def construct(self):
words = TextMobject("Order 3 Pseudo-Hilbert Curve")
words.highlight(GREEN)
words.to_edge(UP)
grid4 = Mobject(Grid(2, 2), Grid(4, 4, point_thickness=2))
grid8 = Grid(8, 8, point_thickness=1)
order_3_curve = HilbertCurve(order=3)
mini_curves = [
HilbertCurve(order=2).scale(0.5).shift(1.5 * vect)
for vect in [LEFT + DOWN, LEFT + UP, RIGHT + UP, RIGHT + DOWN]
]
self.add(words, grid4)
self.dither()
self.play(ShowCreation(grid8))
self.dither()
self.play(*map(GrowFromCenter, mini_curves))
self.dither()
self.clear()
self.add(words, grid8, *mini_curves)
self.play(*[
ApplyMethod(curve.rotate_in_place, np.pi, axis)
for curve, axis in [(mini_curves[0],
UP + RIGHT), (mini_curves[3], UP + LEFT)]
])
self.play(ShowCreation(order_3_curve, run_time=5))
self.dither()
def construct(self):
words, s = TextMobject(["Pseudo-Hilbert Curve", "s"]).split()
pre_words = TextMobject("Order 1")
pre_words.next_to(words, LEFT, buff = 0.5)
s.next_to(words, RIGHT, buff = 0.05, aligned_edge = DOWN)
cluster = Mobject(pre_words, words, s)
cluster.center()
cluster.scale(0.7)
cluster.to_edge(UP, buff = 0.3)
cluster.highlight(GREEN)
grid1 = Grid(1, 1)
grid2 = Grid(2, 2)
curve = HilbertCurve(order = 1)
self.add(words, s)
self.dither()
self.play(Transform(
s, pre_words,
path_func = path_along_arc(-np.pi/3)
))
self.dither()
self.play(ShowCreation(grid1))
self.dither()
self.play(ShowCreation(grid2))
self.dither()
kwargs = {
"run_time" : 5,
"rate_func" : None
}
self.play(ShowCreation(curve, **kwargs))
self.dither()
def construct(self):
scale_factor = 0.9
grid = Grid(4, 4, stroke_width=1)
curve = HilbertCurve(order=2)
for mob in grid, curve:
mob.scale(scale_factor)
words = TextMobject("""
Sequence of curves is stable
$\\leftrightarrow$ existence of limit curve
""",
size="\\normal")
words.scale(1.25)
words.to_edge(UP)
self.add(curve, grid)
self.wait()
for n in range(3, 7):
if n == 5:
self.play(ShimmerIn(words))
new_grid = Grid(2**n, 2**n, stroke_width=1)
new_curve = HilbertCurve(order=n)
for mob in new_grid, new_curve:
mob.scale(scale_factor)
self.play(ShowCreation(new_grid), Animation(curve))
self.remove(grid)
grid = new_grid
self.play(Transform(curve, new_curve))
self.wait()
def construct(self):
words = TextMobject("Order 2 Pseudo-Hilbert Curve")
words.to_edge(UP, buff=0.3)
words.highlight(GREEN)
grid2 = Grid(2, 2)
grid4 = Grid(4, 4, point_thickness=2)
# order_1_curve = HilbertCurve(order = 1)
# squaggle_curve = order_1_curve.copy().apply_function(
# lambda (x, y, z) : (x + np.cos(3*y), y + np.sin(3*x), z)
# )
# squaggle_curve.show()
mini_curves = [
HilbertCurve(order=1).scale(0.5).shift(1.5 * vect)
for vect in [LEFT + DOWN, LEFT + UP, RIGHT + UP, RIGHT + DOWN]
]
last_curve = mini_curves[0]
naive_curve = Mobject(last_curve)
for mini_curve in mini_curves[1:]:
line = Line(last_curve.points[-1], mini_curve.points[0])
naive_curve.add(line, mini_curve)
last_curve = mini_curve
naive_curve.ingest_sub_mobjects()
naive_curve.gradient_highlight(RED, GREEN)
order_2_curve = HilbertCurve(order=2)
self.add(words, grid2)
self.dither()
self.play(ShowCreation(grid4))
self.play(*[ShowCreation(mini_curve) for mini_curve in mini_curves])
self.dither()
self.play(ShowCreation(naive_curve, run_time=5))
self.remove(*mini_curves)
self.dither()
self.play(Transform(naive_curve, order_2_curve))
self.dither()
def construct(self):
words = TextMobject([
"One does not simply define the limit \\\\ \
of a sequence of", "curves", "\\dots"
])
top_words = TextMobject(["curves", "are functions"]).to_edge(UP)
curves1 = words.split()[1]
curves2 = top_words.split()[0]
words.ingest_submobjects()
number = TexMobject("0.27")
pair = TexMobject("(0.53, 0.02)")
pair.next_to(number, buff=2)
arrow = Arrow(number, pair)
Mobject(number, arrow, pair).center().shift(UP)
