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Merge pull request #176 from 3b1b/winding-number-grant
Winding number grant
This commit is contained in:
Grant Sanderson
GitHub
commit
873fa49fbe
3 changed files with 291 additions and 13 deletions
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@ -619,7 +619,8 @@ def walker_animation_with_display(
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display.background_rectangle.fill_color = GREY
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display.background_rectangle.scale(1.2)
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displaycement = 0.5 * DOWN # How about that pun, eh?
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display.move_to(walker.get_center() + displaycement)
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# display.move_to(walker.get_center() + displaycement)
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display.next_to(walker, DOWN+RIGHT, SMALL_BUFF)
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display_anim = ChangingDecimal(display,
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number_update_func,
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tracked_mobject = walker_anim.compound_walker.walker,
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@ -1928,6 +1928,20 @@ class TinyLoopAroundRed(TinyLoop):
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"target_coords" : (-1, 1),
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}
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class ConfusedPiCreature(PiCreatureScene):
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def construct(self):
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morty = self.pi_creature
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morty.set_color(YELLOW_E)
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morty.flip()
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morty.center()
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self.play(morty.change, "awe", DOWN+3*RIGHT)
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self.wait(2)
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self.play(morty.change, "confused")
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self.wait(2)
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self.play(morty.change, "pondering")
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self.wait(2)
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class FailureOfComposition(ColorMappedObjectsScene):
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CONFIG = {
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"func" : lambda p : (
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@ -2180,7 +2194,11 @@ class TransitionFromPathsToBoundaries(ColorMappedObjectsScene):
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CONFIG = {
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"func" : plane_func_by_wind_spec(
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(-2, 0, 2), (2, 0, 1)
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)
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),
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"dot_fill_opacity" : 1,
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"dot_stroke_width" : 1,
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"include_walkers" : True,
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"include_question_mark" : True,
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}
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def construct(self):
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ColorMappedObjectsScene.construct(self)
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@ -2210,7 +2228,10 @@ class TransitionFromPathsToBoundaries(ColorMappedObjectsScene):
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#Setup region labels
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for square, tex in (left_square, "x"), (right_square, "y"), (joint_rect, "x+y \\, ?"):
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sum_tex = "x+y"
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if self.include_question_mark:
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sum_tex += "\\, ?"
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for square, tex in (left_square, "x"), (right_square, "y"), (joint_rect, sum_tex):
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square.label = TextMobject("Winding = ", "$%s$"%tex)
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square.label.move_to(square)
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@ -2264,7 +2285,7 @@ class TransitionFromPathsToBoundaries(ColorMappedObjectsScene):
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path_arc = TAU/6
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),
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FadeIn(joint_rect.label[1][1]),
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FadeIn(joint_rect.label[1][3]),
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FadeIn(joint_rect.label[1][3:]),
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FadeOut(right_square.label[0]),
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Transform(
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right_square.label[1], joint_rect.label[1][2],
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@ -2313,9 +2334,12 @@ class TransitionFromPathsToBoundaries(ColorMappedObjectsScene):
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#Setup dot, arrow and label
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dot = self.dot = Dot(radius = 0.1)
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dot.set_stroke(WHITE, 1)
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dot.set_stroke(WHITE, self.dot_stroke_width)
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update_dot_color = ContinualUpdateFromFunc(
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dot, lambda d : d.set_fill(get_output_color())
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dot, lambda d : d.set_fill(
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get_output_color(),
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self.dot_fill_opacity
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)
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)
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label = DecimalNumber(0, num_decimal_points = 1)
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@ -2326,7 +2350,7 @@ class TransitionFromPathsToBoundaries(ColorMappedObjectsScene):
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arrow_length = 0.75
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arrow = Vector(arrow_length*RIGHT)
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arrow.set_stroke(WHITE, 1)
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arrow.set_stroke(WHITE, self.dot_stroke_width)
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def arrow_update_func(arrow):
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arrow.set_fill(get_output_color(), 1)
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arrow.rotate(-TAU*get_output_rev() - arrow.get_angle())
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@ -2335,7 +2359,8 @@ class TransitionFromPathsToBoundaries(ColorMappedObjectsScene):
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return arrow
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update_arrow = ContinualUpdateFromFunc(arrow, arrow_update_func)
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self.add(update_arrow, update_dot_color, label_upadte)
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if self.include_walkers:
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self.add(update_arrow, update_dot_color, label_upadte)
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return dot
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def position_dot(self, point):
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@ -2343,6 +2368,17 @@ class TransitionFromPathsToBoundaries(ColorMappedObjectsScene):
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self.start_rev = self.get_output_rev()
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self.curr_winding = 0
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class TransitionFromPathsToBoundariesArrowless(TransitionFromPathsToBoundaries):
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CONFIG = {
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"func" : plane_func_by_wind_spec(
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(-2, 0, 2), (2, 0, 1)
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),
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"dot_fill_opacity" : 0,
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"dot_stroke_width" : 0,
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"include_walkers" : False,
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"include_question_mark" : False,
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}
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class BreakDownLoopWithNonzeroWinding(TransitionFromPathsToBoundaries):
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def construct(self):
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zero_point = 2*LEFT
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@ -2621,9 +2657,7 @@ class PolynomialTerms(MonomialTerm):
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class SearchSpacePerimeterVsArea(EquationSolver2d):
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CONFIG = {
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"func" : plane_func_by_wind_spec(
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(-3, -1.3, 2), (0.1, 0.2, 1), (2.8, -2, 1)
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),
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"func" : example_plane_func,
