
def cos(x): return math.cos(math.radians(x))
def tan(d): return math.tan(math.radians(d))
@lru_cache(maxsize=None)
def negate(ls): return [-e for e in ls]
@lru_cache(maxsize=None)
def rotate(point, deg):
x, y = point
x1 = x * cos(deg) - y * sin(deg)
y1 = y * cos(deg) + x * sin(deg)
return (x1, y1)
@lru_cache(maxsize=None)
def square_vertices(init_pos, side, direction):
x0, y0 = init_pos
hd, vd = direction
x1 = x0 + side if hd == 'right' else x0 - side
y1 = y0 + side if vd == 'up' else y0 - side
return [(x0, y0), (x1, y0), (x1, y1), (x0, y1)]
def make_square(init_pos, side, direction, color=None):
vertices = square_vertices(init_pos, side, direction)
x, y = zip(*vertices)
minx = min(x)
miny = min(y)
if not color:
color = '#'+random.randbytes(3).hex()
return Rectangle((minx, miny), side, side, color=color, alpha=0.5), vertices
@lru_cache(maxsize=None)
def first_level(number):
assert number > 2
half = 180 / number
x1 = (16 / (1 + tan(half) ** 2)) ** .5
y1 = (16 - x1 ** 2) ** .5
k = y1 - tan(half - 90) * x1
x2 = -k / tan(half - 90)
r = ((x1 - x2) ** 2 + y1 ** 2) ** .5
return (x2, r)
@lru_cache(maxsize=None)
def next_level(distance, radius):
base = (distance - radius) / (distance + radius)
new_radius = radius * base
new_distance = distance - radius - new_radius
return (new_distance, new_radius)
@lru_cache(maxsize=None)
def fade_color(color, strength):
assert re.match(r'^#[0-9a-f]{6}$', color)
assert 0 < strength <= 1
color = int(color[1:], 16)
r, g, b = [round(((color>>i)%256)*strength) for i in (16, 8, 0)]
return f'#{r:02x}{g:02x}{b:02x}'
def fibonacci_spiral(n, c=12, bsquare=True, barc=True, reverse=False):
assert barc or bsquare
colors = [spectrum_position(int(1530*i/c)) for i in range(c)]
sequence = fibonacci(n+1)
init_pos = (0, 0)
vertexes = []
shapes = []
for i, n in enumerate(sequence[1:]):
direction = DIRECTIONS[i % 4] if not reverse else DIRECTIONS[::-1][i % 4]
angle = ANGLES[i % 4] if not reverse else CCW_ANGLES[i % 4]
square, vertices = make_square(init_pos, n, direction, colors[i%c])
center = vertices[[1, 3][i % 2]]
a, b = angle
arc = Arc(center, 2*n, 2*n, theta1=a, theta2=b, edgecolor='k', lw=3.0)
if bsquare: shapes.append(square)
if barc: shapes.append(arc)
init_pos = vertices[-2]
vertexes.extend(vertices)
return shapes, vertexes
def fibonacci_arcs(n, degree, color='k' ,reverse=False, alpha=0.75, lw=2):
sequence = fibonacci(n)
init_pos = (0, 0)
vertexes = []
shapes = []
for i, n in enumerate(sequence[1:]):
direction = DIRECTIONS[i % 4] if not reverse else DIRECTIONS[::-1][i % 4]
angle = ANGLES[i % 4] if not reverse else CCW_ANGLES[i % 4]
vertices = square_vertices(init_pos, n, direction)
rotated_vertices = [rotate(pos, degree) for pos in vertices]
index = [1, 3][i % 2]
center = rotated_vertices[index]
a, b = [(i + degree) % 360 for i in angle]
arc = Arc(center, 2*n, 2*n, theta1=a, theta2=b, edgecolor=color, lw=lw, alpha=alpha)
init_pos = vertices[-2]
shapes.append(arc)
vertexes.extend(rotated_vertices)
return shapes, vertexes
################################################################
# plotting functions
def plot_hailstone(n, real=False):
assert n > 1
fig = plt.figure(frameon=False)
ax = fig.add_subplot(111)
ax.set_axis_off()
numbers = list(range(-n, 0)) + list(range(1, n))
for i in numbers:
ax.plot(hailstone(i))
fig.subplots_adjust(left=0, bottom=0, right=1, top=1, wspace=0, hspace=0)
if real:
plt.axis('scaled')
plt.box(False)
plt.set_cmap('rainbow')
plt.show()
def plot_spiral(n, c, barc=True, bsquare=True, reverse=False):
shapes, vertexes = fibonacci_spiral(n, c, barc=barc, bsquare=bsquare, reverse=reverse)
x, y = zip(*vertexes)
fig = plt.figure(frameon=False)
ax = fig.add_subplot(111)
ax.set_axis_off()
for shape in shapes:
ax.add_patch(shape)
fig.subplots_adjust(left=0, bottom=0, right=1, top=1, wspace=0, hspace=0)
plt.axis('scaled')
plt.box(False)
ax = plt.gca()
ax.set_xlim([min(x), max(x)])
ax.set_ylim([min(y), max(y)])
plt.show()
def plot_lolipop(n, arms, lw=1):
arcs = []
points = []
