def construct(self):
        logo = ImageMobject("LogoGeneration", invert = False)
        name_mob = TextMobject("3Blue1Brown").center()
        name_mob.highlight("grey")
        name_mob.shift(2*DOWN)
        self.add(name_mob, logo)

        new_text = TextMobject(["with ", "Steven Strogatz"])
        new_text.next_to(name_mob, DOWN)
        self.play(*[
            ShimmerIn(part)
            for part in new_text.split()
        ])
        self.dither()
        with_word, steve = new_text.split()
        steve_copy = steve.copy().center().to_edge(UP)
        # logo.sort_points(lambda p : -np.linalg.norm(p))
        sort_by_color(logo)
        self.play(
            Transform(steve, steve_copy),
            DelayByOrder(Transform(logo, Point())),
            FadeOut(with_word),
            FadeOut(name_mob),
            run_time = 3
        )

    def construct(self):
        definition = TexMobject([
            "\\text{HC}(", "x", ")",
            "=\\lim_{n \\to \\infty}\\text{PHC}_n(", "x", ")"
        ])
        definition.to_edge(UP)
        definition.split()[1].highlight(BLUE)
        definition.split()[-2].highlight(BLUE)
        intro = TextMobject("Three things need to be proven")
        prove_that = TextMobject("Prove that HC is $\\dots$")
        prove_that.scale(0.7)
        prove_that.to_edge(LEFT)
        items = TextMobject([
            "\\begin{enumerate}",
            "\\item Well-defined: ",
            "Points on Pseudo-Hilbert-curves really do converge",
            "\\item A Curve: ",
            "HC is continuous",
            "\\item Space-filling: ",
            "Each point in the unit square is an output of HC",
            "\\end{enumerate}",
        ]).split()
        items[1].highlight(GREEN)
        items[3].highlight(YELLOW_C)
        items[5].highlight(MAROON)
        Mobject(*items).to_edge(RIGHT)

        self.add(definition)
        self.play(ShimmerIn(intro))
        self.dither()
        self.play(Transform(intro, prove_that))
        for item in items[1:-1]:
            self.play(ShimmerIn(item))
            self.dither()

    def construct(self):
        self.knob = Circle(color = BLUE_D)
        self.knob.add_line(UP, DOWN)
        self.knob.to_corner(UP+RIGHT)
        self.knob.shift(0.5*DOWN)
        self.last_angle = np.pi/2
        arrow = Vector(ORIGIN, RIGHT)
        arrow.next_to(self.knob, LEFT)
        words = TextMobject("Turn this knob over time to define the curve")
        words.next_to(arrow, LEFT)
        self.path = self.get_path()
        self.path.shift(1.5*DOWN)
        self.path.show()
        self.path.highlight(BLACK)        

        randy = Randolph()
        randy.scale(RANDY_SCALE_VAL)
        randy.shift(-randy.get_bottom())

        self.play(ShimmerIn(words))
        self.play(ShowCreation(arrow))
        self.play(ShowCreation(self.knob))
        self.dither()
        self.add(self.path)

        self.slide(randy, self.path)
        self.dither()

    def construct(self):
        glass = Region(lambda x, y : y < 0, color = BLUE_E)
        self.generate_start_and_end_points()
        straight = Line(self.start_point, self.end_point)
        slow = TextMobject("Slow")
        slow.rotate(np.arctan(straight.get_slope()))
        slow.shift(straight.points[int(0.7*straight.get_num_points())])
        slow.shift(0.5*DOWN)
        too_long = TextMobject("Too long")
        too_long.shift(UP)
        air = TextMobject("Air").shift(2*UP)
        water = TextMobject("Water").shift(2*DOWN)

        self.add(glass)
        self.play(GrowFromCenter(air))
        self.play(GrowFromCenter(water))
        self.dither()
        self.remove(air, water)
        ShowMultiplePathsScene.construct(self)
        self.play(
            Transform(self.path, straight)
        )
        self.dither()
        self.play(GrowFromCenter(slow))
        self.dither()
        self.remove(slow)
        self.leftmost.ingest_submobjects()
        self.play(Transform(self.path, self.leftmost, run_time = 3))
        self.dither()
        self.play(ShimmerIn(too_long))
        self.dither()

    def construct(self):
        words, s = TextMobject(["Pseudo-Hilbert Curve", "s"]).split()
        pre_words = TextMobject("Order 1")
        pre_words.next_to(words, LEFT, buff = 0.5)
        s.next_to(words, RIGHT, buff = 0.05, aligned_edge = DOWN)
        cluster = Mobject(pre_words, words, s)
        cluster.center()
        cluster.scale(0.7)
        cluster.to_edge(UP, buff = 0.3)
        cluster.highlight(GREEN)
        grid1 = Grid(1, 1)
        grid2 = Grid(2, 2)
        curve = HilbertCurve(order = 1)