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"num_iterations" : 15,
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"display_in_parallel" : False,
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"use_fancy_lines" : True,
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@ -2686,6 +2720,223 @@ class SearchSpacePerimeterVsArea(EquationSolver2d):
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self.play(FadeOut(full_rect))
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self.wait()
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class ShowPolynomialFinalState(SolveX5MinusXMinus1):
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CONFIG = {
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"num_iterations" : 15,
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}
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def construct(self):
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self.force_skipping()
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SolveX5MinusXMinus1.construct(self)
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self.revert_to_original_skipping_status()
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class PiCreatureInAwe(Scene):
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def construct(self):
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randy = Randolph()
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self.play(randy.change, "awe")
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self.play(Blink(randy))
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self.play(randy.look, UP, run_time = 2)
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self.play(
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randy.look, RIGHT,
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run_time = 4,
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rate_func = there_and_back,
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path_arc = -TAU/4
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)
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self.wait()
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class ShowComplexFunction(Scene):
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def construct(self):
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plane = ComplexPlane()
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plane.add_coordinates()
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four_i = plane.coordinate_labels[-1]
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plane.coordinate_labels.remove(four_i)
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plane.remove(four_i)
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title = TextMobject("Complex Plane")
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title.to_edge(UP, buff = MED_SMALL_BUFF)
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rect = BackgroundRectangle(title, fill_opacity = 1, buff = MED_SMALL_BUFF)
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x = complex(1, 0.4)
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f = lambda x : x**5 - x - 1
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x_point = plane.number_to_point(x)
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fx_point = plane.number_to_point(f(x))
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x_dot = Dot(x_point)
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fx_dot = Dot(fx_point, color = YELLOW)
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arrow = Arrow(
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x_point, fx_point,
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use_rectangular_stem = False,
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path_arc = TAU/3,
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color = YELLOW
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)
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arrow.pointwise_become_partial(arrow, 0, 0.95)
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x_label = TexMobject("x = %d+%.1fi"%(x.real, x.imag))
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x_label.next_to(x_dot, RIGHT)
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x_label.add_background_rectangle()
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fx_label = TexMobject("f(x) = x^5 - x - 1")
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fx_label.next_to(fx_dot, DOWN, SMALL_BUFF)
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fx_label.highlight(YELLOW)
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fx_label.add_background_rectangle()
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fx_label.generate_target()
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fx_label.target.move_to(title)
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fx_label.target[1].highlight(WHITE)
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self.play(
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Write(plane),
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FadeIn(rect),
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LaggedStart(FadeIn, title)
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)
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self.play(*map(FadeIn, [x_dot, x_label]))
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self.wait()
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self.play(
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ReplacementTransform(x_dot.copy(), fx_dot, path_arc = arrow.path_arc),
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ShowCreation(arrow, rate_func = squish_rate_func(smooth, 0.2, 1))
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)
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self.play(FadeIn(fx_label))
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self.wait(2)
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self.play(
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MoveToTarget(fx_label),
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*map(FadeOut, [title, x_dot, x_label, arrow, fx_dot])
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)
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self.play(FadeOut(plane.coordinate_labels))
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self.wait()
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class WindingNumbersInInputOutputContext(PathContainingZero):
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CONFIG = {
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"in_loop_center_coords" : (-2, -1),
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"run_time" : 10,
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}
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def construct(self):
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self.remove(self.pi_creature)
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self.setup_planes()
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in_loop = Circle()
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in_loop.flip(RIGHT)
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# in_loop = Square(side_length = 2)
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in_loop.insert_n_anchor_points(100)
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in_loop.move_to(self.input_plane.coords_to_point(
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*self.in_loop_center_coords
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))
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in_loop.match_background_image_file(self.input_coloring)
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out_loop = in_loop.copy()
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out_loop.match_background_image_file(self.output_coloring)
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update_out_loop = ContinualUpdateFromFunc(
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out_loop,
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lambda m : m.set_points(in_loop.points).apply_function(self.point_function)
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)
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# self.add(update_out_loop)
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in_dot = Dot(radius = 0.04)
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update_in_dot = ContinualUpdateFromFunc(
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in_dot, lambda d : d.move_to(in_loop.point_from_proportion(1))
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)
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self.add(update_in_dot)
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out_arrow = Arrow(LEFT, RIGHT)
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update_out_arrow = ContinualUpdateFromFunc(
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out_arrow,
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lambda a : a.put_start_and_end_on(
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self.output_plane.coords_to_point(0, 0),
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out_loop.point_from_proportion(1)
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)
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)
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update_out_arrow_color = ContinualUpdateFromFunc(
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out_arrow,
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lambda a : a.highlight(rev_to_color(a.get_angle()/TAU))
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)
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self.add(update_out_arrow, update_out_arrow_color)
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words = TextMobject(
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"How many times does \\\\ the output wind around?"
