def construct(self):
curve = HilbertCurve(order = 6)
curve.highlight(WHITE)
colored_curve = curve.copy()
colored_curve.thin_out(3)
lion = ImageMobject("lion", invert = False)
lion.replace(curve, stretch = True)
sparce_lion = lion.copy()
sparce_lion.thin_out(100)
distance_matrix = cdist(colored_curve.points, sparce_lion.points)
closest_point_indices = np.apply_along_axis(
np.argmin, 1, distance_matrix
)
colored_curve.rgbs = sparce_lion.rgbs[closest_point_indices]
line = Line(5*LEFT, 5*RIGHT)
Mobject.align_data(line, colored_curve)
line.rgbs = colored_curve.rgbs
self.add(lion)
self.play(ShowCreation(curve, run_time = 3))
self.play(
FadeOut(lion),
Transform(curve, colored_curve),
run_time = 3
)
self.dither()
self.play(Transform(curve, line, run_time = 5))
self.dither()
def __init__(self, **kwargs):
x_axis = NumberLine(**kwargs)
y_axis = NumberLine(**kwargs).rotate(np.pi/2, OUT)
Mobject.__init__(self, x_axis, y_axis)
def play(self, *animations, **kwargs):
self.num_animations += 1
if "run_time" in kwargs:
run_time = kwargs["run_time"]
else:
run_time = animations[0].run_time
for animation in animations:
animation.set_run_time(run_time)
moving_mobjects = [mobject for anim in animations for mobject in anim.mobject.submobject_family()]
bundle = Mobject(*self.mobjects)
static_mobjects = filter(lambda m: m not in moving_mobjects, bundle.submobject_family())
background = disp.paint_mobjects(static_mobjects, self.background, include_sub_mobjects=False)
times = np.arange(0, run_time, self.frame_duration)
time_progression = ProgressDisplay(times)
time_progression.set_description(
"Animation %d: " % self.num_animations + str(animations[0]) + (", etc." if len(animations) > 1 else "")
)
for t in time_progression:
for animation in animations:
animation.update(t / animation.run_time)
new_frame = disp.paint_mobjects([anim.mobject for anim in animations], background)
self.frames.append(new_frame)
for animation in animations:
animation.clean_up()
self.add(*[anim.mobject for anim in animations])
self.repaint_mojects()
return self
def snells_law_at_every_point(self, cycloid, chopped_cycloid):
square = Square(side_length = 0.2, color = WHITE)
words = TextMobject(["Snell's law ", "everywhere"])
snells, rest = words.split()
colon = TextMobject(":")
words.next_to(square)
words.shift(0.3*UP)
combo = Mobject(square, words)
combo.get_center = lambda : square.get_center()
new_snells = snells.copy().center().to_edge(UP, buff = 1.5)
colon.next_to(new_snells)
colon.shift(0.05*DOWN)
self.play(MoveAlongPath(
combo, cycloid,
run_time = 5
))
self.play(MoveAlongPath(
combo, chopped_cycloid,
run_time = 4
))
dot = Dot(combo.get_center())
self.play(Transform(square, dot))
self.play(
Transform(snells, new_snells),
Transform(rest, colon)
)
self.dither()
return colon
class Flash(Animation):
CONFIG = {
"color" : "white",
"slow_factor" : 0.01,
"run_time" : 0.1,
"rate_func" : None,
}
def __init__(self, mobject, **kwargs):
self.intermediate = Mobject(color = self.color)
self.intermediate.add_points([
point + (x, y, 0)
for point in self.mobject.points
for x in [-1, 1]
for y in [-1, 1]
])
Animation.__init__(self, mobject, **kwargs)
def update_mobject(self, alpha):
#Makes alpha go from 0 to slow_factor to 0 instead of 0 to 1
alpha = self.slow_factor * (1.0 - 4 * (alpha - 0.5)**2)
self.mobject.interpolate(
self.starting_mobject,
self.intermediate,
alpha
)
def __init__(self, *args, **kwargs):
Mobject.__init__(self, *args, **kwargs)
complex_power = 0.9
radius = self.INITIAL_WIDTH/2
circle = Circle(density = radius*DEFAULT_POINT_DENSITY_1D)
circle.apply_complex_function(lambda z : z**complex_power)
circle.scale(radius)
boundary_point_as_complex = radius*complex(-1)**complex_power
boundary_points = [
[
boundary_point_as_complex.real,
unit*boundary_point_as_complex.imag,
0
]
for unit in -1, 1
]
tip = radius*(1.5*LEFT+UP)
self.add(
circle,
Line(boundary_points[0], tip),
Line(boundary_points[1], tip)
)
self.highlight("white")
self.rotate(np.pi/2)
self.points[:,1] *= float(self.INITIAL_HEIGHT)/self.INITIAL_WIDTH
Bubble.__init__(self, direction = LEFT)
def get_circles_and_points(self, min_input, max_input):
input_left, input_right = [
self.interval.number_to_point(num)
for num in min_input, max_input
]
input_circle = Circle(
radius = np.linalg.norm(input_left-input_right)/2,
color = WHITE
)
input_circle.shift((input_left+input_right)/2)
input_points = Line(
input_left, input_right,
color = self.input_color
)
output_points = Mobject(color = self.output_color)
n = self.output.get_num_points()
output_points.add_points(
self.output.points[int(min_input*n):int(max_input*n)]
)
output_center = output_points.points[int(0.5*output_points.get_num_points())]
max_distance = np.linalg.norm(output_center-output_points.points[-1])
output_circle = Circle(
