3b1b-videos/_2020/ldm.py

from imports_3b1b import *
import math


class StreamIntro(Scene):
    def construct(self):
        # Add logo
        logo = Logo()
        spikes = VGroup(*[
            spike
            for layer in logo.spike_layers
            for spike in layer
        ])
        self.add(*logo.family_members_with_points())

        # Add label
        label = TextMobject("The lesson will\\\\begin shortly")
        label.scale(2)
        label.next_to(logo, DOWN)
        self.add(label)

        self.camera.frame.move_to(DOWN)

        for spike in spikes:
            point = spike.get_start()
            spike.angle = angle_of_vector(point)

        anims = []
        for spike in spikes:
            anims.append(Rotate(
                spike, spike.angle * 28 * 2,
                about_point=ORIGIN,
                rate_func=linear,
            ))
        self.play(*anims, run_time=60 * 5)
        self.wait(20)


class OldStreamIntro(Scene):
    def construct(self):
        morty = Mortimer()
        morty.flip()
        morty.set_height(2)
        morty.to_corner(DL)
        self.play(PiCreatureSays(
            morty, "The lesson will\\\\begin soon.",
            bubble_kwargs={
                "height": 2,
                "width": 3,
            },
            target_mode="hooray",
        ))
        bound = AnimatedBoundary(morty.bubble.content, max_stroke_width=1)
        self.add(bound, morty.bubble, morty.bubble.content)
        self.remove(morty.bubble.content)
        morty.bubble.set_fill(opacity=0)

        self.camera.frame.scale(0.6, about_edge=DL)

        self.play(Blink(morty))
        self.wait(5)
        self.play(Blink(morty))
        self.wait(3)
        return

        text = TextMobject("The lesson will\\\\begin soon.")
        text.set_height(1.5)
        text.to_corner(DL, buff=LARGE_BUFF)
        self.add(text)


class QuadraticFormula(TeacherStudentsScene):
    def construct(self):
        formula = TexMobject(
            "\\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}",
        )
        formula.next_to(self.students, UP, buff=MED_LARGE_BUFF, aligned_edge=LEFT)
        self.add(formula)

        self.change_student_modes(
            "angry", "tired", "sad",
            look_at_arg=formula,
        )
        self.teacher_says(
            "It doesn't have\\\\to be this way.",
            bubble_kwargs={
                "width": 4,
                "height": 3,
            }
        )
        self.wait(5)
        self.change_student_modes(
            "pondering", "thinking", "erm",
            look_at_arg=formula
        )
        self.wait(12)


class SimplerQuadratic(Scene):
    def construct(self):
        tex = TexMobject("m \\pm \\sqrt{m^2 - p}")
        tex.set_stroke(BLACK, 12, background=True)
        tex.scale(1.5)
        self.add(tex)


class CosGraphs(Scene):
    def construct(self):
        axes = Axes(
            x_min=-0.75 * TAU,
            x_max=0.75 * TAU,
            y_min=-1.5,
            y_max=1.5,
            x_axis_config={
                "tick_frequency": PI / 4,
                "include_tip": False,
            },
            y_axis_config={
                "tick_frequency": 0.5,
                "include_tip": False,
                "unit_size": 1.5,
            }
        )

        graph1 = axes.get_graph(np.cos)
        graph2 = axes.get_graph(lambda x: np.cos(x)**2)

        graph1.set_stroke(YELLOW, 5)
        graph2.set_stroke(BLUE, 5)

        label1 = TexMobject("\\cos(x)")
        label2 = TexMobject("\\cos^2(x)")

        label1.match_color(graph1)
        label1.set_height(0.75)
        label1.next_to(axes.input_to_graph_point(-PI, graph1), DOWN)

        label2.match_color(graph2)
        label2.set_height(0.75)
        label2.next_to(axes.input_to_graph_point(PI, graph2), UP)

        for mob in [graph1, graph2, label1, label2]:
            mc = mob.copy()
            mc.set_stroke(BLACK, 10, background=True)
            self.add(mc)

        self.add(axes)
        self.add(graph1)
        self.add(graph2)
        self.add(label1)
        self.add(label2)

        self.embed()


class SineWave(Scene):
    def construct(self):
        w_axes = self.get_wave_axes()
        square, circle, c_axes = self.get_edge_group()

        self.add(w_axes)
        self.add(square, circle, c_axes)

        theta_tracker = ValueTracker(0)
        c_dot = Dot(color=YELLOW)
        c_line = Line(DOWN, UP, color=GREEN)
        w_dot = Dot(color=YELLOW)
        w_line = Line(DOWN, UP, color=GREEN)

        def update_c_dot(dot, axes=c_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            dot.move_to(axes.c2p(
                np.cos(theta),
                np.sin(theta),
            ))

