def tilted_sqares():
p1 = gt.Point(0, 0)
p2 = gt.Point(0, 100)
p3 = gt.Point(100, 100)
p4 = gt.Point(100, 0)
rectangle = Structure([p1, p2, p3, p4])
submat = [[0]]
subpoints = [[[[0.8, 0.2, 0, 0], [0, 0.8, 0.2, 0], [0, 0, 0.8, 0.2],
[0.2, 0.0, 0.0, 0.8]]]]
t = gt.Tiling(rectangle, submat, subpoints)
t.draw()
def triangles():
p1 = gt.Point(0, 0)
p2 = gt.Point(100, 0)
p3 = gt.Point(50, 86)
triangle = gt.Structure([p1, p2, p3])
submat = [[0, 0, 0, 0]]
substates = [[1, 0, 0, 0]]
subpoints = [[[[1, 0, 0], [0.5, 0.5, 0], [0.5, 0, 0.5]],
[[0.5, 0.5, 0], [0, 1, 0], [0, 0.5, 0.5]],
[[0.5, 0, 0.5], [0, 0.5, 0.5], [0, 0, 1]],
[[0, 0.5, 0.5], [0.5, 0., 0.5], [0.5, 0.5, 0]]]]
t = gt.Tiling(triangle, submat, subpoints, substates)
t.draw(depth=6,
filename='geometric_tilings/test_triangle_c2.png',
fill=True)
def snowflake():
p1 = gt.Point(0, 0)
p2 = gt.Point(100, 0)
p3 = gt.Point(50, 86)
triangle = gt.Structure([p1, p2, p3])
submat = [[1, 1, 1], [1, 1]]
subpoints = [
[ # substructures of initial triangle:
[[1 / 3, 2 / 3, 0], [2 / 3, 1 / 3, 0], [2 / 3, 2 / 3, -1 / 3]],
[[0, 1 / 3, 2 / 3], [0, 2 / 3, 1 / 3], [-1 / 3, 2 / 3, 2 / 3]],
[[2 / 3, 0, 1 / 3], [1 / 3, 0, 2 / 3], [2 / 3, -1 / 3, 2 / 3]]
],
[[[0, 1 / 3, 2 / 3], [0, 2 / 3, 1 / 3], [-1 / 3, 2 / 3, 2 / 3]],
[[2 / 3, 0, 1 / 3], [1 / 3, 0, 2 / 3], [2 / 3, -1 / 3, 2 / 3]]]
]
t = gt.Tiling(triangle, submat, subpoints)
t.draw(depth=5, filename='geometric_tilings/test_snowflake.png')
def rectangles():
p1 = gt.Point(0, 0)
p2 = gt.Point(0, 100)
p3 = gt.Point(100, 100)
p4 = gt.Point(100, 0)
eps = 0.0
rectangle = gt.Structure([p1, p2, p3, p4])
# 0: rectangle
# 1: triangle
# rectangle is split into four triangles (in the corners) and a rectangle in the center (rotated 45° to the initial rectangle)
# triangle is split into two triangles and a rectangle
submat = [[1, 1, 1, 1, 0], [1, 1, 0]]
sp1 = [0.5 - eps, 0.5 - eps, eps, eps]
sp2 = [eps, 0.5 - eps, 0.5 - eps, eps]
sp3 = [eps, eps, 0.5 - eps, 0.5 - eps]
sp4 = [0.5 - eps, eps, eps, 0.5 - eps]
c1 = [1, 0, 0, 0]
c2 = [0, 1, 0, 0]
c3 = [0, 0, 1, 0]
c4 = [0, 0, 0, 1]
subpoints = [
[ # substructures of rectangle:
[sp1, sp2, c2], # first triangle
[sp2, sp3, c3], # second triangle
