3b1b-manim/borsuk.py

from helpers import *

from mobject.tex_mobject import TexMobject
from mobject import Mobject
from mobject.image_mobject import ImageMobject
from mobject.vectorized_mobject import *

from animation.animation import Animation
from animation.transform import *
from animation.simple_animations import *
from animation.playground import *
from topics.geometry import *
from topics.characters import *
from topics.functions import *
from topics.fractals import *
from topics.number_line import *
from topics.combinatorics import *
from topics.numerals import *
from topics.three_dimensions import *
from scene import Scene
from camera import Camera, ShadingCamera
from mobject.svg_mobject import *
from mobject.tex_mobject import *

from eoc.graph_scene import GraphScene

class Jewel(VMobject):
    CONFIG = {
        "color" : WHITE,
        "fill_opacity" : 0.75,
        "stroke_width" : 0,
        "propogate_style_to_family" : True,
        "height" : 0.5,
        "num_equator_points" : 5,
        "sun_vect" : OUT+LEFT+UP,
    }
    def generate_points(self):
        for vect in OUT, IN:
            compass_vects = list(compass_directions(self.num_equator_points))
            if vect is IN:
                compass_vects.reverse()
            for vect_pair in adjascent_pairs(compass_vects):
                self.add(Polygon(vect, *vect_pair))
        self.scale_to_fit_height(self.height)
        self.rotate(-np.pi/2-np.pi/24, RIGHT)        
        self.rotate(-np.pi/12, UP)
        self.submobjects.sort(lambda m1, m2 : cmp(-m1.get_center()[2], -m2.get_center()[2]))
        return self

class Necklace(VMobject):
    CONFIG = {
        "height" : 2*SPACE_WIDTH - 1,
        "jewel_buff" : MED_SMALL_BUFF,
        "chain_color" : GREY,
        "default_colors" : [(4, BLUE), (6, WHITE), (4, GREEN)]
    }
    def __init__(self, *colors, **kwargs):
        digest_config(self, kwargs, locals())
        if len(colors) == 0:
            self.colors = self.get_default_colors()
        VMobject.__init__(self, **kwargs)

    def get_default_colors(self):
        result = list(it.chain(*[
            num*[color]
            for num, color in self.default_colors
        ]))
        random.shuffle(result)
        return result

    def generate_points(self):
        jewels = VGroup(*[
            Jewel(color = color)
            for color in self.colors
        ])
        jewels.arrange_submobjects(buff = self.jewel_buff)
        jewels.scale_to_fit_width(self.width)
        jewels.center()
        j_to_j_dist = (jewels[1].get_center()-jewels[0].get_center())[0]

        chain = Line(
            jewels[0].get_center() + j_to_j_dist*LEFT/2, 
            jewels[-1].get_center() + j_to_j_dist*RIGHT/2, 
            color = self.chain_color,
        )
        self.add(chain, *jewels)
        self.chain = chain
        self.jewels = jewels

################

class CheckOutMathologer(PiCreatureScene):
    CONFIG = {
        "logo_height" : 1.5,
        "screen_height" : 5,
        "channel_name" : "Mathologer",
        "logo_file" : "mathologer_logo",
        "logo_color" : None,
    }
    def construct(self):
        logo = ImageMobject(self.logo_file)
        logo.scale_to_fit_height(self.logo_height)
        logo.to_corner(UP+LEFT)
        if self.logo_color is not None:
            logo.highlight(self.logo_color)
            logo.stroke_width = 1
        name = TextMobject(self.channel_name)
        name.next_to(logo, RIGHT)

        rect = Rectangle(height = 9, width = 16)
        rect.scale_to_fit_height(self.screen_height)
        rect.next_to(logo, DOWN)
        rect.to_edge(LEFT)

        logo.save_state()
        logo.shift(DOWN)
        logo.highlight(BLACK)
        self.play(
            logo.restore,
            self.pi_creature.change_mode, "hooray",
        )
        self.play(
            ShowCreation(rect),
            Write(name)
        )
        self.dither(2)
        self.change_mode("happy")
        self.dither(2)