        def update_c_line(line, axes=c_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            x = np.cos(theta)
            y = np.sin(theta)
            if y == 0:
                y = 1e-6
            line.put_start_and_end_on(
                axes.c2p(x, 0),
                axes.c2p(x, y),
            )

        def update_w_dot(dot, axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            dot.move_to(axes.c2p(theta, np.sin(theta)))

        def update_w_line(line, axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            x = theta
            y = np.sin(theta)
            if y == 0:
                y = 1e-6
            line.put_start_and_end_on(
                axes.c2p(x, 0),
                axes.c2p(x, y),
            )

        def get_partial_circle(circle=circle, tracker=theta_tracker):
            result = circle.copy()
            theta = tracker.get_value()
            result.pointwise_become_partial(
                circle, 0, clip(theta / TAU, 0, 1),
            )
            result.set_stroke(RED, width=3)
            return result

        def get_partial_wave(axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            graph = axes.get_graph(np.sin, x_min=0, x_max=theta, step_size=0.025)
            graph.set_stroke(BLUE, 3)
            return graph

        def get_h_line(axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            return Line(
                axes.c2p(0, 0),
                axes.c2p(theta, 0),
                stroke_color=RED
            )

        c_dot.add_updater(update_c_dot)
        c_line.add_updater(update_c_line)
        w_dot.add_updater(update_w_dot)
        w_line.add_updater(update_w_line)
        partial_circle = always_redraw(get_partial_circle)
        partial_wave = always_redraw(get_partial_wave)
        h_line = always_redraw(get_h_line)

        self.add(partial_circle)
        self.add(partial_wave)
        self.add(h_line)
        self.add(c_line, c_dot)
        self.add(w_line, w_dot)

        sin_label = TexMobject(
            "\\sin\\left(\\theta\\right)",
            tex_to_color_map={"\\theta": RED}
        )
        sin_label.next_to(w_axes.get_top(), UR)
        self.add(sin_label)

        self.play(
            theta_tracker.set_value, 1.25 * TAU,
            run_time=15,
            rate_func=linear,
        )

    def get_wave_axes(self):
        wave_axes = Axes(
            x_min=0,
            x_max=1.25 * TAU,
            y_min=-1.0,
            y_max=1.0,
            x_axis_config={
                "tick_frequency": TAU / 8,
                "unit_size": 1.0,
            },
            y_axis_config={
                "tick_frequency": 0.5,
                "include_tip": False,
                "unit_size": 1.5,
            }
        )
        wave_axes.y_axis.add_numbers(
            -1, 1, number_config={"num_decimal_places": 1}
        )
        wave_axes.to_edge(RIGHT, buff=MED_SMALL_BUFF)

        pairs = [
            (PI / 2, "\\frac{\\pi}{2}"),
            (PI, "\\pi"),
            (3 * PI / 2, "\\frac{3\\pi}{2}"),
            (2 * PI, "2\\pi"),
        ]
        syms = VGroup()
        for val, tex in pairs:
            sym = TexMobject(tex)
            sym.scale(0.5)
            sym.next_to(wave_axes.c2p(val, 0), DOWN, MED_SMALL_BUFF)
            syms.add(sym)
        wave_axes.add(syms)

        theta = TexMobject("\\theta")
        theta.set_color(RED)
        theta.next_to(wave_axes.x_axis.get_end(), UP)
        wave_axes.add(theta)

        return wave_axes

    def get_edge_group(self):
        axes_max = 1.25
        radius = 1.5
        axes = Axes(
            x_min=-axes_max,
            x_max=axes_max,
            y_min=-axes_max,
            y_max=axes_max,
            axis_config={
                "tick_frequency": 0.5,
                "include_tip": False,
                "numbers_with_elongated_ticks": [-1, 1],
                "tick_size": 0.05,
                "unit_size": radius,
            },
        )
        axes.to_edge(LEFT, buff=MED_LARGE_BUFF)

        background = SurroundingRectangle(axes, buff=MED_SMALL_BUFF)
        background.set_stroke(WHITE, 1)
        background.set_fill(GREY_E, 1)

        circle = Circle(radius=radius)
        circle.move_to(axes)
        circle.set_stroke(WHITE, 1)

        nums = VGroup()
        for u in 1, -1:
            num = Integer(u)
            num.set_height(0.2)
            num.set_stroke(BLACK, 3, background=True)
            num.next_to(axes.c2p(u, 0), DOWN + u * RIGHT, SMALL_BUFF)
            nums.add(num)

        axes.add(nums)

        return background, circle, axes


class CosWave(SineWave):
    CONFIG = {
        "include_square": False,
    }

    def construct(self):
        w_axes = self.get_wave_axes()
        square, circle, c_axes = self.get_edge_group()