number_line = UnitInterval()
number_line.stretch_to_fit_width(5)
number_line.to_edge(LEFT).shift(DOWN)
grid = Grid(4, 4).scale(0.4)
grid.next_to(number_line, buff=2)
low_arrow = Arrow(number_line, grid)
self.play(ShimmerIn(words))
self.dither()
self.play(FadeOut(words), ApplyMethod(curves1.replace, curves2),
ShimmerIn(top_words.split()[1]))
self.dither()
self.play(FadeIn(number))
self.play(ShowCreation(arrow))
self.play(FadeIn(pair))
self.dither()
self.play(ShowCreation(number_line))
self.play(ShowCreation(low_arrow))
self.play(ShowCreation(grid))
self.dither()
def construct(self):
curve = HilbertCurve(order=1)
grid = Grid(2, 2, stroke_width=1)
self.add(grid, curve)
for order in range(2, 6):
self.dither()
new_grid = Grid(2**order, 2**order, stroke_width=1)
self.play(ShowCreation(new_grid), Animation(curve))
self.remove(grid)
grid = new_grid
self.play(Transform(curve, HilbertCurve(order=order)))
square = Square(side_length=6, color=WHITE)
square.corner = Mobject1D()
square.corner.add_line(3 * DOWN, ORIGIN)
square.corner.add_line(ORIGIN, 3 * RIGHT)
square.digest_mobject_attrs()
square.scale(2**(-5))
square.corner.highlight(
Color(rgb=curve.rgbs[curve.get_num_points() / 3]))
square.shift(
grid.get_corner(UP+LEFT)-\
square.get_corner(UP+LEFT)
)
self.dither()
self.play(FadeOut(grid), FadeOut(curve), FadeIn(square))
self.play(ApplyMethod(square.replace, grid))
self.dither()
def construct(self, order):
if order == 2:
result_tex = "(0.125, 0.75)"
elif order == 3:
result_tex = "(0.0758, 0.6875)"
phc, arg, result = TexMobject([
"\\text{PHC}_%d"%order,
"(0.3)",
"= %s"%result_tex
]).to_edge(UP).split()
function = TextMobject("Function", size = "\\normal")
function.shift(phc.get_center()+DOWN+2*LEFT)
function_arrow = Arrow(function, phc)
line = Line(5*LEFT, 5*RIGHT)
curve = HilbertCurve(order = order)
line.match_colors(curve)
grid = Grid(2**order, 2**order)
grid.fade()
for mob in curve, grid:
mob.scale(0.7)
index = int(0.3*line.get_num_points())
dot1 = Dot(line.points[index])
arrow1 = Arrow(arg, dot1, buff = 0.1)
dot2 = Dot(curve.points[index])
arrow2 = Arrow(result.get_bottom(), dot2, buff = 0.1)
self.add(phc)
self.play(
ShimmerIn(function),
ShowCreation(function_arrow)
)
self.wait()
self.remove(function_arrow, function)
self.play(ShowCreation(line))
self.wait()
self.play(
ShimmerIn(arg),
ShowCreation(arrow1),
ShowCreation(dot1)
)
self.wait()
self.remove(arrow1)
self.play(
FadeIn(grid),
Transform(line, curve),
Transform(dot1, dot2),
run_time = 2
)
self.wait()
self.play(
ShimmerIn(result),
ShowCreation(arrow2)
)
self.wait()
def construct(self, order):
if order == 2:
result_tex = "(0.125, 0.75)"
elif order == 3:
result_tex = "(0.0758, 0.6875)"
phc, arg, result = TexMobject([
"\\text{PHC}_%d"%order,
"(0.3)",
"= %s"%result_tex
]).to_edge(UP).split()
function = TextMobject("Function", size = "\\normal")
function.shift(phc.get_center()+DOWN+2*LEFT)
function_arrow = Arrow(function, phc)
line = Line(5*LEFT, 5*RIGHT)
curve = HilbertCurve(order = order)
line.match_colors(curve)
grid = Grid(2**order, 2**order)
grid.fade()
for mob in curve, grid:
mob.scale(0.7)
index = int(0.3*line.get_num_points())
dot1 = Dot(line.points[index])
arrow1 = Arrow(arg, dot1, buff = 0.1)
dot2 = Dot(curve.points[index])
arrow2 = Arrow(result.get_bottom(), dot2, buff = 0.1)
self.add(phc)
self.play(
ShimmerIn(function),
ShowCreation(function_arrow)
)
self.dither()
self.remove(function_arrow, function)
self.play(ShowCreation(line))
self.dither()
self.play(
ShimmerIn(arg),
ShowCreation(arrow1),
ShowCreation(dot1)
)
self.dither()
self.remove(arrow1)
self.play(
FadeIn(grid),
Transform(line, curve),
Transform(dot1, dot2),
run_time = 2
)
self.dither()
self.play(
ShimmerIn(result),
ShowCreation(arrow2)
)
self.dither()
def construct(self):
big_grid_dim = 20.