radius = max_distance,
color = WHITE
)
output_circle.shift(output_center)
return (
input_circle,
input_points,
output_circle,
output_points
)
def show_sin_thetas(self):
pc = Line(self.p_point, self.c_point)
mob = Mobject(self.theta, self.d_mob).copy()
mob.ingest_submobjects()
triplets = [
(pc, "D\\sin(\\theta)", 0.5),
(self.y_line, "D\\sin^2(\\theta)", 0.7),
]
for line, tex, scale in triplets:
trig_mob = TexMobject(tex)
trig_mob.scale_to_fit_width(
scale*line.get_length()
)
trig_mob.shift(-1.2*trig_mob.get_top())
trig_mob.rotate(line.get_angle())
trig_mob.shift(line.get_center())
if line is self.y_line:
trig_mob.shift(0.1*UP)
self.play(Transform(mob, trig_mob))
self.add(trig_mob)
self.dither()
self.remove(mob)
self.d_sin_squared_theta = trig_mob
def setup(self):
self.input_color = YELLOW_C
self.output_color = RED
def spiril(t):
theta = 2*np.pi*t
return t*np.cos(theta)*RIGHT+t*np.sin(theta)*UP
self.spiril1 = ParametricFunction(
lambda t : 1.5*RIGHT + DOWN + 2*spiril(t),
density = 5*DEFAULT_POINT_DENSITY_1D,
)
self.spiril2 = ParametricFunction(
lambda t : 5.5*RIGHT + UP - 2*spiril(1-t),
density = 5*DEFAULT_POINT_DENSITY_1D,
)
Mobject.align_data(self.spiril1, self.spiril2)
self.output = Mobject(self.spiril1, self.spiril2)
self.output.ingest_sub_mobjects()
self.output.highlight(GREEN_A)
self.interval = UnitInterval()
self.interval.scale_to_fit_width(SPACE_WIDTH-1)
self.interval.to_edge(LEFT)
self.input_dot = Dot(color = self.input_color)
self.output_dot = self.input_dot.copy().highlight(self.output_color)
left, right = self.interval.get_left(), self.interval.get_right()
self.input_homotopy = lambda (x, y, z, t) : (x, y, t) + interpolate(left, right, t)
output_size = self.output.get_num_points()-1
output_points = self.output.points
self.output_homotopy = lambda (x, y, z, t) : (x, y, z) + output_points[int(t*output_size)]
def construct(self):
coords_set = [ORIGIN]
for n in range(int(2*SPACE_WIDTH)):
for vect in UP, RIGHT:
for k in range(n):
new_coords = coords_set[-1]+((-1)**n)*vect
coords_set.append(new_coords)
square = Square(side_length = 1, color = WHITE)
squares = Mobject(*[
square.copy().shift(coords)
for coords in coords_set
]).ingest_sub_mobjects()
self.play(
DelayByOrder(FadeIn(squares)),
run_time = 3
)
curve = HilbertCurve(order = 6).scale(1./6)
all_curves = Mobject(*[
curve.copy().shift(coords)
for coords in coords_set
]).ingest_sub_mobjects()
all_curves.thin_out(10)
self.play(ShowCreation(
all_curves,
rate_func = None,
run_time = 15
))
def __init__(self, **kwargs):
Mobject.__init__(self, **kwargs)
for part_name in self.PART_NAMES:
mob = ImageMobject(
part_name_to_directory(part_name),
should_center = False
)
if part_name not in self.WHITE_PART_NAMES:
mob.highlight(self.color)
setattr(self, part_name, mob)
self.eyes = Mobject(self.left_eye, self.right_eye)
self.legs = Mobject(self.left_leg, self.right_leg)
mouth_center = self.get_mouth_center()
self.mouth.center()
self.smile = self.mouth
self.frown = self.mouth.copy().rotate(np.pi, RIGHT)
self.straight_mouth = TexMobject("-").scale(0.7)
for mouth in self.smile, self.frown, self.straight_mouth:
mouth.sort_points(lambda p : p[0])
mouth.highlight(self.color) ##to blend into background
mouth.shift(mouth_center)
self.digest_mobject_attrs()
self.give_smile()
self.add(self.mouth)
self.scale(PI_CREATURE_SCALE_VAL)
def __init__(self, mobject1, mobject2,
run_time = DEFAULT_TRANSFORM_RUN_TIME,
interpolation_function = straight_path,
black_out_extra_points = False,
*args, **kwargs):
self.interpolation_function = interpolation_function
count1, count2 = mobject1.get_num_points(), mobject2.get_num_points()
if count2 == 0:
mobject2 = Point((SPACE_WIDTH, SPACE_HEIGHT, 0))
count2 = mobject2.get_num_points()
Mobject.align_data(mobject1, mobject2)
Animation.__init__(self, mobject1, run_time = run_time, *args, **kwargs)
self.ending_mobject = mobject2
self.mobject.SHOULD_BUFF_POINTS = \
mobject1.SHOULD_BUFF_POINTS and mobject2.SHOULD_BUFF_POINTS
self.reference_mobjects.append(mobject2)
self.name += "To" + str(mobject2)
if black_out_extra_points and count2 < count1:
#Ensure redundant pixels fade to black
indices = np.arange(
0, count1-1, float(count1) / count2
).astype('int')
temp = np.zeros(mobject2.points.shape)
temp[indices] = mobject2.rgbs[indices]
mobject2.rgbs = temp
self.non_redundant_m2_indices = indices
def __init__(self, function, **kwargs):
self.function = function
if self.density:
self.epsilon = 1.0 / self.density
elif self.dim == 1:
self.epsilon = 1.0 / self.expected_measure / DEFAULT_POINT_DENSITY_1D
else:
self.epsilon = 1.0 / np.sqrt(self.expected_measure) / DEFAULT_POINT_DENSITY_2D
Mobject.__init__(self, **kwargs)
def __init__(self, expression, **kwargs):
if "size" not in kwargs:
#Todo, make this more sophisticated.
