def construct(self):
formula = TexMobject(*([
"{%d \\over %d} \\cdot" % (wallis_numer(n), wallis_denom(n))
for n in range(1, self.n_terms + 1)
] + ["\\cdots"]))
result = TexMobject("=", "{\\pi \\over 2}")
result.scale(2)
pi = result[-1][0]
pi.set_color(YELLOW)
circle = Circle(color=YELLOW)
circle.surround(pi)
question = TexMobject("?", color=YELLOW)
question.scale(2).next_to(circle, RIGHT, buff=0.4)
result_group = VGroup(result, circle, question)
result_group.next_to(formula, DOWN)
group = VGroup(formula, result_group)
group.center().set_width(10)
bg_rect = BackgroundRectangle(group, fill_opacity=0.5, buff=0.2)
random_walks = VGroup(*[
RandomWalk1DArrow(random_walk_string(30), step_size=0.5)
for k in range(self.n_walks)
])
for k, walk in enumerate(random_walks):
walk.shift(random.randrange(-5, 5) * 2 * walk.step_size * UP)
random_walks.center()
text = TextMobject(self.part).scale(3).set_color(YELLOW)
text.to_corner(RIGHT + DOWN)
self.add(random_walks)
self.add(FullScreenFadeRectangle())
self.add(bg_rect, group, text)
def get_order_n_self(self, order):
if order == 0:
result = self.get_seed_shape()
else:
lower_order = self.get_order_n_self(order - 1)
subparts = [lower_order.copy() for x in range(self.num_subparts)]
self.arrange_subparts(*subparts)
result = VGroup(*subparts)
result.set_height(self.height)
result.center()
return result
def get_order_n_self(self, order):
if order == 0:
result = self.get_seed_shape()
else:
lower_order = self.get_order_n_self(order - 1)
subparts = [
lower_order.copy()
for x in range(self.num_subparts)
]
self.arrange_subparts(*subparts)
result = VGroup(*subparts)
result.scale_to_fit_height(self.height)
result.center()
return result
def setup(self):
pos_axis = NumberLine(x_min=-4.5, x_max=2.5, color=WHITE)
pos_axis.rotate(np.pi / 2)
pos_axis.add_tip()
time_axis = NumberLine(x_min=0, x_max=9.5, color=WHITE)
time_axis.add_tip()
vec = pos_axis.number_to_point(0) - time_axis.number_to_point(0)
time_axis.shift(vec)
pos_text = TextMobject("位置")
pos_text.next_to(pos_axis.number_to_point(2.5), LEFT)
time_text = TextMobject("时间")
time_text.next_to(time_axis.number_to_point(9.5), DOWN)
axes_group = VGroup(pos_axis, time_axis, pos_text, time_text)
axes_group.center()
title_pq = TexMobject("P_4", "570", "Q_4")
title_pq.scale(1.5)
title_pq.to_corner(UP + RIGHT)
for part, color in zip(title_pq, [ORANGE, BLACK, GREEN]):
part.set_color(color)
r_arrow = TexMobject("\\rightarrow")
l_arrow = TexMobject("\\leftarrow")
for arrow in (r_arrow, l_arrow):
arrow.scale(1.5)
arrow.move_to(title_pq[1])
sequence_p, sequence_q = sequences = ["UDUUDUDD", "DDUDDUDD"]
colors = [ORANGE, GREEN]
walk_p, walk_q = walks = [
RandomWalk1DArrow(sequence, up_color = color, down_color = color) \
.move_start_to(pos_axis.number_to_point(0))
for sequence, color in zip(sequences, colors)
]
parts_p, parts_q = [walk.split_at(3) for walk in walks]
self.axes_group = axes_group
self.title_pq = title_pq
self.walk_p = walk_p
self.walk_q = walk_q
self.parts_p = parts_p
self.parts_q = parts_q
self.r_arrow = r_arrow
self.l_arrow = l_arrow
def construct(self):
# Chart on the left
colors = [WHITE, ORANGE, GREEN]
titles = VGroup(*[
TexMobject(text).set_color(color)
for text, color in zip(["n", "p_n", "q_n"], colors)
])
contents = VGroup(*[
VGroup(*[
TexMobject("%d" % num) for num in
[k, central_binomial_coeff(k),
central_binomial_coeff(k)]
]) for k in range(8)
])
titles.arrange_submobjects(RIGHT, buff=1)
for num, line in enumerate(contents):
for k, element in enumerate(line):
buff = 0.6 + 0.8 * num
element.next_to(titles[k], DOWN, aligned_edge=LEFT, buff=buff)
