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https://github.com/3b1b/manim.git
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PascalsTriangle in eop/combinations
This commit is contained in:
Grant Sanderson
parent
0e7fbe73b8
commit
0399b54b9a
1 changed files with 411 additions and 4 deletions
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@ -1146,10 +1146,417 @@ class BuildFiveFromFour(ProbabilityOfThreeWomenInGroupOfFive):
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self.play(Write(new_numbers, run_time = 3))
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self.dither()
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class BuildUpFromStart(Scene):
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CONFIG = {
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"n_iterations" : 7,
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}
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def construct(self):
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stacks = VGroup(VGroup(Male()), VGroup(Female()))
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stacks.arrange_submobjects(RIGHT, buff = LARGE_BUFF)
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stacks.numbers = self.get_numbers(stacks)
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max_width = 2*SPACE_WIDTH - 3
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max_height = SPACE_HEIGHT - 1
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self.add(stacks, stacks.numbers)
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for x in range(self.n_iterations):
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if x < 2:
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dither_time = 1
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else:
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dither_time = 0.2
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#Divide
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low_stacks = stacks
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low_group = VGroup(low_stacks, low_stacks.numbers)
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top_stacks = stacks.deepcopy()
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top_group = VGroup(top_stacks, top_stacks.numbers)
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for group, vect in (top_group, UP), (low_group, DOWN):
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group.generate_target()
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if group[0].get_height() > max_height:
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group.target[0].stretch_to_fit_height(max_height)
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for stack, num in zip(*group.target):
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num.next_to(stack, UP)
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group.target.next_to(ORIGIN, vect)
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self.play(*map(MoveToTarget, [top_group, low_group]))
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self.dither(dither_time)
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#Expand
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for stacks, i in (low_stacks, 0), (top_stacks, -1):
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sym = stacks[i][i][i]
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new_stacks = VGroup()
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for stack in stacks:
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new_stack = VGroup()
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for line in stack:
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new_line = line.copy()
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new_sym = sym.copy()
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buff = 0.3*line.get_height()
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new_sym.next_to(line, RIGHT, buff = buff)
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new_line.add(new_sym)
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line.add(VectorizedPoint(line[-1].get_center()))
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new_stack.add(new_line)
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new_stacks.add(new_stack)
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new_stacks.arrange_submobjects(
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RIGHT, buff = LARGE_BUFF, aligned_edge = DOWN
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)
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if new_stacks.get_width() > max_width:
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new_stacks.stretch_to_fit_width(max_width)
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if new_stacks.get_height() > max_height:
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new_stacks.stretch_to_fit_height(max_height)
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new_stacks.move_to(stacks, DOWN)
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stacks.target = new_stacks
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stacks.numbers.generate_target()
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for num, stack in zip(stacks.numbers.target, new_stacks):
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num.next_to(stack, UP)
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self.play(*map(MoveToTarget, [
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top_stacks, low_stacks,
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top_stacks.numbers, low_stacks.numbers,
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]))
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self.dither(dither_time)
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#Shift
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dist = top_stacks[1].get_center()[0] - top_stacks[0].get_center()[0]
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self.play(
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top_group.shift, dist*RIGHT/2,
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low_group.shift, dist*LEFT/2,
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)
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self.dither(dither_time)
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#Stack
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all_movers = VGroup()
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plusses = VGroup()
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expressions = VGroup(low_stacks.numbers[0])
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stacks = VGroup(low_stacks[0])
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v_buff = 0.25*stacks[0][0].get_height()
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for i, top_stack in enumerate(top_stacks[:-1]):
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low_stack = low_stacks[i+1]
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top_num = top_stacks.numbers[i]
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low_num = low_stacks.numbers[i+1]
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movers = [top_stack, top_num, low_num]
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for mover in movers:
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mover.generate_target()
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plus = TexMobject("+")
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expr = VGroup(top_num.target, plus, low_num.target)
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expr.arrange_submobjects(RIGHT, buff = SMALL_BUFF)
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top_stack.target.next_to(low_stack, UP, buff = v_buff)
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expr.next_to(top_stack.target, UP)
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all_movers.add(*movers)
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plusses.add(plus)
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expressions.add(VGroup(top_num, plus, low_num))
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stacks.add(VGroup(*it.chain(low_stack, top_stack)))
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last_group = VGroup(top_stacks[-1], top_stacks.numbers[-1])
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last_group.generate_target()
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last_group.target.align_to(low_stacks, DOWN)
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all_movers.add(last_group)
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stacks.add(top_stacks[-1])
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expressions.add(top_stacks.numbers[-1])
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self.play(*it.chain(
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map(MoveToTarget, all_movers),
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map(Write, plusses),
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))
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#Add
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new_numbers = self.get_numbers(stacks)
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self.play(ReplacementTransform(
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expressions, VGroup(*map(VGroup, new_numbers))
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))
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self.dither(dither_time)
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stacks.numbers = new_numbers
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####
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def get_numbers(self, stacks):
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return VGroup(*[
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TexMobject(str(len(stack))).next_to(stack, UP)
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for stack in stacks