        self.add(w_axes)
        self.add(square, circle, c_axes)

        theta_tracker = ValueTracker(0)
        c_dot = Dot(color=YELLOW)
        c_line = Line(DOWN, UP, color=GREEN)
        w_dot = Dot(color=YELLOW)
        w_line = Line(DOWN, UP, color=GREEN)

        def update_c_dot(dot, axes=c_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            dot.move_to(axes.c2p(
                np.cos(theta),
                np.sin(theta),
            ))

        def update_c_line(line, axes=c_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            x = np.cos(theta)
            y = np.sin(theta)
            line.set_points_as_corners([
                axes.c2p(0, y),
                axes.c2p(x, y),
            ])

        def update_w_dot(dot, axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            dot.move_to(axes.c2p(theta, np.cos(theta)))

        def update_w_line(line, axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            x = theta
            y = np.cos(theta)
            if y == 0:
                y = 1e-6
            line.set_points_as_corners([
                axes.c2p(x, 0),
                axes.c2p(x, y),
            ])

        def get_partial_circle(circle=circle, tracker=theta_tracker):
            result = circle.copy()
            theta = tracker.get_value()
            result.pointwise_become_partial(
                circle, 0, clip(theta / TAU, 0, 1),
            )
            result.set_stroke(RED, width=3)
            return result

        def get_partial_wave(axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            graph = axes.get_graph(np.cos, x_min=0, x_max=theta, step_size=0.025)
            graph.set_stroke(PINK, 3)
            return graph

        def get_h_line(axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            return Line(
                axes.c2p(0, 0),
                axes.c2p(theta, 0),
                stroke_color=RED
            )

        def get_square(line=c_line):
            square = Square()
            square.set_stroke(WHITE, 1)
            square.set_fill(MAROON_B, opacity=0.5)
            square.match_width(line)
            square.move_to(line, DOWN)
            return square

        def get_square_graph(axes=w_axes, tracker=theta_tracker):
            theta = tracker.get_value()
            graph = axes.get_graph(
                lambda x: np.cos(x)**2, x_min=0, x_max=theta, step_size=0.025
            )
            graph.set_stroke(MAROON_B, 3)
            return graph

        c_dot.add_updater(update_c_dot)
        c_line.add_updater(update_c_line)
        w_dot.add_updater(update_w_dot)
        w_line.add_updater(update_w_line)
        h_line = always_redraw(get_h_line)
        partial_circle = always_redraw(get_partial_circle)
        partial_wave = always_redraw(get_partial_wave)

        self.add(partial_circle)
        self.add(partial_wave)
        self.add(h_line)
        self.add(c_line, c_dot)
        self.add(w_line, w_dot)

        if self.include_square:
            self.add(always_redraw(get_square))
            self.add(always_redraw(get_square_graph))

        cos_label = TexMobject(
            "\\cos\\left(\\theta\\right)",
            tex_to_color_map={"\\theta": RED}
        )
        cos_label.next_to(w_axes.get_top(), UR)
        self.add(cos_label)

        self.play(
            theta_tracker.set_value, 1.25 * TAU,
            run_time=15,
            rate_func=linear,
        )


class CosSquare(CosWave):
    CONFIG = {
        "include_square": True
    }


class ComplexNumberPreview(Scene):
    def construct(self):
        plane = ComplexPlane(axis_config={"stroke_width": 4})
        plane.add_coordinates()

        z = complex(2, 1)
        dot = Dot()
        dot.move_to(plane.n2p(z))
        label = TexMobject("2+i")
        label.set_color(YELLOW)
        dot.set_color(YELLOW)
        label.next_to(dot, UR, SMALL_BUFF)
        label.set_stroke(BLACK, 5, background=True)

        line = Line(plane.n2p(0), plane.n2p(z))
        arc = Arc(start_angle=0, angle=np.log(z).imag, radius=0.5)

        self.add(plane)
        self.add(line, arc)
        self.add(dot)
        self.add(label)

        self.embed()


class ComplexMultiplication(Scene):
    def construct(self):
        # Add plane
        plane = ComplexPlane()
        plane.add_coordinates()

        z = complex(2, 1)
        z_dot = Dot(color=PINK)
        z_dot.move_to(plane.n2p(z))
        z_label = TexMobject("z")
        z_label.next_to(z_dot, UR, buff=0.5 * SMALL_BUFF)
        z_label.match_color(z_dot)

        self.add(plane)
        self.add(z_dot)
        self.add(z_label)