        self.add(words, s)
        self.dither()
        self.play(Transform(
            s, pre_words, 
            path_func = path_along_arc(-np.pi/3)
        ))
        self.dither()
        self.play(ShowCreation(grid1))
        self.dither()
        self.play(ShowCreation(grid2))
        self.dither()
        kwargs = {
            "run_time" : 5,
            "rate_func" : None
        }
        self.play(ShowCreation(curve, **kwargs))
        self.dither()

    def construct(self):
        scale_factor = 0.9
        grid = Grid(4, 4, point_thickness = 1)
        curve = HilbertCurve(order = 2)
        for mob in grid, curve:
            mob.scale(scale_factor)
        words = TextMobject("""
            Sequence of curves is stable 
            $\\leftrightarrow$ existence of limit curve
        """, size = "\\normal")
        words.scale(1.25)
        words.to_edge(UP)

        self.add(curve, grid)
        self.dither()
        for n in range(3, 7):
            if n == 5:
                self.play(ShimmerIn(words))
            new_grid = Grid(2**n, 2**n, point_thickness = 1)
            new_curve = HilbertCurve(order = n)
            for mob in new_grid, new_curve:
                mob.scale(scale_factor)
            self.play(
                ShowCreation(new_grid),
                Animation(curve)
            )
            self.remove(grid)
            grid = new_grid
            self.play(Transform(curve, new_curve))
            self.dither()

    def construct(self):
        point_a, point_b = 3*LEFT, 3*RIGHT
        dots = []
        for point, char in [(point_a, "A"), (point_b, "B")]:
            dot = Dot(point)
            letter = TexMobject(char)
            letter.next_to(dot, UP+LEFT)
            dot.add(letter)
            dots.append(dot)

        path = ParametricFunction(
            lambda t : (t/2 + np.cos(t))*RIGHT + np.sin(t)*UP,
            start = -2*np.pi,
            end = 2*np.pi
        )
        path.scale(6/(2*np.pi))
        path.shift(point_a - path.points[0])
        path.highlight(RED)
        line = Line(point_a, point_b)
        words = TextMobject("Shortest path from $A$ to $B$")
        words.to_edge(UP)

        self.play(
            ShimmerIn(words),
            *map(GrowFromCenter, dots)
        )
        self.play(ShowCreation(path))
        self.play(Transform(
            path, line,
            path_func = path_along_arc(np.pi)
        ))
        self.dither()

    def construct(self):
        europe = ImageMobject("Europe", use_cache = False)
        self.add(europe)
        self.freeze_background()

        mathematicians = [
            ("Newton", [-1.75, -0.75, 0]),
            ("Jacob_Bernoulli",[-0.75, -1.75, 0]),
            ("Ehrenfried_von_Tschirnhaus",[0.5, -0.5, 0]),
            ("Gottfried_Wilhelm_von_Leibniz",[0.2, -1.75, 0]),
            ("Guillaume_de_L'Hopital", [-1.75, -1.25, 0]),
        ]

        for name, point in mathematicians:
            man = ImageMobject(name, invert = False)
            if name == "Newton":
                name = "Isaac_Newton"
            name_mob = TextMobject(name.replace("_", " "))
            name_mob.to_corner(UP+LEFT, buff=0.75)
            self.add(name_mob)
            man.scale_to_fit_height(4)
            mobject = Point(man.get_corner(UP+LEFT))
            self.play(Transform(mobject, man))
            man.scale(0.2)
            man.shift(point)
            self.play(Transform(mobject, man))
            self.remove(name_mob)

    def show_pendulum(self, arc_angle = np.pi, arc_color = GREEN):
        words = TextMobject(": Instantaneous center of rotation")
        words.next_to(self.c_label)
        line = Line(self.p_point, self.c_point)
        line_angle = line.get_angle()+np.pi
        line_length = line.get_length()
        line.add(self.p_dot.copy())
        line.get_center = lambda : self.c_point
        tangent_line = Line(3*LEFT, 3*RIGHT)
        tangent_line.rotate(line_angle-np.pi/2)
        tangent_line.shift(self.p_point)
        tangent_line.highlight(arc_color)
        right_angle_symbol = Mobject(
            Line(UP, UP+RIGHT),
            Line(UP+RIGHT, RIGHT)
        )
        right_angle_symbol.scale(0.3)
        right_angle_symbol.rotate(tangent_line.get_angle()+np.pi)
        right_angle_symbol.shift(self.p_point)