if len("".join(expression)) < MAX_LEN_FOR_HUGE_TEX_FONT:
size = "\\Huge"
else:
size = "\\large"
digest_locals(self)
Mobject.__init__(self, **kwargs)
def __init__(self, condition = (lambda x, y : True), **kwargs):
"""
Condition must be a function which takes in two real
arrays (representing x and y values of space respectively)
and return a boolean array. This can essentially look like
a function from R^2 to {True, False}, but & and | must be
used in place of "and" and "or"
"""
Mobject.__init__(self, **kwargs)
self.condition = condition
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 __init__(self, image_file, **kwargs):
digest_locals(self)
Mobject.__init__(self, **kwargs)
self.name = to_camel_case(
os.path.split(image_file)[-1].split(".")[0]
)
path = get_full_image_path(image_file)
self.generate_points_from_file(path)
self.scale(self.scale_factorue)
if self.should_center:
self.center()
def play(self, *args, **kwargs):
if not self.has_started:
self.has_started = True
everything = Mobject(*self.mobjects)
vect = 2*SPACE_WIDTH*RIGHT
everything.shift(vect)
self.play(ApplyMethod(
everything.shift, -vect,
rate_func = rush_from
))
Scene.play(self, *args, **kwargs)
def separate_moving_and_static_mobjects(self, *animations):
moving_mobjects = list(it.chain(*[
anim.mobject.submobject_family()
for anim in animations
]))
bundle = Mobject(*self.mobjects)
static_mobjects = filter(
lambda m : m not in moving_mobjects,
bundle.submobject_family()
)
return moving_mobjects, static_mobjects
def __init__(self, **kwargs):
digest_config(self, kwargs, locals())
if self.file_name is None:
raise Exception("Must invoke Bubble subclass")
SVGMobject.__init__(self, **kwargs)
self.center()
self.stretch_to_fit_height(self.height)
self.stretch_to_fit_width(self.width)
if self.direction[0] > 0:
Mobject.flip(self)
self.direction_was_specified = ("direction" in kwargs)
self.content = Mobject()
def solve_energy(self):
loss_in_potential = TextMobject("Loss in potential: ")
loss_in_potential.shift(2*UP)
potential = TexMobject("m g y".split())
potential.next_to(loss_in_potential)
kinetic = TexMobject([
"\\dfrac{1}{2}","m","v","^2","="
])
kinetic.next_to(potential, LEFT)
nudge = 0.1*UP
kinetic.shift(nudge)
loss_in_potential.shift(nudge)
ms = Mobject(kinetic.split()[1], potential.split()[0])
two = TexMobject("2")
two.shift(ms.split()[1].get_center())
half = kinetic.split()[0]
sqrt = TexMobject("\\sqrt{\\phantom{2mg}}")
sqrt.shift(potential.get_center())
nudge = 0.2*LEFT
sqrt.shift(nudge)
squared = kinetic.split()[3]
equals = kinetic.split()[-1]
new_eq = equals.copy().next_to(kinetic.split()[2])
self.play(
Transform(
Point(loss_in_potential.get_left()),
loss_in_potential
),
*map(GrowFromCenter, potential.split())
)
self.dither(2)
self.play(
FadeOut(loss_in_potential),
GrowFromCenter(kinetic)
)
self.dither(2)
self.play(ApplyMethod(ms.shift, 5*UP))
self.dither()
self.play(Transform(
half, two,
path_func = counterclockwise_path()
))
self.dither()
self.play(
Transform(
squared, sqrt,
path_func = clockwise_path()
),
Transform(equals, new_eq)
)
self.dither(2)
def get_triangles(self):
triangle = Polygon(
LEFT/np.sqrt(3),
UP,
RIGHT/np.sqrt(3),
color = GREEN
)
triangles = Mobject(
triangle.copy().scale(0.5).shift(LEFT),
triangle,
triangle.copy().scale(0.3).shift(0.5*UP+RIGHT)
)
triangles.center()
return triangles
def __init__(self, mobject, ending_mobject, **kwargs):
mobject, ending_mobject = map(instantiate, [mobject, ending_mobject])
digest_config(self, kwargs, locals())
count1, count2 = mobject.get_num_points(), ending_mobject.get_num_points()
if count2 == 0:
ending_mobject.add_points([SPACE_WIDTH*RIGHT+SPACE_HEIGHT*UP])
count2 = ending_mobject.get_num_points()
Mobject.align_data(mobject, ending_mobject)
if self.should_black_out_extra_points and count2 < count1:
self.black_out_extra_points(count1, count2)
Animation.__init__(self, mobject, **kwargs)
self.name += "To" + str(ending_mobject)
self.mobject.point_thickness = ending_mobject.point_thickness
def __init__(self, *args, **kwargs):
Mobject.__init__(self, *args, **kwargs)
self.add(Circle().scale(0.15).shift(2.5*DOWN+2*LEFT))
self.add(Circle().scale(0.3).shift(2*DOWN+1.5*LEFT))
for n in range(self.NUM_BULGES):
theta = 2*np.pi*n/self.NUM_BULGES
self.add(Circle().shift((np.cos(theta), np.sin(theta), 0)))
self.filter_out(lambda p : np.linalg.norm(p) < self.INITIAL_INNER_RADIUS)