        # Show 1
        one_vect = Vector(RIGHT)
        one_vect.set_color(YELLOW)
        one_vect.target = Vector(plane.n2p(z))
        one_vect.target.match_style(one_vect)

        z_rhs = TexMobject("=", "z \\cdot 1")
        z_rhs[1].match_color(one_vect)
        z_rhs.next_to(z_label, RIGHT, 1.5 * SMALL_BUFF, aligned_edge=DOWN)
        z_rhs.set_stroke(BLACK, 3, background=True)

        one_label, i_label = [l for l in plane.coordinate_labels if l.get_value() == 1]

        self.play(GrowArrow(one_vect))
        self.wait()
        self.add(one_vect, z_dot)
        self.play(
            MoveToTarget(one_vect),
            TransformFromCopy(one_label, z_rhs),
        )
        self.wait()

        # Show i
        i_vect = Vector(UP, color=GREEN)
        zi_point = plane.n2p(z * complex(0, 1))
        i_vect.target = Vector(zi_point)
        i_vect.target.match_style(i_vect)
        i_vect_label = TexMobject("z \\cdot i")
        i_vect_label.match_color(i_vect)
        i_vect_label.set_stroke(BLACK, 3, background=True)
        i_vect_label.next_to(zi_point, UL, SMALL_BUFF)

        self.play(GrowArrow(i_vect))
        self.wait()
        self.play(
            MoveToTarget(i_vect),
            TransformFromCopy(i_label, i_vect_label),
            run_time=1,
        )
        self.wait()

        self.play(
            TransformFromCopy(one_vect, i_vect.target, path_arc=-90 * DEGREES),
        )
        self.wait()

        # Transform plane
        plane.generate_target()
        for mob in plane.target.family_members_with_points():
            if isinstance(mob, Line):
                mob.set_stroke(GREY, opacity=0.5)
        new_plane = ComplexPlane(faded_line_ratio=0)

        self.remove(plane)
        self.add(plane, new_plane, *self.mobjects)

        new_plane.generate_target()
        new_plane.target.apply_complex_function(lambda w, z=z: w * z)

        self.play(
            MoveToTarget(plane),
            MoveToTarget(new_plane),
            run_time=6,
            rate_func=there_and_back_with_pause
        )
        self.wait()

        # Show Example Point
        w = complex(2, -1)
        w_dot = Dot(plane.n2p(w), color=WHITE)
        one_vects = VGroup(*[Vector(RIGHT) for x in range(2)])
        one_vects.arrange(RIGHT, buff=0)
        one_vects.move_to(plane.n2p(0), LEFT)
        one_vects.set_color(YELLOW)
        new_i_vect = Vector(DOWN)
        new_i_vect.move_to(plane.n2p(2), UP)
        new_i_vect.set_color(GREEN)
        vects = VGroup(*one_vects, new_i_vect)
        vects.set_opacity(0.8)

        w_group = VGroup(*vects, w_dot)
        w_group.target = VGroup(
            one_vect.copy().set_opacity(0.8),
            one_vect.copy().shift(plane.n2p(z)).set_opacity(0.8),
            i_vect.copy().rotate(PI, about_point=ORIGIN).shift(2 * plane.n2p(z)).set_opacity(0.8),
            Dot(plane.n2p(w * z), color=WHITE)
        )

        self.play(FadeInFromLarge(w_dot))
        self.wait()
        self.play(ShowCreation(vects))
        self.wait()

        self.play(
            MoveToTarget(plane),
            MoveToTarget(new_plane),
            MoveToTarget(w_group),
            run_time=2,
            path_arc=np.log(z).imag,
        )
        self.wait()


class RotatePiCreature(Scene):
    def construct(self):
        randy = Randolph(mode="thinking")
        randy.set_height(6)

        plane = ComplexPlane(x_min=-12, x_max=12)
        plane.add_coordinates()

        self.camera.frame.move_to(3 * RIGHT)

        self.add(randy)
        self.wait()
        self.play(Rotate(randy, 30 * DEGREES, run_time=3))
        self.wait()
        self.play(Rotate(randy, -30 * DEGREES))

        self.add(plane, randy)
        self.play(
            ShowCreation(plane),
            randy.set_opacity, 0.75,
        )
        self.wait()

        dots = VGroup()
        for mob in randy.family_members_with_points():
            for point in mob.get_anchors():
                dot = Dot(point)
                dot.set_height(0.05)
                dots.add(dot)

        self.play(ShowIncreasingSubsets(dots))
        self.wait()

        label = VGroup(
            TexMobject("(x + iy)"),
            Vector(DOWN),
            TexMobject("(\\cos(30^\\circ) + i\\sin(30^\\circ))", "(x + iy)"),
        )
        label[2][0].set_color(YELLOW)
        label.arrange(DOWN)
        label.to_corner(DR)
        label.shift(3 * RIGHT)

        for mob in label:
            mob.add_background_rectangle()

        self.play(FadeIn(label))
        self.wait()

        randy.add(dots)
        self.play(Rotate(randy, 30 * DEGREES), run_time=3)
        self.wait()


class ExpMeaning(Scene):
    CONFIG = {
        "include_circle": True
    }

    def construct(self):
        # Plane
        plane = ComplexPlane(y_min=-6, y_max=6)
        plane.shift(1.5 * DOWN)
        plane.add_coordinates()
        if self.include_circle:
            circle = Circle(radius=1)
            circle.set_stroke(RED, 1)
            circle.move_to(plane.n2p(0))
            plane.add(circle)