small_grid_dim = 6.
big_grid = Grid(64, 64, height=big_grid_dim, width=big_grid_dim)
big_grid.to_corner(UP + RIGHT, buff=2)
small_grid = big_grid.copy()
small_grid.scale(small_grid_dim / big_grid_dim)
small_grid.center()
pixel = MobjectFromRegion(
region_from_polygon_vertices(
*0.2 *
np.array([RIGHT + DOWN, RIGHT + UP, LEFT + UP, LEFT + DOWN])))
pixel.set_color(WHITE)
pixel_width = big_grid.width / big_grid.columns
pixel.scale_to_fit_width(pixel_width)
pixel.to_corner(UP + RIGHT, buff=2)
pixel.shift(5 * pixel_width * (2 * LEFT + DOWN))
freq_line = get_freq_line()
dot = Dot()
dot.shift(freq_line.get_center() + 2 * RIGHT)
string = Line(LEFT, RIGHT, color=GREEN)
arrow = Arrow(dot, string.get_center())
vibration_config = {
"overtones": 1,
"spatial_period": 2,
}
vibration, loud_vibration, quiet_vibration = [
Vibrate(string.copy(), amplitude=a, **vibration_config)
for a in [0.5, 1., 0.25]
]
self.add(small_grid)
self.dither()
self.play(Transform(small_grid, big_grid))
self.play(FadeIn(pixel))
self.dither()
self.play(FadeOut(small_grid), ShowCreation(freq_line))
self.remove(small_grid)
self.play(Transform(pixel, dot), )
self.dither()
self.play(ShowCreation(arrow))
self.play(loud_vibration)
self.play(TransformAnimations(loud_vibration, quiet_vibration),
ApplyMethod(dot.fade, 0.9))
self.clear()
self.add(freq_line, dot, arrow)
self.play(quiet_vibration)
def construct(self):
grids = [
Grid(
2**order, 2**order,
stroke_width = 1
).shift(0.3*DOWN)
for order in 6, 7
]
grid = grids[0]
side_brace = Brace(grid, LEFT)
top_brace = Brace(grid, UP)
top_words = TextMobject("256")
new_top_words = TextMobject("512")
side_words = top_words.copy()
new_side_words = new_top_words.copy()
for words in top_words, new_top_words:
words.next_to(top_brace, UP, buff = 0.1)
for words in side_words, new_side_words:
words.next_to(side_brace, LEFT)
self.add(grid)
self.play(
GrowFromCenter(side_brace),
GrowFromCenter(top_brace),
ShimmerIn(top_words),
ShimmerIn(side_words)
)
self.dither()
self.play(
DelayByOrder(Transform(*grids)),
Transform(top_words, new_top_words),
Transform(side_words, new_side_words)
)
self.dither()
def construct(self):
curve = HilbertCurve(order = 1)
grid = Grid(2, 2, point_thickness=1)
self.add(grid, curve)
for order in range(2, 6):
self.dither()
new_grid = Grid(2**order, 2**order, point_thickness=1)
self.play(
ShowCreation(new_grid),
Animation(curve)
)
self.remove(grid)
grid = new_grid
self.play(Transform(
curve, HilbertCurve(order = order)
))
square = Square(side_length = 6, color = WHITE)
square.corner = Mobject1D()
square.corner.add_line(3*DOWN, ORIGIN)
square.corner.add_line(ORIGIN, 3*RIGHT)
square.digest_mobject_attrs()
square.scale(2**(-5))
square.corner.highlight(
Color(rgb = curve.rgbs[curve.get_num_points()/3])
)
square.shift(
grid.get_corner(UP+LEFT)-\
square.get_corner(UP+LEFT)
)
self.dither()
self.play(
FadeOut(grid),
FadeOut(curve),
FadeIn(square)
)
self.play(
ApplyMethod(square.replace, grid)
)
self.dither()
def construct(self):
grid = Grid(16, 16).fade()
snake_curve = SnakeCurve(order=4)
words = TextMobject("``Snake Curve''")
words.next_to(grid, UP)
self.add(grid)
self.play(ShowCreation(snake_curve, run_time=7, rate_func=None))
self.dither()
self.play(ShimmerIn(words))
self.dither()
def construct(self):
words = TextMobject(
["One does not simply define the limit \\\\ \
of a sequence of","curves","\\dots"]
)
top_words = TextMobject([
"curves", "are functions"