[sp3, sp4, c4], # third triangle
[sp4, sp1, c1], # fourth triangle
[sp1, sp2, sp3, sp4] # center rectangle
],
[ # substructers of triangles:
[[1, 0, 0], [0.5, 0.5, 0], [0.5, 0, 0.5]], # first triangle
[[0, 1, 0], [0.5, 0.5, 0], [0, 0.5, 0.5]], # second triangle
[[0, 0, 1], [0, 0.5, 0.5], [0.5, 0.5, 0], [0.5, 0.,
0.5]] # rectangle
]
]
substates = [[0, 0, 0, 0, 1], [0, 0, 1]]
t = gt.Tiling(rectangle, submat, subpoints, substates)
t.draw(depth=7,
fill=True,
filename='geometric_tilings/test_rect_triang_c2_e' + str(eps) +
'.png')
def triangles_2():
p1 = gt.Point(0, 0)
p2 = gt.Point(100, 0)
p3 = gt.Point(50, 86)
eps = 0.02
sp1 = [1, 0, 0]
sp2 = [0.5 - eps, 0.5 - eps, 2 * eps]
sp3 = [0.5 - eps, 2 * eps, 0.5 - eps]
sp4 = [2 * eps, 0.5 - eps, 0.5 - eps]
sp5 = [0, 1, 0]
sp6 = [0, 0, 1]
triangle = gt.Structure([p1, p2, p3])
submat = [[0, 0, 0, 0]]
substates = [[1, 1, 1, 2]]
subpoints = [[[sp1, sp2, sp3], [sp2, sp5, sp4], [sp3, sp4, sp6],
[sp4, sp3, sp2]]]
t = gt.Tiling(triangle, submat, subpoints, substates)
t.draw(depth=7,
filename='geometric_tilings/test_triangles_2_c1_e' + str(eps) +
'.png',
fill=True)
def chair_tiling():
# p6 ------ p5
# I I
# I s8 - s3
# I I I
# s4 - s9 p4 - s2 - p3
# I I I I
# I s5 - s6 - s7 I
# I I I
# p1 ------ s1 ------ p2
p1 = gt.Point(0, 0)
p2 = gt.Point(100, 0)
p3 = gt.Point(100, 50)
p4 = gt.Point(50, 50)
p5 = gt.Point(50, 100)
p6 = gt.Point(0, 100)
chair = gt.Structure([p1, p2, p3, p4, p5, p6])
p1 = [1, 0, 0, 0, 0, 0]
p2 = [0, 1, 0, 0, 0, 0]
p3 = [0, 0, 1, 0, 0, 0]
p4 = [0, 0, 0, 1, 0, 0]
p5 = [0, 0, 0, 0, 1, 0]
p6 = [0, 0, 0, 0, 0, 1]
s1 = [0.5, 0.5, 0, 0, 0, 0]
s2 = [0, 0, 0.5, 0.5, 0, 0]
s3 = [0, 0, 0, 0.5, 0.5, 0]
s4 = [0.5, 0, 0, 0, 0, 0.5]
s5 = [0.5, 0, 0, 0.5, 0, 0]
s6 = [0.5, 0, 0.5, 0, 0, 0]
s7 = [0, 0.5, 0, 0.5, 0, 0]
s8 = [0, 0, 0, 0.5, 0, 0.5]
s9 = [0.5, 0, 0, 0, 0.5, 0]
submat = [[0, 0, 0, 0]]
subpoints = [[[p1, s1, s6, s5, s9, s4], [p2, p3, s2, s7, s6, s1],
[p6, s4, s9, s8, s3, p5], [s5, s7, s2, p4, s3, s8]]]
substates = [[0, 1, 0, 2]]
t = gt.Tiling(chair, submat, subpoints, substates)
t.draw(depth=6,
filename='geometric_tilings/chair_1_c2.png',
only_last_it=True,
fill=True)
def penrose():