        # Equation
        equation = TexMobject(
            "\\text{exp}(i\\theta) = ",
            "1 + ",
            "i\\theta + ",
            "{(i\\theta)^2 \\over 2} + ",
            "{(i\\theta)^3 \\over 6} + ",
            "{(i\\theta)^4 \\over 24} + ",
            "\\cdots",
            tex_to_color_map={
                "\\theta": YELLOW,
                "i": GREEN,
            },
        )
        equation.add_background_rectangle(buff=MED_SMALL_BUFF, opacity=1)
        equation.to_edge(UL, buff=0)

        # Label
        theta_tracker = ValueTracker(0)
        theta_label = VGroup(
            TexMobject("\\theta = "),
            DecimalNumber(0, num_decimal_places=4)
        )
        theta_decimal = theta_label[1]
        theta_decimal.add_updater(
            lambda m, tt=theta_tracker: m.set_value(tt.get_value())
        )
        theta_label.arrange(RIGHT, buff=SMALL_BUFF)
        theta_label.set_color(YELLOW)
        theta_label.add_to_back(BackgroundRectangle(
            theta_label,
            buff=MED_SMALL_BUFF,
            fill_opacity=1,
        ))
        theta_label.next_to(equation, DOWN, aligned_edge=LEFT, buff=0)

        # Vectors
        def get_vectors(n_vectors=20, plane=plane, tracker=theta_tracker):
            last_tip = plane.n2p(0)
            z = complex(0, tracker.get_value())
            vects = VGroup()
            colors = color_gradient([GREEN, YELLOW, RED], 6)
            for i, color in zip(range(n_vectors), it.cycle(colors)):
                vect = Vector(complex_to_R3(z**i / math.factorial(i)))
                vect.set_color(color)
                vect.shift(last_tip)
                last_tip = vect.get_end()
                vects.add(vect)
            return vects

        vectors = always_redraw(get_vectors)
        dot = Dot()
        dot.set_height(0.03)
        dot.add_updater(lambda m, vs=vectors: m.move_to(vs[-1].get_end()))

        self.add(plane)
        self.add(vectors)
        self.add(dot)
        self.add(equation)
        self.add(theta_label)

        self.play(
            theta_tracker.set_value, 1,
            run_time=3,
            rate_func=smooth,
        )
        self.wait()
        for target in PI, TAU:
            self.play(
                theta_tracker.set_value, target,
                run_time=10,
            )
            self.wait()

        self.embed()


class ExpMeaningWithoutCircle(ExpMeaning):
    CONFIG = {
        "include_circle": False,
    }


class PositionAndVelocityExample(Scene):
    def construct(self):
        plane = NumberPlane()

        self.add(plane)

        self.embed()


class EulersFormula(Scene):
    def construct(self):
        kw = {"tex_to_color_map": {"\\theta": YELLOW}}
        formula = TexMobject(
            "&e^{i\\theta} = \\\\ &\\cos\\left(\\theta\\right) + i\\cdot\\sin\\left(\\theta\\right)",
        )[0]
        formula[:4].scale(2, about_edge=UL)
        formula[:4].shift(SMALL_BUFF * RIGHT + MED_LARGE_BUFF * UP)
        VGroup(formula[2], formula[8], formula[17]).set_color(YELLOW)
        formula.scale(1.5)
        formula.set_stroke(BLACK, 5, background=True)
        self.add(formula)


class EtoILimit(Scene):
    def construct(self):
        tex = TexMobject(
            "\\lim_{n \\to \\infty} \\left(1 + \\frac{it}{n}\\right)^n",
        )[0]
        VGroup(tex[3], tex[12], tex[14]).set_color(YELLOW)
        tex[9].set_color(BLUE)
        tex.scale(1.5)
        tex.set_stroke(BLACK, 5, background=True)
        # self.add(tex)

        text = TextMobject("Interest rate\\\\of ", "$\\sqrt{-1}$")
        text[1].set_color(BLUE) 
        text.scale(1.5)
        text.set_stroke(BLACK, 5, background=True)
        self.add(text)