u = 360/arms
for i in range(arms):
shapes, vertices = fibonacci_arcs(n, i*u, color='k', alpha=1, lw=lw)
arcs.extend(shapes)
points.extend(vertices)
x, y = zip(*points)
a, b = points[-2]
radius = (a*a+b*b)**.5
fig = plt.figure(frameon=False)
ax = fig.add_subplot(111)
ax.set_axis_off()
for arc in arcs:
ax.add_patch(arc)
ax.add_patch(Arc((0, 0), 2*radius, 2*radius, theta1=0, theta2=360, edgecolor='k', lw=lw))
fig.subplots_adjust(left=0, bottom=0, right=1, top=1, wspace=0, hspace=0)
plt.axis('scaled')
plt.box(False)
ax = plt.gca()
ax.set_xlim([min(x), max(x)])
ax.set_ylim([min(y), max(y)])
plt.show()
def plot_sunflower(n, cw, color=True, alpha=0.75):
assert cw in FIBONACCI and cw <= 144
i = FIBONACCI.index(cw)
ccw = FIBONACCI[i+1]
arcs = []
points = []
cwu = 360/cw
ccwu = 360/ccw
color_cwu = 1530/cw
color_ccwu = 1530/ccw
for i in range(cw):
color1 = spectrum_position(round(i*color_cwu)) if color else 'k'
shapes, vertices = fibonacci_arcs(n, i*cwu, color=color1, alpha=alpha)
arcs.extend(shapes)
points.extend(vertices)
for i in range(ccw):
color2 = spectrum_position(round(1530-i*color_ccwu)) if color else 'k'
shapes, vertices = fibonacci_arcs(n, (90+i*ccwu)%360, color=color2, reverse=True, alpha=alpha)
arcs.extend(shapes)
points.extend(vertices)
x, y = zip(*points)
fig = plt.figure(frameon=False)
ax = fig.add_subplot(111)
ax.set_axis_off()
for arc in arcs:
ax.add_patch(arc)
fig.subplots_adjust(left=0, bottom=0, right=1, top=1, wspace=0, hspace=0)
plt.axis('scaled')
plt.box(False)
ax = plt.gca()
ax.set_xlim([min(x), max(x)])
ax.set_ylim([min(y), max(y)])
plt.show()
def recur_plot_circle(division, levels, fade=0.9):
assert division > 2
fig = plt.figure(frameon=False)
ax = fig.add_subplot(111)
ax.set_axis_off()
rotation_diff = 90 / division
unit_rotation = 360 / division
color_increment = 1530 / division
distance, radius = first_level(division)
max_distance = distance + radius
for level in range(levels):
for index in range(division):
center = rotate((0, distance), level * rotation_diff + index * unit_rotation)
color = spectrum_position(round(index * color_increment))
color = fade_color(color, fade**level)
ax.add_patch(Circle(center, radius, color=color))
distance, radius = next_level(distance, radius)
fig.subplots_adjust(left=0, bottom=0, right=1, top=1, wspace=0, hspace=0)
plt.axis('scaled')
plt.box(False)
ax = plt.gca()
ax.set_xlim([-max_distance, max_distance])
ax.set_ylim([-max_distance, max_distance])
plt.show()
def plot_star(number):
assert number > 2
fig = plt.figure(frameon=False)
ax = fig.add_subplot(111)
ax.set_axis_off()
unit_rotation = 360 / number
points = []
for i in range(number):
points.append(rotate((0, 4), unit_rotation*i))
for a, b in combinations(points, r=2):
x, y = zip(*[a, b])
ax.plot(x, y)
fig.subplots_adjust(left=0, bottom=0, right=1, top=1, wspace=0, hspace=0)
plt.axis('scaled')
plt.box(False)
ax = plt.gca()
ax.set_xlim([-4, 4])
ax.set_ylim([-4, 4])
plt.set_cmap('rainbow')
plt.show()
def plot_tetragram(n):
assert n > 2 and not (n - 2) % 3
averages = [dice_average(i) for i in range(2, n)]
fig = plt.figure(frameon=False)
ax = fig.add_subplot(111)
ax.set_axis_off()
for a, b, c in zip(averages[::3], averages[1::3], averages[2::3]):
x, y = zip(*[a, b, c])
ax.triplot(x, y)
ax.triplot(negate(x), y)
ax.triplot(negate(x), negate(y))
ax.triplot(x, negate(y))
ax.triplot(y, x)
ax.triplot(negate(y), x)
ax.triplot(negate(y), negate(x))
ax.triplot(y, negate(x))
fig.subplots_adjust(left=0, bottom=0, right=1, top=1, wspace=0, hspace=0)
plt.axis('scaled')
plt.box(False)
plt.set_cmap('rainbow')
plt.show()
if __name__ == '__main__':
plot_hailstone(128)
plot_hailstone(12, 1)
plot_spiral(24, 6)
plot_spiral(24, 6, reverse=True)
plot_lolipop(25, 12)
for i in (2, 3, 5, 8, 13):
plot_sunflower(24, i)
for i in (3, 6, 12, 24):
recur_plot_circle(i, round(i/2))
for i in range(4, 13):
plot_star(i)
for i in range(14, 32, 3):
plot_tetragram(i)
All examples:
How can the script be improved? I am really new to all these stuffs.