self.stretch_to_fit_width(self.INITIAL_WIDTH)
self.highlight("white")
Bubble.__init__(
self,
index_of_tip = np.argmin(self.points[:,1]),
**kwargs
)
def get_edge_mobject(self, image_array):
edged_image = get_edges(image_array)
individual_edges = get_connected_components(edged_image)
colored_edges = [
color_region(edge, image_array)
for edge in individual_edges
]
colored_edge_mobject_list = [
MobjectFromPixelArray(colored_edge)
for colored_edge in colored_edges
]
random.shuffle(colored_edge_mobject_list)
edge_mobject = Mobject(*colored_edge_mobject_list)
edge_mobject.ingest_submobjects()
return edge_mobject
def slide_system(self, ring):
equilibrium_slide_kwargs = dict(self.slide_kwargs)
def jiggle_to_equilibrium(t):
return 0.7*(1+((1-t)**2)*(-np.cos(10*np.pi*t)))
equilibrium_slide_kwargs = {
"rate_func" : jiggle_to_equilibrium,
"run_time" : 3
}
start = Mobject(ring, self.start_springs)
end = Mobject(
ring.copy().shift(self.ring_shift_val),
self.end_springs
)
for kwargs in self.slide_kwargs, equilibrium_slide_kwargs:
self.play(Transform(start, end, **kwargs))
self.dither()
def generate_points(self):
phi = (1 + np.sqrt(5)) / 2
x = np.array([1, 0, 0])
y = np.array([0, 1, 0])
z = np.array([0, 0, 1])
v1, v2 = (phi, 1/phi, 0), (phi, -1/phi, 0)
vertex_pairs = [
(v1, v2),
(x+y+z, v1),
(x+y-z, v1),
(x-y+z, v2),
(x-y-z, v2),
]
five_lines_points = Mobject(*[
Line(pair[0], pair[1], density = 1.0/self.epsilon)
for pair in vertex_pairs
]).points
#Rotate those 5 edges into all 30.
for i in range(3):
perm = map(lambda j : j%3, range(i, i+3))
for b in [-1, 1]:
matrix = b*np.array([x[perm], y[perm], z[perm]])
self.add_points(np.dot(five_lines_points, matrix))
self.pose_at_angle()
self.set_color(GREEN)
def show_equation(self, chopped_cycloid, ref_mob):
point2, point1 = chopped_cycloid.points[-2:]
arc, theta, vert_line, tangent_line = self.get_marks(point1, point2)
equation = TexMobject([
"\\sin(\\theta)",
"\\over \\sqrt{y}",
])
sin, sqrt_y = equation.split()
equation.next_to(ref_mob)
const = TexMobject(" = \\text{constant}")
const.next_to(equation)
ceil_point = np.array(point1)
ceil_point[1] = self.top[1]
brace = Brace(Mobject(Point(point1), Point(ceil_point)), RIGHT)
y_mob = TexMobject("y").next_to(brace)
self.play(GrowFromCenter(sin), ShowCreation(arc),
GrowFromCenter(theta))
self.play(ShowCreation(vert_line))
self.play(ShowCreation(tangent_line))
self.dither()
self.play(GrowFromCenter(sqrt_y), GrowFromCenter(brace),
GrowFromCenter(y_mob))
self.dither()
self.play(Transform(Point(const.get_left()), const))
self.dither()
def finite_analog(self, left_mob, arrow, right_mob):
self.clear()
self.add(left_mob, arrow, right_mob)
ex = TextMobject("\\times")
ex.highlight(RED)
# ex.shift(arrow.get_center())
middle = TexMobject("\\sum_{n=0}^N 2^n \\equiv -1 \\mod 2^{N+1}")
finite_analog = TextMobject("Finite analog")
finite_analog.scale(0.8)
brace = Brace(middle, UP)
finite_analog.next_to(brace, UP)
new_left = left_mob.copy().to_edge(LEFT)
new_right = right_mob.copy().to_edge(RIGHT)
left_arrow, right_arrow = [
Arrow(mob1.get_right()[0] * RIGHT,
mob2.get_left()[0] * RIGHT,
buff=0)
for mob1, mob2 in [(new_left, middle), (middle, new_right)]
]
for mob in ex, middle:
mob.sort_points(np.linalg.norm)
self.play(GrowFromCenter(ex))
self.dither()
self.play(Transform(left_mob, new_left),
Transform(arrow.copy(), left_arrow),
DelayByOrder(Transform(ex, middle)),
Transform(arrow, right_arrow),
Transform(right_mob, new_right))
self.play(GrowFromCenter(brace), ShimmerIn(finite_analog))
self.dither()
self.equivalence(left_mob, left_arrow,
Mobject(middle, brace, finite_analog))
def __init__(self, **kwargs):
digest_config(self, kwargs, locals())
if self.file_name is None:
raise Exception("Must invoke Bubble subclass")
try:
SVGMobject.__init__(self, **kwargs)
except IOError as err:
self.file_name = os.path.join(FILE_DIR, self.file_name)
SVGMobject.__init__(self, **kwargs)
self.center()
self.stretch_to_fit_height(self.height)
self.stretch_to_fit_width(self.width)
if self.direction[0] > 0:
Mobject.flip(self)
self.direction_was_specified = ("direction" in kwargs)
self.content = Mobject()
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):
words = TextMobject([
"Fermat's principle:", """
If a beam of light travels
from point $A$ to $B$, it does so along the
fastest path possible.