class ImaginaryInterestRates(Scene):
    def construct(self):
        plane = ComplexPlane(x_min=-20, x_max=20, y_min=-20, y_max=20)
        plane.add_coordinates()
        circle = Circle(radius=1)
        circle.set_stroke(YELLOW, 1)
        self.add(plane, circle)

        frame = self.camera.frame
        frame.save_state()
        frame.generate_target()
        frame.target.set_width(25)
        frame.target.move_to(8 * RIGHT + 2 * DOWN)

        dt_tracker = ValueTracker(1)

        def get_vectors(tracker=dt_tracker, plane=plane, T=8):
            dt = tracker.get_value()
            last_z = 1
            vects = VGroup()
            for t in np.arange(0, T, dt):
                next_z = last_z + complex(0, 1) * last_z * dt
                vects.add(Arrow(
                    plane.n2p(last_z),
                    plane.n2p(next_z),
                    buff=0,
                ))
                last_z = next_z
            vects.set_submobject_colors_by_gradient(YELLOW, GREEN, BLUE)
            return vects

        vects = get_vectors()

        line = Line()
        line.add_updater(lambda m, v=vects: m.put_start_and_end_on(
            ORIGIN, v[-1].get_start() if len(v) > 0 else RIGHT,
        ))

        self.add(line)
        self.play(
            ShowIncreasingSubsets(
                vects,
                rate_func=linear,
                int_func=np.ceil,
            ),
            MoveToTarget(
                frame,
                rate_func=squish_rate_func(smooth, 0.5, 1),
            ),
            run_time=8,
        )
        self.wait()
        self.play(FadeOut(line))

        self.remove(vects)
        vects = always_redraw(get_vectors)
        self.add(vects)
        self.play(
            Restore(frame),
            dt_tracker.set_value, 0.2,
            run_time=5,
        )
        self.wait()
        self.play(dt_tracker.set_value, 0.01, run_time=3)
        vects.clear_updaters()
        self.wait()

        theta_tracker = ValueTracker(0)

        def get_arc(tracker=theta_tracker):
            theta = tracker.get_value()
            arc = Arc(
                radius=1,
                stroke_width=3,
                stroke_color=RED,
                start_angle=0,
                angle=theta
            )
            return arc

        arc = always_redraw(get_arc)
        dot = Dot()
        dot.add_updater(lambda m, arc=arc: m.move_to(arc.get_end()))

        label = VGroup(
            DecimalNumber(0, num_decimal_places=3),
            TextMobject("Years")
        )
        label.arrange(RIGHT, aligned_edge=DOWN)
        label.move_to(3 * LEFT + 1.5 * UP)

        label[0].set_color(RED)
        label[0].add_updater(lambda m, tt=theta_tracker: m.set_value(tt.get_value()))

        self.add(BackgroundRectangle(label), label, arc, dot)
        for n in range(1, 5):
            target = n * PI / 2
            self.play(
                theta_tracker.set_value, target,
                run_time=3
            )
            self.wait(2)


class Logs(Scene):
    def construct(self):
        log = TexMobject(
            "&\\text{log}(ab) = \\\\ &\\text{log}(a) + \\text{log}(b)",
            tex_to_color_map={"a": BLUE, "b": YELLOW},
            alignment="",
        )
        
        log.scale(1.5)
        log.set_stroke(BLACK, 5, background=True)

        self.add(log)


class LnX(Scene):
    def construct(self):
        sym = TexMobject("\\ln(x)")
        sym.scale(3)
        sym.shift(UP)
        sym.set_stroke(BLACK, 5, background=True)

        word = TextMobject("Natural?")
        word.scale(1.5)
        word.set_color(YELLOW)
        word.set_stroke(BLACK, 5, background=True)
        word.next_to(sym, DOWN, buff=0.5)
        arrow = Arrow(word.get_top(), sym[0][1].get_bottom())

        self.add(sym, word, arrow)


class HarmonicSum(Scene):
    def construct(self):
        axes = Axes(
            x_min=0,
            x_max=13,
            y_min=0,
            y_max=1.25,
            y_axis_config={
                "unit_size": 4,
                "tick_frequency": 0.25,
            }
        )
        axes.to_corner(DL, buff=1)
        axes.x_axis.add_numbers()
        axes.y_axis.add_numbers(
            *np.arange(0.25, 1.25, 0.25),
            number_config={"num_decimal_places": 2},
        )
        self.add(axes)

        graph = axes.get_graph(lambda x: 1 / x, x_min=0.1, x_max=15)
        graph_fill = graph.copy()
        graph_fill.add_line_to(axes.c2p(15, 0))
        graph_fill.add_line_to(axes.c2p(1, 0))
        graph_fill.add_line_to(axes.c2p(1, 1))
        graph.set_stroke(WHITE, 3)
        graph_fill.set_fill(BLUE_E, 0.5)
        graph_fill.set_stroke(width=0)
        self.add(graph, graph_fill)