element.set_color(colors[k])
sep_line = Line(ORIGIN, 4.5 * RIGHT, stroke_width=5)
sep_line.next_to(titles, DOWN)
chart = VGroup(titles, contents, sep_line)
chart.set_height(7)
chart.center().to_edge(LEFT)
self.add(chart)
# Figures on the right
std_zero_pos_axis = NumberLine(x_min=-2,
x_max=2,
color=GREY,
unit_size=0.25,
tick_size=0.05)
std_zero_pos_axis.rotate(np.pi / 2)
std_nocross_pos_axis = NumberLine(x_min=-4,
x_max=4,
color=GREY,
unit_size=0.25,
tick_size=0.05)
std_nocross_pos_axis.rotate(np.pi / 2)
std_time_axis = NumberLine(x_min=0,
x_max=5.5,
color=GREY,
unit_size=0.25,
tick_size=0.05)
std_zero_axes = VGroup(std_zero_pos_axis, std_time_axis)
std_nocross_axes = VGroup(std_nocross_pos_axis, std_time_axis)
zero_walks = VGroup()
for sequence in ["UUDD", "UDUD", "UDDU", "DDUU", "DUDU", "DUUD"]:
axes = std_zero_axes.copy()
zero_walk = RandomWalk1DArrow(sequence, step_size=0.25)
zero_walk.move_start_to(axes[0].number_to_point(0))
zero_walks.add(VGroup(axes, zero_walk))
zero_walks.arrange_submobjects_in_grid(2, 3, buff=0.5)
zero_rect = SurroundingRectangle(zero_walks, color=ORANGE, buff=0.4)
zero_walks.add(zero_rect)
nocross_walks = VGroup()
for sequence in ["DDDD", "DDUD", "DDDU", "UUUU", "UUDU", "UUUD"]:
axes = std_nocross_axes.copy()
nocross_walk = RandomWalk1DArrow(sequence, step_size=0.25)
nocross_walk.move_start_to(axes[0].number_to_point(0))
nocross_walks.add(VGroup(axes, nocross_walk))
nocross_walks.arrange_submobjects_in_grid(2, 3, buff=0.5)
nocross_rect = SurroundingRectangle(nocross_walks,
color=GREEN,
buff=0.4)
nocross_walks.add(nocross_rect)
relation = TexMobject("p_2", "=", "q_2", "=", "6")
relation[0].set_color(ORANGE)
relation[2].set_color(GREEN)
relation.scale(1.5)
figure = VGroup(zero_walks, relation, nocross_walks)
figure.arrange_submobjects(DOWN)
figure.set_height(7)
figure.center().to_edge(RIGHT)
self.add(figure)
self.wait()
def add_figure(self):
# Main figure
wallis_rects = WallisRectangles(order=10)
order = wallis_rects.get_order()
# Variables and braces on the left
left_rects = [
wallis_rects.get_rectangle(k, 0) for k in [0, 1, 2, 3, -1]
]
left_braces = [
Brace(rect, direction=LEFT, buff=0.1) for rect in left_rects
]
left_lengths = [
TexMobject(symbol)
for symbol in ["a_0", "a_1", "a_2", "a_3", "a_{n-1}"]
]
left_lengths[-1].scale(0.7)
for brace, length in zip(left_braces, left_lengths):
brace.put_at_tip(length, buff=0.15)
left_vdots = TexMobject("\\vdots")
left_vdots.scale(2)
left_vdots.move_to(
(left_braces[-1].get_center() + left_braces[-2].get_center()) / 2)
# Variables and braces on the bottom
down_rects = [
wallis_rects.get_rectangle(0, k) for k in [0, 1, 2, 3, -1]
]
down_braces = [
Brace(rect, direction=DOWN, buff=0.1) for rect in down_rects
]
down_lengths = [
TexMobject(symbol)
for symbol in ["a_0", "a_1", "a_2", "a_3", "a_{n-1}"]
]
down_lengths[-1].scale(0.7)
for brace, length in zip(down_braces, down_lengths):
brace.put_at_tip(length, buff=0.15)
down_cdots = TexMobject("\\cdots")
down_cdots.scale(2)
down_cdots.move_to(
(down_braces[-1].get_center() + down_braces[-2].get_center()) / 2)
# The quarter circle
quarter_circle = Sector(
outer_radius=wallis_rects.get_height(),
stroke_color=GREY,
stroke_width=5.70,
fill_opacity=0,
)
quarter_circle.move_arc_center_to(wallis_rects.get_bottom_left())
# Add everthing
figure_group = VGroup(
wallis_rects,
VGroup(*left_braces),
VGroup(*left_lengths),
left_vdots,
VGroup(*down_braces),
VGroup(*down_lengths),
down_cdots,
quarter_circle,
)
figure_group.center().to_edge(LEFT, buff=0.15)
self.add(figure_group)