"""
])
words.split()[0].highlight(BLUE)
everything = MobjectFromRegion(Region())
everything.scale(0.9)
angles = np.apply_along_axis(angle_of_vector, 1, everything.points)
norms = np.apply_along_axis(np.linalg.norm, 1, everything.points)
norms -= np.min(norms)
norms /= np.max(norms)
alphas = 0.25 + 0.75 * norms * (1 + np.sin(12 * angles)) / 2
everything.rgbas = alphas.repeat(3).reshape((len(alphas), 3))
Mobject(everything, words).show()
everything.sort_points(np.linalg.norm)
self.add(words)
self.play(DelayByOrder(FadeIn(everything, run_time=3)),
Animation(words))
self.play(ApplyMethod(everything.highlight, WHITE), )
self.dither()
def construct(self):
definition = TexMobject([
"\\text{HC}(", "x", ")", "=\\lim_{n \\to \\infty}\\text{PHC}_n(",
"x", ")"
])
definition.to_edge(UP)
definition.split()[1].highlight(BLUE)
definition.split()[-2].highlight(BLUE)
intro = TextMobject("Three things need to be proven")
prove_that = TextMobject("Prove that HC is $\\dots$")
prove_that.scale(0.7)
prove_that.to_edge(LEFT)
items = TextMobject([
"\\begin{enumerate}",
"\\item Well-defined: ",
"Points on Pseudo-Hilbert-curves really do converge",
"\\item A Curve: ",
"HC is continuous",
"\\item Space-filling: ",
"Each point in the unit square is an output of HC",
"\\end{enumerate}",
]).split()
items[1].highlight(GREEN)
items[3].highlight(YELLOW_C)
items[5].highlight(MAROON)
Mobject(*items).to_edge(RIGHT)
self.add(definition)
self.play(ShimmerIn(intro))
self.dither()
self.play(Transform(intro, prove_that))
for item in items[1:-1]:
self.play(ShimmerIn(item))
self.dither()
def show_geometry(self, slider, vector):
point_a = self.point_a.get_center()
horiz_line = Line(point_a, point_a + 6 * RIGHT)
ceil_point = point_a
ceil_point[0] = slider.get_center()[0]
vert_brace = Brace(Mobject(Point(ceil_point),
Point(slider.get_center())),
RIGHT,
buff=0.5)
vect_brace = Brace(slider)
vect_brace.stretch_to_fit_width(vector.get_length())
vect_brace.rotate(np.arctan(vector.get_slope()))
vect_brace.center().shift(vector.get_center())
nudge = 0.2 * (DOWN + LEFT)
vect_brace.shift(nudge)
y_mob = TexMobject("y")
y_mob.next_to(vert_brace)
sqrt_y = TexMobject("k\\sqrt{y}")
sqrt_y.scale(0.5)
sqrt_y.shift(vect_brace.get_center())
sqrt_y.shift(3 * nudge)
self.play(ShowCreation(horiz_line))
self.play(GrowFromCenter(vert_brace), ShimmerIn(y_mob))
self.play(GrowFromCenter(vect_brace), ShimmerIn(sqrt_y))
self.dither(3)
self.solve_energy()
def construct(self):
curve = PeanoCurve(order=5)
curve.stretch_to_fit_width(2 * SPACE_WIDTH)
curve.stretch_to_fit_height(2 * SPACE_HEIGHT)
curve_start = curve.copy()
curve_start.apply_over_attr_arrays(lambda arr: arr[:200])
time_line = get_time_line()
time_line.shift(-time_line.number_to_point(2000))
self.add(time_line)
self.play(
ApplyMethod(time_line.shift,
-time_line.number_to_point(1900),
run_time=3))
brace = Brace(
Mobject(
Point(time_line.number_to_point(1865)),
Point(time_line.number_to_point(1888)),
), UP)
words = TextMobject("""
Cantor drives himself (and the \\\\
mathematical community at large) \\\\
crazy with research on infinity.
""")
words.next_to(brace, UP)
self.play(GrowFromCenter(brace), ShimmerIn(words))
self.dither()
self.play(Transform(time_line, curve_start), FadeOut(brace),
FadeOut(words))
self.play(ShowCreation(curve, run_time=5, rate_func=None))
self.dither()
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, stroke_width=2))
grid8 = Grid(8, 8, stroke_width=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.wait()
self.play(ShowCreation(grid8))
self.wait()
self.play(*map(GrowFromCenter, mini_curves))
self.wait()
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.wait()
def __init__(self, direction = LEFT, index_of_tip = -1, center = ORIGIN):
self.direction = direction
self.content = Mobject()
self.index_of_tip = index_of_tip
self.center_offset = center - Mobject.get_center(self)
if direction[0] > 0:
self.rotate(np.pi, UP)
def show_snells(self, index, frame):
left_text, right_text = [
"\\dfrac{\\sin(\\theta_%d)}{\\phantom{\\sqrt{y_1}}}"%x
for x in index, index+1
]
left, equals, right = TexMobject(
[left_text, "=", right_text]
).split()
vs = []
sqrt_ys = []
for x, numerator in [(index, left), (index+1, right)]:
v, sqrt_y = [
TexMobject(
text, size = "\\Large"
).next_to(numerator, DOWN)
for text in "v_%d"%x, "\\sqrt{y_%d}"%x
]
vs.append(v)
sqrt_ys.append(sqrt_y)
start, end = [
Mobject(
left.copy(), mobs[0], equals.copy(), right.copy(), mobs[1]
).replace(frame)
for mobs in vs, sqrt_ys
]
self.add(start)
self.dither(2)
self.play(Transform(
start, end,