        bars = VGroup()
        bar_labels = VGroup()
        for x in range(1, 15):
            line = Line(axes.c2p(x, 0), axes.c2p(x + 1, 1 / x))
            bar = Rectangle()
            bar.set_fill(GREEN_E, 1)
            bar.replace(line, stretch=True)
            bars.add(bar)

            label = TexMobject(f"1 \\over {x}")
            label.set_height(0.7)
            label.next_to(bar, UP, SMALL_BUFF)
            bar_labels.add(label)

        bars.set_submobject_colors_by_gradient(GREEN_C, GREEN_E)
        bars.set_stroke(WHITE, 1)
        bars.set_fill(opacity=0.25)

        self.add(bars)
        self.add(bar_labels)


        self.embed()


class PowerTower(Scene):
    def construct(self):
        mob = TexMobject("4 = x^{x^{{x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}}}}")
        mob[0][-1].shift(0.1 * DL)
        mob[0][-2].shift(0.05 * DL)

        mob.set_height(4)
        mob.set_stroke(BLACK, 5, background=True)

        self.add(mob)


class ItoTheI(Scene):
    def construct(self):
        tex = TexMobject("i^i")
        # tex = TexMobject("\\sqrt{-1}^{\\sqrt{-1}}")
        tex.set_height(3)
        tex.set_stroke(BLACK, 8, background=True)
        self.add(tex)


class ComplexExponentialPlay(Scene):
    def setup(self):
        self.transform_alpha = 0

    def construct(self):
        # Plane
        plane = ComplexPlane(
            x_min=-2 * FRAME_WIDTH,
            x_max=2 * FRAME_WIDTH,
            y_min=-2 * FRAME_HEIGHT,
            y_max=2 * FRAME_HEIGHT,
        )
        plane.add_coordinates()
        self.add(plane)

        # R Dot
        r_dot = Dot(color=YELLOW)

        def update_r_dot(dot, point_tracker=self.mouse_drag_point):
            point = point_tracker.get_location()
            if abs(point[0]) < 0.1:
                point[0] = 0
            if abs(point[1]) < 0.1:
                point[1] = 0
            dot.move_to(point)

        r_dot.add_updater(update_r_dot)
        self.mouse_drag_point.move_to(plane.n2p(1))

        # Transformed sample dots
        def func(z, dot=r_dot, plane=plane):
            r = plane.p2n(dot.get_center())
            result = np.exp(r * z)
            if abs(result) > 20:
                result *= 20 / abs(result)
            return result

        sample_dots = VGroup()
        dot_template = Dot(radius=0.05)
        dot_template.set_opacity(0.8)
        spacing = 0.05
        for x in np.arange(-7, 7, spacing):
            dot = dot_template.copy()
            dot.set_color(TEAL)
            dot.z = x
            dot.move_to(plane.n2p(dot.z))
            sample_dots.add(dot)
        for y in np.arange(-6, 6, spacing):
            dot = dot_template.copy()
            dot.set_color(MAROON)
            dot.z = complex(0, y)
            dot.move_to(plane.n2p(dot.z))
            sample_dots.add(dot)

        special_values = [1, complex(0, 1), -1, complex(0, -1)]
        special_dots = VGroup(*[
            list(filter(lambda d: abs(d.z - x) < 0.01, sample_dots))[0]
            for x in special_values
        ])
        for dot in special_dots:
            dot.set_fill(opacity=1)
            dot.scale(1.2)
            dot.set_stroke(WHITE, 2)

        sample_dots.save_state()

        def update_sample(sample, f=func, plane=plane, scene=self):
            sample.restore()
            sample.apply_function_to_submobject_positions(
                lambda p: interpolate(
                    p,
                    plane.n2p(f(plane.p2n(p))),
                    scene.transform_alpha,
                )
            )
            return sample

        sample_dots.add_updater(update_sample)

        # Sample lines
        x_line = Line(plane.n2p(plane.x_min), plane.n2p(plane.x_max))
        y_line = Line(plane.n2p(plane.y_min), plane.n2p(plane.y_max))
        y_line.rotate(90 * DEGREES)
        x_line.set_color(GREEN)
        y_line.set_color(PINK)
        axis_lines = VGroup(x_line, y_line)
        for line in axis_lines:
            line.insert_n_curves(50)
        axis_lines.save_state()

        def update_axis_liens(lines=axis_lines, f=func, plane=plane, scene=self):
            lines.restore()
            lines.apply_function(
                lambda p: interpolate(
                    p,
                    plane.n2p(f(plane.p2n(p))),
                    scene.transform_alpha,
                )
            )
            lines.make_smooth()

        axis_lines.add_updater(update_axis_liens)