]).to_edge(UP)
curves1 = words.split()[1]
curves2 = top_words.split()[0]
words.ingest_sub_mobjects()
number = TexMobject("0.27")
pair = TexMobject("(0.53, 0.02)")
pair.next_to(number, buff = 2)
arrow = Arrow(number, pair)
Mobject(number, arrow, pair).center().shift(UP)
number_line = UnitInterval()
number_line.stretch_to_fit_width(5)
number_line.to_edge(LEFT).shift(DOWN)
grid = Grid(4, 4).scale(0.4)
grid.next_to(number_line, buff = 2)
low_arrow = Arrow(number_line, grid)
self.play(ShimmerIn(words))
self.dither()
self.play(
FadeOut(words),
ApplyMethod(curves1.replace, curves2),
ShimmerIn(top_words.split()[1])
)
self.dither()
self.play(FadeIn(number))
self.play(ShowCreation(arrow))
self.play(FadeIn(pair))
self.dither()
self.play(ShowCreation(number_line))
self.play(ShowCreation(low_arrow))
self.play(ShowCreation(grid))
self.dither()
def construct(self):
grid = Grid(64, 64)
space_region = Region()
space_mobject = MobjectFromRegion(space_region, DARK_GREY)
curve = PeanoCurve(order = 5).replace(space_mobject)
line = Line(5*LEFT, 5*RIGHT)
line.gradient_highlight(curve.start_color, curve.end_color)
for mob in grid, space_mobject:
mob.sort_points(np.linalg.norm)
infinitely = TextMobject("Infinitely")
detailed = TextMobject("detailed")
extending = TextMobject("extending")
detailed.next_to(infinitely, RIGHT)
extending.next_to(infinitely, RIGHT)
Mobject(extending, infinitely, detailed).center()
arrows = Mobject(*[
Arrow(2*p, 4*p)
for theta in np.arange(np.pi/6, 2*np.pi, np.pi/3)
for p in [rotate_vector(RIGHT, theta)]
])
self.add(grid)
self.wait()
self.play(Transform(grid, space_mobject, run_time = 5))
self.remove(grid)
self.highlight_region(space_region, DARK_GREY)
self.wait()
self.add(infinitely, detailed)
self.wait()
self.play(DelayByOrder(Transform(detailed, extending)))
self.play(ShowCreation(arrows))
self.wait()
self.clear()
self.highlight_region(space_region, DARK_GREY)
self.play(ShowCreation(line))
self.play(Transform(line, curve, run_time = 5))
def get_grid():
return Grid(64, 64)
def construct(self):
big_grid_dim = 20.
small_grid_dim = 6.
big_grid = Grid(64, 64, height = big_grid_dim, width = big_grid_dim)
big_grid.to_corner(UP+RIGHT, buff = 2)
small_grid = big_grid.copy()
small_grid.scale(small_grid_dim/big_grid_dim)
small_grid.center()
pixel = MobjectFromRegion(
region_from_polygon_vertices(*0.2*np.array([
RIGHT+DOWN,
RIGHT+UP,
LEFT+UP,
LEFT+DOWN
]))
)
pixel.set_color(WHITE)
pixel_width = big_grid.width/big_grid.columns
pixel.scale_to_fit_width(pixel_width)
pixel.to_corner(UP+RIGHT, buff = 2)
pixel.shift(5*pixel_width*(2*LEFT+DOWN))
freq_line = get_freq_line()
dot = Dot()
dot.shift(freq_line.get_center() + 2*RIGHT)
string = Line(LEFT, RIGHT, color = GREEN)
arrow = Arrow(dot, string.get_center())
vibration_config = {
"overtones" : 1,
"spatial_period" : 2,
}
vibration, loud_vibration, quiet_vibration = [
Vibrate(string.copy(), amplitude = a, **vibration_config)
for a in [0.5, 1., 0.25]
]
self.add(small_grid)
self.dither()
self.play(
Transform(small_grid, big_grid)
)
self.play(FadeIn(pixel))
self.dither()
self.play(
FadeOut(small_grid),
ShowCreation(freq_line)
)
self.remove(small_grid)
self.play(
Transform(pixel, dot),
)
self.dither()
self.play(ShowCreation(arrow))
self.play(loud_vibration)
self.play(
TransformAnimations(loud_vibration, quiet_vibration),
ApplyMethod(dot.fade, 0.9)
)
self.clear()
self.add(freq_line, dot, arrow)
self.play(quiet_vibration)