path_func = counterclockwise_path()
))
self.dither(2)
self.remove(start, end)
def get_paths(self):
return [
Mobject(Line(self.start_point, midpoint),
Line(midpoint, self.end_point)).highlight(color)
for midpoint, color in zip([2 * UP, 2 * DOWN],
Color(YELLOW).range_to(WHITE, 2))
]
def construct(self):
number_line = NumberLine(
numerical_radius = 5,
number_at_center = 5,
leftmost_tick = 0,
density = 2*DEFAULT_POINT_DENSITY_1D
)
number_line.shift(2*RIGHT)
number_line.add_numbers()
number_line.scale(2)
brace = Brace(Mobject(
*number_line.sub_mobjects[:2]
))
self.add(number_line)
for n in range(0, 10, 2):
if n == 0:
brace_anim = GrowFromCenter(brace)
else:
brace_anim = ApplyMethod(brace.shift, 2*RIGHT)
self.play(
ApplyMethod(
number_line.highlight,
RED,
lambda p : p[0] > n-6.2 and p[0] < n-4 and p[1] > -0.4
),
brace_anim
)
def show_infinite_objects(self):
sigma, summand, equals, result = TexMobject([
"\\sum_{n = 1}^{\\infty}",
"\\dfrac{1}{n^2}",
"=",
"\\dfrac{\pi^2}{6}"
]).split()
alt_summand = TexMobject("n").replace(summand)
alt_result = TexMobject("-\\dfrac{1}{12}").replace(result)
rationals, other_equals, naturals = TexMobject([
"|\\mathds{Q}|",
"=",
"|\\mathds{N}|"
]).scale(2).split()
infinity = TexMobject("\\infty").scale(2)
local_mobjects = filter(
lambda m : isinstance(m, Mobject),
locals().values(),
)
for mob in local_mobjects:
mob.sort_points(np.linalg.norm)
self.play(ShimmerIn(infinity))
self.dither()
self.play(
ShimmerIn(summand),
ShimmerIn(equals),
ShimmerIn(result),
DelayByOrder(Transform(infinity, sigma))
)
self.dither()
self.play(
Transform(summand, alt_summand),
Transform(result, alt_result),
)
self.dither()
self.remove(infinity)
self.play(*[
CounterclockwiseTransform(
Mobject(summand, equals, result, sigma),
Mobject(rationals, other_equals, naturals)
)
])
self.dither()
self.clear()
self.add(self.bubble)
def construct(self):
start_words = TextMobject([
"``", "Space Filling", "Curve ''",
]).to_edge(TOP, buff = 0.25)
quote, space_filling, curve_quote = start_words.copy().split()
curve_quote.shift(
space_filling.get_left()-\
curve_quote.get_left()
)
space_filling = Point(space_filling.get_center())
end_words = Mobject(*[
quote, space_filling, curve_quote
]).center().to_edge(TOP, buff = 0.25)
space_filling_fractal = TextMobject("""
``Space Filling Fractal''
""").to_edge(UP)
curve = HilbertCurve(order = 2).shift(DOWN)
fine_curve = HilbertCurve(order = 8)
fine_curve.replace(curve)
dots = Mobject(*[
Dot(
curve.points[n*curve.get_num_points()/15],
color = YELLOW_C
)
for n in range(1, 15)
if n not in [4, 11]
])
start_words.shift(2*(UP+LEFT))
self.play(
ApplyMethod(start_words.shift, 2*(DOWN+RIGHT))
)
self.wait()
self.play(Transform(start_words, end_words))
self.wait()
self.play(ShowCreation(curve))
self.wait()
self.play(ShowCreation(
dots,
run_time = 3,
))
self.wait()
self.clear()
self.play(ShowCreation(fine_curve, run_time = 5))
self.wait()
self.play(ShimmerIn(space_filling_fractal))
self.wait()
def show_pendulum(self, arc_angle = np.pi, arc_color = GREEN):
words = TextMobject(": Instantaneous center of rotation")
words.next_to(self.c_label)
line = Line(self.p_point, self.c_point)
line_angle = line.get_angle()+np.pi
line_length = line.get_length()
line.add(self.p_dot.copy())
line.get_center = lambda : self.c_point
tangent_line = Line(3*LEFT, 3*RIGHT)
tangent_line.rotate(line_angle-np.pi/2)
tangent_line.shift(self.p_point)
tangent_line.highlight(arc_color)
right_angle_symbol = Mobject(
Line(UP, UP+RIGHT),
Line(UP+RIGHT, RIGHT)
)
right_angle_symbol.scale(0.3)
right_angle_symbol.rotate(tangent_line.get_angle()+np.pi)
right_angle_symbol.shift(self.p_point)
self.play(ShowCreation(line))
self.play(ShimmerIn(words))
self.wait()
pairs = [
(line_angle, arc_angle/2),
(line_angle+arc_angle/2, -arc_angle),
(line_angle-arc_angle/2, arc_angle/2),
]
arcs = []
for start, angle in pairs:
arc = Arc(
angle = angle,
radius = line_length,
start_angle = start,
color = GREEN
)
arc.shift(self.c_point)
self.play(
ShowCreation(arc),
ApplyMethod(
line.rotate_in_place,
angle,
path_func = path_along_arc(angle)
),
run_time = 2
)
arcs.append(arc)
self.wait()
self.play(Transform(arcs[1], tangent_line))
self.add(tangent_line)
self.play(ShowCreation(right_angle_symbol))
self.wait()
self.tangent_line = tangent_line
self.right_angle_symbol = right_angle_symbol
self.pc_line = line
self.remove(words, *arcs)
def construct(self):
names = [
"Johann_Bernoulli2", "Jacob_Bernoulli",
"Gottfried_Wilhelm_von_Leibniz", "Newton"
]
guys = [ImageMobject(name, invert=False) for name in names]
johann = guys[0]