        # Labels
        labels = VGroup(
            TexMobject("f(1)"),
            TexMobject("f(i)"),
            TexMobject("f(-1)"),
            TexMobject("f(-i)"),
        )
        for label, dot in zip(labels, special_dots):
            label.set_height(0.3)
            label.match_color(dot)
            label.set_stroke(BLACK, 3, background=True)
            label.add_background_rectangle(opacity=0.5)

        def update_labels(labels, dots=special_dots, scene=self):
            for label, dot in zip(labels, dots):
                label.next_to(dot, UR, 0.5 * SMALL_BUFF)
                label.set_opacity(self.transform_alpha)

        labels.add_updater(update_labels)

        # Titles
        title = TexMobject(
            "f(x) =", "\\text{exp}(r\\cdot x)",
            tex_to_color_map={"r": YELLOW}
        )
        title.to_corner(UL)
        title.set_stroke(BLACK, 5, background=True)
        brace = Brace(title[1:], UP, buff=SMALL_BUFF)
        e_pow = TexMobject("e^{rx}", tex_to_color_map={"r": YELLOW})
        e_pow.add_background_rectangle()
        e_pow.next_to(brace, UP, buff=SMALL_BUFF)
        title.add(brace, e_pow)

        r_eq = VGroup(
            TexMobject("r=", tex_to_color_map={"r": YELLOW}),
            DecimalNumber(1)
        )
        r_eq.arrange(RIGHT, aligned_edge=DOWN)
        r_eq.next_to(title, DOWN, aligned_edge=LEFT)
        r_eq[0].set_stroke(BLACK, 5, background=True)
        r_eq[1].set_color(YELLOW)
        r_eq[1].add_updater(lambda m: m.set_value(plane.p2n(r_dot.get_center())))

        self.add(title)
        self.add(r_eq)

        # self.add(axis_lines)
        self.add(sample_dots)
        self.add(r_dot)
        self.add(labels)

        # Animations
        def update_transform_alpha(mob, alpha, scene=self):
            scene.transform_alpha = alpha

        frame = self.camera.frame
        frame.set_height(10)
        r_dot.clear_updaters()
        r_dot.move_to(plane.n2p(1))

        self.play(
            UpdateFromAlphaFunc(
                VectorizedPoint(),
                update_transform_alpha,
            )
        )
        self.play(r_dot.move_to, plane.n2p(2))
        self.wait()
        self.play(r_dot.move_to, plane.n2p(PI))
        self.wait()
        self.play(r_dot.move_to, plane.n2p(np.log(2)))
        self.wait()
        self.play(r_dot.move_to, plane.n2p(complex(0, np.log(2))), path_arc=90 * DEGREES, run_time=2)
        self.wait()
        self.play(r_dot.move_to, plane.n2p(complex(0, PI / 2)))
        self.wait()
        self.play(r_dot.move_to, plane.n2p(np.log(2)), run_time=2)
        self.wait()
        self.play(frame.set_height, 14)
        self.play(r_dot.move_to, plane.n2p(complex(np.log(2), PI)), run_time=3)
        self.wait()
        self.play(r_dot.move_to, plane.n2p(complex(np.log(2), TAU)), run_time=3)
        self.wait()

        self.embed()

    def on_mouse_scroll(self, point, offset):
        frame = self.camera.frame
        if self.zoom_on_scroll:
            factor = 1 + np.arctan(10 * offset[1])
            frame.scale(factor, about_point=ORIGIN)
        else:
            self.transform_alpha = clip(self.transform_alpha + 5 * offset[1], 0, 1)


class LDMEndScreen(PatreonEndScreen):
    CONFIG = {
        "scroll_time": 20,
    }


class ProbDiagram(Scene):
    def construct(self):
        square = Square(side_length=1)
        square.move_to(ORIGIN, DL)
        square.set_stroke(width=0)
        square.set_fill(BLUE_E, 1)

        frame = self.camera.frame
        frame.set_height(1.5)
        frame.move_to(square)

        tri = Polygon(ORIGIN, UP, UR)
        tri.set_fill(BLUE_E, 1)
        tri.set_stroke(width=0)

        tris = VGroup(tri)
        N = 1000
        for n in range(1, N):
            tri = Polygon(
                ORIGIN,
                RIGHT + UP / n,
                RIGHT + UP / (n + 1),
            )
            tri.set_stroke(width=0)
            color = BLUE_E if (n % 2 == 0) else RED_D
            tri.set_fill(color, 1)
            tris.add(tri)

        self.add(tris)