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])
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class PascalsTriangle(Scene):
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CONFIG = {
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"max_n" : 9,
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}
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def construct(self):
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self.show_triangle()
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self.show_sum_of_two_over_rule()
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self.keep_in_mind_what_these_mean()
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self.issolate_9_choose_4_term()
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self.show_9_choose_4_pattern()
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self.cap_off_triangle()
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def show_triangle(self):
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distance = 0.8
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max_width = 0.7*distance
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angle = 0.2*np.pi
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t_down = rotate_vector(distance*DOWN, -angle)
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t_right = 2*distance*np.sin(angle)*RIGHT
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rows = VGroup()
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for n in range(self.max_n + 1):
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row = VGroup()
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for k in range(n+1):
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num = TexMobject(str(choose(n, k)))
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# if num.get_width() > max_width:
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# num.scale_to_fit_width(max_width)
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num.shift(n*t_down + k*t_right)
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row.add(num)
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rows.add(row)
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rows.to_edge(UP)
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self.play(FadeIn(rows[1]))
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for last_row, curr_row in zip(rows[1:], rows[2:]):
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self.play(*[
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Transform(
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last_row.copy(), VGroup(*mobs),
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remover = True
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)
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for mobs in curr_row[1:], curr_row[:-1]
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])
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self.add(curr_row)
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self.dither()
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self.rows = rows
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def show_sum_of_two_over_rule(self):
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rows = self.rows
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example = rows[5][3]
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ex_top1 = rows[4][2]
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ex_top2 = rows[4][3]
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rects = VGroup()
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for mob, color in (example, GREEN), (ex_top1, BLUE), (ex_top2, YELLOW):
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mob.rect = SurroundingRectangle(mob, color = color)
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rects.add(mob.rect)
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rows_to_fade = VGroup(*rows[1:4] + rows[6:])
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rows_to_fade.save_state()
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top_row = rows[4]
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low_row = rows[5]
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top_row_copy = top_row.copy()
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top_row.save_state()
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top_row.add(ex_top2.rect)
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top_row_copy.add(ex_top1.rect)
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h_line = Line(LEFT, RIGHT)
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h_line.stretch_to_fit_width(low_row.get_width() + 2)
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h_line.next_to(low_row, UP, 1.5*SMALL_BUFF)
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plus = TexMobject("+")
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plus.next_to(h_line.get_left(), UP+RIGHT, buff = 1.5*SMALL_BUFF)
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self.play(ShowCreation(example.rect))
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self.play(
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ReplacementTransform(example.rect.copy(), ex_top1.rect),
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ReplacementTransform(example.rect.copy(), ex_top2.rect),
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)
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self.dither(2)
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self.play(rows_to_fade.fade, 1)
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self.play(
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top_row.align_to, low_row, LEFT,
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top_row_copy.next_to, top_row, UP,
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top_row_copy.align_to, low_row, RIGHT,
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)
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self.play(
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ShowCreation(h_line),
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Write(plus)
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)
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self.dither(2)
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for row in top_row, top_row_copy:
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row.remove(row[-1])
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self.play(
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rows_to_fade.restore,
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top_row.restore,
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Transform(
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top_row_copy, top_row.saved_state,
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remover = True
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),
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FadeOut(VGroup(h_line, plus)),
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FadeOut(rects),
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)
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self.dither()
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def keep_in_mind_what_these_mean(self):
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morty = Mortimer().flip()
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morty.scale(0.7)
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morty.to_edge(LEFT)
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morty.shift(DOWN)
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numbers = VGroup(*it.chain(*self.rows[1:]))
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random.shuffle(numbers.submobjects)
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self.play(FadeIn(morty))
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self.play(PiCreatureSays(
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morty, "Keep in mind \\\\ what these mean.",
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bubble_kwargs = {
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"width" : 3.5,
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"height" : 2.5,
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}
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))
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self.play(
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Blink(morty),
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LaggedStart(
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Indicate, numbers,
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rate_func = wiggle,
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color = PINK,
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)
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)
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self.play(*map(FadeOut, [
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morty, morty.bubble, morty.bubble.content
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]))
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def issolate_9_choose_4_term(self):
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rows = self.rows
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for n in range(1, self.max_n+1):
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num = rows[n][0]
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line = get_stack(Female(), Male(), n, 0)[0]
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if n < self.max_n:
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line.next_to(num, LEFT)
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else:
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line.next_to(num, DOWN, MED_LARGE_BUFF)
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self.highlight_num(num)
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self.add(line)
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if n < self.max_n:
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self.dither(0.25)
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else:
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self.dither(1.25)
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self.dehighlight_num(num)
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self.remove(line)
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for k in range(1, 5):
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num = rows[self.max_n][k]
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line = get_stack(Female(), Male(), self.max_n, k)[0]
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line.next_to(num, DOWN, MED_LARGE_BUFF)
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self.highlight_num(num)
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self.add(line)
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self.dither(0.5)
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self.dehighlight_num(num)
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self.remove(line)
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num.highlight(YELLOW)
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num.scale_in_place(1.2)
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self.add(line)
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self.dither()
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self.nine_choose_four_term = num
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self.nine_choose_four_line = line
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def show_9_choose_4_pattern(self):
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rows = VGroup(*self.rows[1:])
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all_stacks = get_stacks(Female(), Male(), 9)
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stack = all_stacks[4]
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all_lines = VGroup(*it.chain(*all_stacks))
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self.play(
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rows.shift, 3*UP,
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self.nine_choose_four_line.shift, 2.5*UP,
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)
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self.remove(self.nine_choose_four_line)
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for n, line in enumerate(stack):
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line.next_to(self.nine_choose_four_term, DOWN, LARGE_BUFF)
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num = Integer(n+1)
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num.next_to(line, DOWN, MED_LARGE_BUFF)
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self.add(line, num)
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self.dither(0.1)
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self.remove(line, num)
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self.add(line, num)
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self.dither()
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self.curr_line = line
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#Probability
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expr = TexMobject(
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"P(4", "\\female", "\\text{ out of }", "9", ")", "="
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)
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expr.move_to(num.get_left())
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expr.highlight_by_tex("female", MAROON_B)
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nine_choose_four_term = self.nine_choose_four_term.copy()
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nine_choose_four_term.generate_target()
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nine_choose_four_term.target.scale(1./1.2)
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over_512 = TexMobject("\\quad \\over 2^9")
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frac = VGroup(nine_choose_four_term.target, over_512)
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frac.arrange_submobjects(DOWN, buff = SMALL_BUFF)
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frac.next_to(expr, RIGHT, SMALL_BUFF)
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eq_result = TexMobject("\\approx 0.246")
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eq_result.next_to(frac, RIGHT)
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def show_random_lines(n, dither_time = 1):
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for x in range(n):
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if x == n-1:
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dither_time = 0
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new_line = random.choice(all_lines)
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new_line.move_to(self.curr_line)
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self.remove(self.curr_line)
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self.curr_line = new_line
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self.add(self.curr_line)
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self.dither(dither_time)
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self.play(FadeOut(num), FadeIn(expr))
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show_random_lines(4)
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self.play(
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MoveToTarget(nine_choose_four_term),
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Write(over_512)
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)
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show_random_lines(4)
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self.play(Write(eq_result))
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show_random_lines(6)
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self.play(
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self.nine_choose_four_term.scale_in_place, 1./1.2,
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self.nine_choose_four_term.highlight, WHITE,
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*map(FadeOut, [
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expr, nine_choose_four_term,
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over_512, eq_result, self.curr_line
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])
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)
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self.play(rows.shift, 3*DOWN)
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def cap_off_triangle(self):
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top_row = self.rows[0]
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circle = Circle(color = YELLOW)
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circle.replace(top_row, dim_to_match = 1)
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circle.scale_in_place(1.5)
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line_groups = VGroup()
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for n in range(4, -1, -1):
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line = VGroup(*[
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random.choice([Male, Female])()
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for k in range(n)
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])
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if n == 0:
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line.add(Line(LEFT, RIGHT).scale(0.1).set_stroke(BLACK, 0))
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line.arrange_submobjects(RIGHT, SMALL_BUFF)
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line.shift(SPACE_WIDTH*RIGHT/2 + SPACE_HEIGHT*UP/2)
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brace = Brace(line, UP)
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if n == 1:
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label = "1 Person"
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else:
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label = "%d People"%n
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brace_text = brace.get_text(label)
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line_group = VGroup(line, brace, brace_text)
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line_groups.add(line_group)
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self.play(ShowCreation(circle))
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self.play(Write(top_row))
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self.dither()
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curr_line_group = line_groups[0]
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self.play(FadeIn(curr_line_group))
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for line_group in line_groups[1:]:
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self.play(ReplacementTransform(
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curr_line_group, line_group
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))
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curr_line_group = line_group
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self.dither()
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###
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def highlight_num(self, num):
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num.highlight(YELLOW)
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num.scale_in_place(1.2)
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def dehighlight_num(self, num):
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num.highlight(WHITE)
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num.scale_in_place(1.0/1.2)
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