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)
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label = self.output_plane.label
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words.move_to(label, UP)
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self.output_plane.remove(label)
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self.add(words)
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decimal = DecimalNumber(0)
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decimal.next_to(self.output_plane.get_corner(UP+RIGHT), DOWN+LEFT)
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self.play(
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ShowCreation(in_loop),
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ShowCreation(out_loop),
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ChangeDecimalToValue(decimal, 2),
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Animation(in_dot),
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run_time = self.run_time,
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rate_func = bezier([0, 0, 1, 1])
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)
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class SolveX5SkipToEnd(SolveX5MinusXMinus1):
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CONFIG = {
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"num_iterations" : 4,
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}
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def construct(self):
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self.force_skipping()
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SolveX5MinusXMinus1.construct(self)
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self.revert_to_original_skipping_status()
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mobjects = VGroup(*self.get_mobjects())
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lines = VGroup()
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rects = VGroup()
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for mob in mobjects:
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if mob.background_image_file is not None:
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mob.set_stroke(width = 2)
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lines.add(mob)
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elif isinstance(mob, Polygon):
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rects.add(mob)
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else:
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self.remove(mob)
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self.clear()
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self.add(lines, rects)
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class ZeroFoundOnBoundary(Scene):
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def construct(self):
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arrow = Vector(DOWN+LEFT, color = WHITE)
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words = TextMobject("Found zero on boundary!")
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words.next_to(arrow.get_start(), UP)
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words.shift(1.5*RIGHT)
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point = VectorizedPoint()
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point.next_to(arrow, DOWN+LEFT)
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self.play(Flash(point))
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self.play(
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GrowArrow(arrow),
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Write(words),
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)
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self.wait()
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class AllOfTheVideos(Scene):
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CONFIG = {
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"camera_config" : {
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"background_alpha" : 255,
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}
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}
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def construct(self):
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thumbnail_dir = os.path.join(MEDIA_DIR, "3b1b_videos/Winding/OldThumbnails")
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n = 4
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images = Group(*[
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ImageMobject(os.path.join(thumbnail_dir, file))
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for file in os.listdir(thumbnail_dir)[:n**2]
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])
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for image in images:
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rect = SurroundingRectangle(image, buff = 0)
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rect.set_stroke(WHITE, 1)
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image.add(rect)
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images.arrange_submobjects_in_grid(n, n, buff = 0)
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images.scale_to_fit_height(2*SPACE_HEIGHT)
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random.shuffle(images.submobjects)
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self.play(LaggedStart(FadeIn, images, run_time = 4))
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self.wait()
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class EndingCredits(Scene):
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def construct(self):
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text = TextMobject(
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@ -2999,8 +3250,33 @@ class EndScreen(PatreonEndScreen, PiCreatureScene):
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self.play(morty.change, mode)
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self.wait(2)
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class Thumbnail(SearchSpacePerimeterVsArea):
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CONFIG = {
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"num_iterations" : 18,
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"func" : plane_func_by_wind_spec(
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(-3, -1.3, 2), (0.1, 0.2, 1), (2.8, -2, 1)
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),
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}
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def construct(self):
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self.force_skipping()
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EquationSolver2d.construct(self)
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self.revert_to_original_skipping_status()
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mobjects = VGroup(*self.get_mobjects())
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lines = VGroup()
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rects = VGroup()
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get_length = lambda mob : max(mob.get_width(), mob.get_height())
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for mob in mobjects:
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if mob.background_image_file is not None:
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mob.set_stroke(width = 4*np.sqrt(get_length(mob)))
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lines.add(mob)
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elif isinstance(mob, Polygon):
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rects.add(mob)
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else:
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self.remove(mob)
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self.clear()
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self.add(lines)
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@ -216,6 +216,7 @@ class ComplexPlane(NumberPlane):
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num_mob.scale_to_fit_height(self.written_coordinate_height)
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num_mob.next_to(point, DOWN+LEFT, SMALL_BUFF)
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result.add(num_mob)
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self.coordinate_labels = result
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return result
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def add_coordinates(self, *numbers):
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