johann.scale(0.8)
pensive_johann = johann.copy()
pensive_johann.scale(0.25)
pensive_johann.to_corner(DOWN + LEFT)
comparitive_johann = johann.copy()
template = Square(side_length=2)
comparitive_johann.replace(template)
comparitive_johann.shift(UP + LEFT)
greater_than = TexMobject(">")
greater_than.next_to(comparitive_johann)
for guy, name in zip(guys, names)[1:]:
guy.replace(template)
guy.next_to(greater_than)
name_mob = TextMobject(name.replace("_", " "))
name_mob.scale(0.5)
name_mob.next_to(guy, DOWN)
guy.name_mob = name_mob
guy.sort_points(lambda p: np.dot(p, DOWN + RIGHT))
bubble = ThoughtBubble(initial_width=12)
bubble.stretch_to_fit_height(6)
bubble.ingest_submobjects()
bubble.pin_to(pensive_johann)
bubble.shift(DOWN)
point = Point(johann.get_corner(UP + RIGHT))
upper_point = Point(comparitive_johann.get_corner(UP + RIGHT))
lightbulb = ImageMobject("Lightbulb", invert=False)
lightbulb.scale(0.1)
lightbulb.sort_points(np.linalg.norm)
lightbulb.next_to(upper_point, RIGHT)
self.add(johann)
self.wait()
self.play(Transform(johann, pensive_johann),
Transform(point, bubble),
run_time=2)
self.remove(point)
self.add(bubble)
weakling = guys[1]
self.play(FadeIn(comparitive_johann), ShowCreation(greater_than),
FadeIn(weakling))
self.wait(2)
for guy in guys[2:]:
self.play(DelayByOrder(Transform(weakling, upper_point)))
self.play(FadeIn(guy), ShimmerIn(guy.name_mob))
self.wait(3)
self.remove(guy.name_mob)
weakling = guy
self.play(FadeOut(weakling), FadeOut(greater_than))
self.play(ShowCreation(lightbulb))
self.wait()
self.play(FadeOut(comparitive_johann), FadeOut(lightbulb))
self.play(ApplyMethod(Mobject(johann, bubble).scale, 10, run_time=3))
def __init__(self, mobject, ending_mobject, **kwargs):
mobject = instantiate(mobject)
#Copy ending_mobject so as to not mess with caller
ending_mobject = instantiate(ending_mobject).copy()
digest_config(self, kwargs, locals())
count1, count2 = mobject.get_num_points(
), ending_mobject.get_num_points()
if count2 == 0:
ending_mobject.add_points([mobject.get_center()], color=BLACK)
count2 = ending_mobject.get_num_points()
Mobject.align_data(mobject, ending_mobject)
if self.should_black_out_extra_points and count2 < count1:
self.black_out_extra_points(count1, count2)
Animation.__init__(self, mobject, **kwargs)
self.name += "To" + str(ending_mobject)
self.mobject.point_thickness = ending_mobject.point_thickness
def get_paths(self):
self.leftmost, self.rightmost = result = [
Mobject(Line(self.start_point, midpoint),
Line(midpoint, self.end_point)).highlight(color)
for midpoint, color in zip([3 * LEFT, 3 * RIGHT],
Color(YELLOW).range_to(WHITE, 2))
]
return result
def __init__(self,
function,
dim = 1,
expected_measure = 10.0,
density = None,
*args,
**kwargs):
self.function = function
self.dim = dim
self.expected_measure = expected_measure
if density:
self.epsilon = 1.0 / density
elif self.dim == 1:
self.epsilon = 1.0 / expected_measure / DEFAULT_POINT_DENSITY_1D
else:
self.epsilon = 1.0 / np.sqrt(expected_measure) / DEFAULT_POINT_DENSITY_2D
Mobject.__init__(self, *args, **kwargs)
def generate_springs(self, ring):
self.start_springs, self.end_springs = [
Mobject(
Spring(self.start_point, r.get_top()),
Spring(self.end_point, r.get_bottom())
)
for r in ring, ring.copy().shift(self.ring_shift_val)
]
def add_extra_dots(self):
dots = self.dots.split()
for vect in UP+LEFT, DOWN+RIGHT:
for n in range(5, 15):
dots.append(
dots[0].copy().center().shift(n*vect)
)
self.dots = Mobject(*dots)
def construct(self):
digest_config(self, {})
## Usually shouldn't need this...
self.frame_duration = self.DEFAULT_CONFIG["frame_duration"]
##
digest_config(self, {})
circle = Circle(
density = self.circle_density,
color = self.circle_blue
)
circle.repeat(self.circle_repeats)
circle.scale(self.radius)
sphere = Sphere(
density = self.sphere_density,
color = self.sphere_blue
)
sphere.scale(self.radius)
sphere.rotate(-np.pi / 7, [1, 0, 0])
sphere.rotate(-np.pi / 7)
iris = Mobject()
iris.interpolate(
circle, sphere,
self.interpolation_factor
)
for mob, color in [(iris, self.sphere_brown), (circle, self.circle_brown)]:
mob.highlight(color, lambda (x, y, z) : x < 0 and y > 0)
mob.highlight(
"black",
lambda point: np.linalg.norm(point) < \
self.inner_radius_ratio*self.radius
)
name = TextMobject("3Blue1Brown").center()
name.highlight("grey")
name.shift(2*DOWN)
self.play(Transform(
circle, iris,
run_time = self.run_time
))
self.frames = drag_pixels(self.frames)
self.save_image(IMAGE_DIR)
self.show_frame()
self.add(name)
self.dither()
print "Dragging pixels..."