        self.play(ShowCreation(line))
        self.play(ShimmerIn(words))
        self.dither()
        pairs = [    
            (line_angle, arc_angle/2),
            (line_angle+arc_angle/2, -arc_angle),
            (line_angle-arc_angle/2, arc_angle/2),
        ]
        arcs = []
        for start, angle in pairs:
            arc = Arc(
                angle = angle,
                radius = line_length,
                start_angle = start,
                color = GREEN
            )
            arc.shift(self.c_point)
            self.play(
                ShowCreation(arc),
                ApplyMethod(
                    line.rotate_in_place, 
                    angle,
                    path_func = path_along_arc(angle)
                ),
                run_time = 2
            )
            arcs.append(arc)
        self.dither()
        self.play(Transform(arcs[1], tangent_line))
        self.add(tangent_line)
        self.play(ShowCreation(right_angle_symbol))
        self.dither()

        self.tangent_line = tangent_line
        self.right_angle_symbol = right_angle_symbol
        self.pc_line = line
        self.remove(words, *arcs)

 def add_title(self, title, scale_factor = 1.5, animate = False):
     if not isinstance(title, Mobject):
         title = TextMobject(title).scale(scale_factor)
     title.to_edge(UP)
     title.add_background_rectangle()
     if animate:
         self.play(Write(title))
     self.add_foreground_mobject(title)
     self.title = title
     return self

    def rearrange(self):
        sqrt_nudge = 0.2*LEFT        
        y, equals = self.y_equals.split()
        d, sin, squared, theta = self.y_expression.split()
        y_sqrt = TexMobject("\\sqrt{\\phantom{y}}")
        d_sqrt = y_sqrt.copy()
        y_sqrt.shift(y.get_center()+sqrt_nudge)
        d_sqrt.shift(d.get_center()+sqrt_nudge)

        self.play(
            ShimmerIn(y_sqrt),
            ShimmerIn(d_sqrt),
            ApplyMethod(squared.shift, 4*UP),
            ApplyMethod(theta.shift, 1.5* squared.get_width()*LEFT)
        )
        self.dither()
        y_sqrt.add(y)
        d_sqrt.add(d)
        sin.add(theta)

        sin_over = TexMobject("\\dfrac{\\phantom{\\sin(\\theta)}}{\\quad}")
        sin_over.next_to(sin, DOWN, 0.15)
        new_eq = equals.copy()
        new_eq.next_to(sin_over, LEFT)
        one_over = TexMobject("\\dfrac{1}{\\quad}")
        one_over.next_to(new_eq, LEFT)
        one_over.shift(
            (sin_over.get_bottom()[1]-one_over.get_bottom()[1])*UP
        )

        self.play(
            Transform(equals, new_eq),
            ShimmerIn(sin_over),
            ShimmerIn(one_over),
            ApplyMethod(
                d_sqrt.next_to, one_over, DOWN,
                path_func = path_along_arc(-np.pi)
            ),
            ApplyMethod(
                y_sqrt.next_to, sin_over, DOWN,
                path_func = path_along_arc(-np.pi)
            ),
            run_time = 2
        )
        self.dither()

        brace = Brace(d_sqrt, DOWN)
        constant = TextMobject("Constant")
        constant.next_to(brace, DOWN)

        self.play(
            GrowFromCenter(brace),
            ShimmerIn(constant)
        )

 def __init__(self, word, **kwargs):
     self.path = Cycloid(end_theta = np.pi)        
     word_mob = TextMobject(list(word))
     end_word = word_mob.copy()
     end_word.shift(-end_word.get_bottom())
     end_word.shift(self.path.get_corner(DOWN+RIGHT))
     end_word.shift(3*RIGHT)
     self.end_letters = end_word.split()
     for letter in word_mob.split():
         letter.center()
         letter.angle = 0
     unit_interval = np.arange(0, 1, 1./len(word))
     self.start_times = 0.5*(1-(unit_interval))
     Animation.__init__(self, word_mob, **kwargs)

    def construct(self):
        words = TextMobject("""
            It might come as a surprise how some well-known
            fractals can be described with curves.
        """)
        words.to_edge(UP)

        self.setup(Sierpinski)
        self.add(TextMobject("Speaking of other fractals\\dots"))
        self.dither(3)
        self.clear()
        self.play(ShimmerIn(words))
        for x in range(9):
            self.increase_order()