# x
# / \
# c d
# x1 __ \
# a / c--x2
# / , \
# / ,b-´ e
# /,-´´ \
# x-------b-------x
#
# (a is the full length of the long sides)
#
#
#
# f ,- x -.
# ,-´ \ `-. f
# ,-´ \ `-.
# x -------f--- x3 --g-- x
#
# (the center diagonal also has length g)
def get_length_big(a):
b = a * (math.sin(math.pi / 5)) / (math.sin(2 * math.pi / 5))
c = a - b
d = c * (math.sin(3 * math.pi / 5)) / (math.sin(math.pi / 5))
e = a - d
return b, c, d, e
def get_h(f):
g = f * (math.sin(math.pi / 5)) / (math.sin(2 * math.pi / 5))
return g
a = 50
# the initial structure: a regular decadon of 10 big triangles centered at 0
p0 = gt.Point(50, 50)
points = [p0]
for i in range(10):
xi = 50 + math.cos(i * math.pi / 5) * a
yi = 50 + math.sin(i * math.pi / 5) * a
points.append(gt.Point(xi, yi))
#print(str(points[-1]))
# struct 1: big Triangle
# struct 2: small triangle
# struct 3: decadon
ratio = (math.sin(math.pi / 5)) / (math.sin(2 * math.pi / 5))
p1 = [1, 0, 0]
p2 = [0, 1, 0]
p3 = [0, 0, 1]
bigT = gt.Structure([points[1], points[2], points[0]])
submat_bigT = [0, 0, 1]
x1 = [1 - ratio, 0, ratio]
x2 = [0, ratio, 1 - ratio]
subpoints_bigT = [[p2, x2, p1], [x1, x2, p1], [p3, x2, x1]]
substates_bigT = [0, 1, 1]
submat_smallT = [0, 1]
x3 = [1 - ratio, ratio, 0]
#subpoints_smallT = [
# [p1, p3, x3],
# [x3, p3, p2]
#]
subpoints_smallT = [[x3, p3, p1], [p2, p3, x3]]
substates_smallT = [0, 0]
decadon = gt.Structure(points[1:], type=2)
submat_decadon = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
c = np.ones(10) * 0.1
p01 = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
p02 = [0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
p03 = [0, 0, 1, 0, 0, 0, 0, 0, 0, 0]
p04 = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
p05 = [0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
p06 = [0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
p07 = [0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
p08 = [0, 0, 0, 0, 0, 0, 0, 1, 0, 0]
p09 = [0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
p10 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
subpoints_decadon = [[p01, p02, c], [p03, p02, c], [p03, p04, c],
[p05, p04, c], [p05, p06, c], [p07, p06, c],
[p07, p08, c], [p09, p08, c], [p09, p10, c],
[p01, p10, c]]
substates_decadon = np.zeros(10)
submat = [submat_bigT, submat_smallT, submat_decadon]
subpoints = [subpoints_bigT, subpoints_smallT, subpoints_decadon]
substates = [substates_bigT, substates_smallT, substates_decadon]
#t = gt.Tiling(decadon, submat, subpoints, substates)
#t.draw(depth=6, only_last_it=True)
for d in [4, 5, 6, 7]:
t = gt.Tiling(decadon, submat, subpoints, substates)
t.draw(depth=d,
only_last_it=True,
filename='geometric_tilings/penrose_' + str(d) + '_c.png',
fill=True)
def rhombs_1():
b = 50
a = b * math.tan(math.pi / 5)
c = (b**2 - a**2) / (2 * a)
d = ((a + c) * a) / (2 * b)
#print (a,b,c,d)
p1 = gt.Point(b, b - a)
p2 = gt.Point(2 * b, b)
p3 = gt.Point(b, b + a)
p4 = gt.Point(0, b)
bigrhomb = gt.Structure([p1, p2, p3, p4], type=0)