def construct(self):
coords_set = [ORIGIN]
for n in range(int(2 * SPACE_WIDTH)):
for vect in UP, RIGHT:
for k in range(n):
new_coords = coords_set[-1] + ((-1)**n) * vect
coords_set.append(new_coords)
square = Square(side_length=1, color=WHITE)
squares = Mobject(
*[square.copy().shift(coords)
for coords in coords_set]).ingest_submobjects()
self.play(DelayByOrder(FadeIn(squares)), run_time=3)
curve = HilbertCurve(order=6).scale(1. / 6)
all_curves = Mobject(
*[curve.copy().shift(coords)
for coords in coords_set]).ingest_submobjects()
all_curves.thin_out(10)
self.play(ShowCreation(all_curves, rate_func=None, run_time=15))
def write_time(self, time):
if hasattr(self, "time_mob"):
self.remove(self.time_mob)
digits = map(TexMobject, "%.2f" % time)
digits[0].next_to(self.t_equals, buff=0.1)
for left, right in zip(digits, digits[1:]):
right.next_to(left, buff=0.1, aligned_edge=DOWN)
self.time_mob = Mobject(*digits)
self.add(self.time_mob)
def __init__(self, image_file, **kwargs):
digest_locals(self)
Mobject.__init__(self, **kwargs)
self.name = to_cammel_case(os.path.split(image_file)[-1].split(".")[0])
possible_paths = [
image_file,
os.path.join(IMAGE_DIR, image_file),
os.path.join(IMAGE_DIR, image_file + ".jpg"),
os.path.join(IMAGE_DIR, image_file + ".png"),
]
for path in possible_paths:
if os.path.exists(path):
self.generate_points_from_file(path)
self.scale(self.scale_value)
if self.should_center:
self.center()
return
raise IOError("File not Found")
def get_q_marks_and_arrows(self, mob, n_marks = 10):
circle = Circle().replace(mob)
q_marks, arrows = result = [Mobject(), Mobject()]
for x in range(n_marks):
index = (x+0.5)*self.cycloid.get_num_points()/n_marks
q_point = self.cycloid.points[index]
vect = q_point-mob.get_center()
start_point = circle.get_boundary_point(vect)
arrow = Arrow(
start_point, q_point,
color = BLUE_E
)
q_marks.add(TextMobject("?").shift(q_point))
arrows.add(arrow)
for mob in result:
mob.ingest_submobjects()
return result
def label_jump(self):
jump_points = Mobject(Point(self.spiril1.points[-1]),
Point(self.spiril2.points[0]))
self.brace = Brace(jump_points, RIGHT)
self.jump = TextMobject("Jump")
self.jump.next_to(self.brace, RIGHT)
self.play(GrowFromCenter(self.brace), ShimmerIn(self.jump))
self.dither()
self.remove(self.brace, self.jump)
def __init__(self, mobject, **kwargs):
self.intermediate = Mobject(color = self.color)
self.intermediate.add_points([
point + (x, y, 0)
for point in self.mobject.points
for x in [-1, 1]
for y in [-1, 1]
])
Animation.__init__(self, mobject, **kwargs)
def construct(self):
words = TextMobject("Would this actually work?")
grid = get_grid()
grid.scale_to_fit_width(6)
grid.to_edge(LEFT)
freq_line = get_freq_line()
freq_line.scale_to_fit_width(6)
freq_line.center().to_edge(RIGHT)
mapping = Mobject(grid, freq_line, Arrow(grid, freq_line))
mapping.ingest_sub_mobjects()
lower_left = Point().to_corner(DOWN + LEFT, buff=0)
lower_right = Point().to_corner(DOWN + RIGHT, buff=0)
self.add(words)
self.dither()
self.play(Transform(words, lower_right),
Transform(lower_left, mapping))
self.dither()
def construct(self):
val = 0.7
text = TexMobject([
"\\text{HC}(", "x", ")",
"=\\lim_{n \\to \\infty}\\text{PHC}_n(", "x", ")"
])
text.to_edge(UP)
x1 = text.split()[1]
x2 = text.split()[-2]
x2.highlight(BLUE)
explanation = TextMobject("Actual Hilbert curve function")
exp_arrow = Arrow(explanation, text.split()[0])
curve = UnitInterval()
dot = Dot(curve.number_to_point(val))
x_arrow = Arrow(x1.get_bottom(), dot, buff = 0)
curve.sort_points(lambda p : p[0])
curve.add_numbers(0, 1)
self.add(*text.split()[:3])
self.play(
ShimmerIn(explanation),
ShowCreation(exp_arrow)
)
self.dither()
self.remove(explanation, exp_arrow)
self.play(ShowCreation(curve))
self.play(
ApplyMethod(x1.highlight, BLUE),
ShowCreation(x_arrow),
ShowCreation(dot)
)
self.dither()
self.remove(x_arrow)
limit = Mobject(*text.split()[3:]).ingest_sub_mobjects()
limit.point_thickness = 1
self.play(ShimmerIn(limit))
for num in range(1, 9):
new_curve = HilbertCurve(order = num)
new_curve.scale(0.8)
new_dot = Dot(new_curve.points[int(val*new_curve.get_num_points())])
self.play(
Transform(curve, new_curve),
Transform(dot, new_dot),
)