class IntroduceStolenNecklaceProblem(Scene):
    CONFIG = {
        "camera_class" : ShadingCamera,
        "jewel_colors" : [BLUE, GREEN, WHITE, RED],
        "num_per_jewel" : [8, 10, 4, 6],
        "num_shuffles" : 1,
        "necklace_center" : UP,
        "random_seed" : 2,
        "forced_binary_choices" : (0, 1, 0, 1, 0),
    }
    def construct(self):
        random.seed(self.random_seed)
        self.add_thieves()
        self.write_title()
        self.introduce_necklace()
        self.divvy_by_cutting_all()
        self.divvy_with_n_cuts()
        self.shuffle_jewels(self.necklace.jewels)
        self.divvy_with_n_cuts()

    def add_thieves(self):
        thieves = VGroup(
            Randolph(),
            Mortimer()
        )
        thieves.arrange_submobjects(RIGHT, buff = 4*LARGE_BUFF)
        thieves.to_edge(DOWN)
        thieves[0].make_eye_contact(thieves[1])

        self.add(thieves)
        self.thieves = thieves

    def write_title(self):
        title = TextMobject("Stolen necklace problem")
        title.to_edge(UP)
        self.play(
            Write(title),
            *[
                ApplyMethod(pi.look_at, title)
                for pi in self.thieves
            ]
        )
        self.title = title

    def introduce_necklace(self):
        necklace = self.get_necklace()
        jewels = necklace.jewels
        jewel_types = self.get_jewels_organized_by_type(jewels)

        enumeration_labels = VGroup()
        for jewel_type in jewel_types:
            num_mob = TexMobject(str(len(jewel_type)))
            jewel_copy = jewel_type[0].copy().scale(2)
            jewel_copy.next_to(num_mob)
            label = VGroup(num_mob, jewel_copy)
            enumeration_labels.add(label)
        enumeration_labels.arrange_submobjects(RIGHT, buff = LARGE_BUFF)
        enumeration_labels.to_edge(UP)

        self.play(
            FadeIn(
                necklace,
                submobject_mode = "lagged_start",
                run_time = 3
            ),
            *it.chain(*[
                [pi.change_mode, "conniving", pi.look_at, necklace]
                for pi in self.thieves
            ])
        )
        self.play(*[
            ApplyMethod(
                jewel.rotate_in_place, np.pi/6, UP, 
                rate_func = there_and_back
            )
            for jewel in jewels
        ])
        self.play(Blink(self.thieves[0]))
        self.dither()
        for x in range(self.num_shuffles):
            self.shuffle_jewels(jewels)
        self.play(FadeOut(self.title))
        for jewel_type, label in zip(jewel_types, enumeration_labels):
            jewel_type.submobjects.sort(lambda m1, m2: cmp(m1.get_center()[0], m2.get_center()[0]))
            jewel_type.save_state()
            jewel_type.generate_target()
            jewel_type.target.arrange_submobjects()
            jewel_type.target.scale(2)
            jewel_type.target.move_to(2*UP)
            self.play(
                MoveToTarget(jewel_type), 
                Write(label)
            )
            self.play(jewel_type.restore)
        self.play(Blink(self.thieves[1]))

        self.enumeration_labels = enumeration_labels
        self.jewel_types = jewel_types

    def divvy_by_cutting_all(self):
        enumeration_labels = self.enumeration_labels
        necklace = self.necklace
        jewel_types = self.jewel_types
        thieves = self.thieves

        both_half_labels = VGroup()
        for thief, vect in zip(self.thieves, [LEFT, RIGHT]):
            half_labels = VGroup()
            for label in enumeration_labels:
                tex, jewel = label
                num = int(tex.get_tex_string())
                half_label = VGroup(
                    TexMobject(str(num/2)),
                    jewel.copy()
                )
                half_label.arrange_submobjects()
                half_labels.add(half_label)
            half_labels.arrange_submobjects(DOWN)
            half_labels.scale_to_fit_height(thief.get_height())
            half_labels.next_to(
                thief, vect, 
                buff = MED_LARGE_BUFF,
                aligned_edge = DOWN
            )
            both_half_labels.add(half_labels)