eps = 0.1
b_p1 = [1, 0, 0, 0]
b_p2 = [0, 1, 0, 0]
b_p3 = [0, 0, 1, 0]
b_p4 = [0, 0, 0, 1]
b_s1 = [
1 - (a - (a + c) / 2) / (2 * a), (b - d) / (2 * b),
(a - (a + c) / 2) / (2 * a), -(b - d) / (2 * b)
]
b_s2 = [(a + c) / (4 * a), d / (2 * b), 1 - ((a + c) / (4 * a)),
-d / (2 * b)]
b_s3 = [(a + c) / (4 * a), -d / (2 * b), 1 - ((a + c) / (4 * a)),
d / (2 * b)]
b_s4 = [
1 - (a - (a + c) / 2) / (2 * a), -(b - d) / (2 * b),
(a - (a + c) / 2) / (2 * a), (b - d) / (2 * b)
]
b_s5 = [1 - ((a - c) / (2 * a)), 0, (a - c) / (2 * a), 0]
e = (a - c) / 2
f = (b - d) - ((e * b) / a)
smallrhomb = gt.Structure([
gt.Point(0, b - d),
gt.Point(-e, 0),
gt.Point(0, -b + d),
gt.Point(e, 0)
],
type=1)
s_p1 = [1, 0, 0, 0]
s_p2 = [0, 1, 0, 0]
s_p3 = []
s_p4 = [0, 0, 0, 1]
s_s1 = [f / (2 * (b - d)), 1, -f / (2 * (b - d)), 0]
s_s2 = [1 / 3, 0, 2 / 3, 0]
s_s2 = [
1 - ((2 * e * b) / a + f) / (2 * (b - d)), 0,
((2 * e * b) / a + f) / (2 * (b - d)), 0
]
s_s3 = [f / (2 * (b - d)), 0, -f / (2 * (b - d)), 1]
s_s4 = [1 - (e * b) / (a * (b - d)), 0, (e * b) / (a * (b - d)), 0]
submat = [[0, 0, 0, 1], [0, 1, 1]]
subpoints = [[[b_s1, b_p2, b_s2, b_s5], [b_s2, b_p3, b_s3, b_s5],
[b_s3, b_p4, b_s4, b_s5], [b_s4, s_p1, b_s1, b_s5]],
[[s_s3, s_p1, s_s1, s_s4], [s_s1, s_p2, s_s2, s_s4],
[s_s2, s_p4, s_s3, s_s4]]]
substates = [[1, 0, 1, 0], [1, 0, 0]]
#t = gt.Tiling(bigrhomb, submat, subpoints, substates)
#t.draw(depth=3, only_last_it=True)#, filename='geometric_tilings/rhombs_c1.png', fill=True)
# the full star:
base = [p1, p2, p3, p4]
origin = (p2.x, p2.y)
points = [p4, p1]
for i in range(1, 5):
#new_points = [p1]
for j in [3, 0]:
px, py = gt.rotate(origin, (base[j].x, base[j].y),
(2 * i / 5) * math.pi)
points.append(gt.Point(px, py))
c = np.array([1, 0, 1, 0, 1, 0, 1, 0, 1, 0]) * 0.2
p11 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
p13 = [0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
p14 = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
p21 = [0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
p23 = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
p24 = [0, 0, 1, 0, 0, 0, 0, 0, 0, 0]
p31 = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
p33 = [0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
p34 = [0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
p41 = [0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
p43 = [0, 0, 0, 0, 0, 0, 0, 1, 0, 0]
p44 = [0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
p51 = [0, 0, 0, 0, 0, 0, 0, 1, 0, 0]
p53 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
p54 = [0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
#print(len(points))
star = gt.Structure(points, type=2)
submat.append([0, 0, 0, 0, 0])
subpoints.append([
[p11, c, p13, p14],
[p21, c, p23, p24],
[p31, c, p33, p34],
[p41, c, p43, p44],
[p51, c, p53, p54],
])
substates.append([0, 0, 0, 0, 0])
t = gt.Tiling(star, submat, subpoints, substates)
t.draw(depth=7,
only_last_it=True,
filename='geometric_tilings/rhombs_circular.png') #, fill=True)