        for half_labels in both_half_labels:
            self.play(ReplacementTransform(
                enumeration_labels.copy(), 
                half_labels
            ))
        self.play(*[ApplyMethod(pi.change_mode, "pondering") for pi in thieves])
        self.dither()

        for type_index, jewel_type in enumerate(jewel_types):
            jewel_type.save_state()
            jewel_type_copy = jewel_type.copy()
            n_jewels = len(jewel_type)
            halves = [
                VGroup(*jewel_type_copy[:n_jewels/2]),
                VGroup(*jewel_type_copy[n_jewels/2:]),
            ]
            for half, thief, vect in zip(halves, thieves, [RIGHT, LEFT]):
                half.arrange_submobjects(DOWN)
                half.next_to(
                    thief, vect, 
                    buff = SMALL_BUFF + type_index*half.get_width(),
                    aligned_edge = DOWN
                )
            self.play(
                Transform(jewel_type, jewel_type_copy),
                *[
                    ApplyMethod(thief.look_at, jewel_type_copy)
                    for thief in thieves
                ]
            )
        self.play(*it.chain(*[
            [thief.change_mode, "happy", thief.look_at, necklace]
            for thief in thieves
        ]))
        self.dither()
        self.play(*[
            jewel_type.restore
            for jewel_type in jewel_types
        ])
        self.play(*it.chain(*[
            [thief.change_mode, "confused", thief.look_at, necklace]
            for thief in thieves
        ]))

    def divvy_with_n_cuts(
        self, 
        with_thieves = True, 
        highlight_groups = True,
        show_matching_after_divvying = True,
        ):
        necklace = self.necklace
        jewel_types = self.jewel_types
        jewels = sorted(
            necklace.jewels, 
            lambda m1, m2 : cmp(m1.get_center()[0], m2.get_center()[0])
        )
        slice_indices, binary_choices = self.find_slice_indices(jewels, jewel_types)
        subgroups = [
            VGroup(*jewels[i1:i2])
            for i1, i2 in zip(slice_indices, slice_indices[1:])
        ]
        buff = (jewels[1].get_left()[0]-jewels[0].get_right()[0])/2
        v_lines = VGroup(*[
            DashedLine(UP, DOWN).next_to(group, RIGHT, buff = buff)
            for group in subgroups[:-1]
        ])
        strand_groups = [VGroup(), VGroup()]
        for group, choice in zip(subgroups, binary_choices):
            strand = Line(
                group[0].get_center(), group[-1].get_center(),
                color = necklace.chain.get_color()
            )
            strand.add(*group)
            strand_groups[choice].add(strand)
            self.add(strand)

        self.play(ShowCreation(v_lines))
        self.play(
            FadeOut(necklace.chain),
            *it.chain(*[
                map(Animation, group)
                for group in strand_groups
            ])
        )
        for group in strand_groups:
            group.save_state()
        self.play(
            strand_groups[0].shift, UP/2.,
            strand_groups[1].shift, DOWN/2.,
        )
        if with_thieves:
            self.play(*it.chain(*[
                [thief.change_mode, "happy", thief.look_at, self.necklace]
                for thief in thieves
            ]))
            self.play(Blink(thieves[1]))
        else:
            self.dither()

        if highlight_groups:
            for group in strand_groups:
                box = Rectangle(
                    width = group.get_width()+2*SMALL_BUFF,
                    height = group.get_height()+2*SMALL_BUFF,
                    stroke_width = 0,
                    fill_color = YELLOW,
                    fill_opacity = 0.3,
                )
                box.move_to(group)
                self.play(FadeIn(box))
                self.dither()
                self.play(FadeOut(box))

        self.dither()
        if show_matching_after_divvying:
            for jewel_type in jewel_types:
                self.play(
                    *[
                        ApplyMethod(jewel.scale_in_place, 1.5)
                        for jewel in jewel_type
                    ],
                    rate_func = there_and_back,
                    run_time = 2
                )
            self.dither()
        self.play(
            FadeOut(v_lines),
            FadeIn(necklace.chain),
            *[
                group.restore for group in strand_groups
            ]
        )
        self.remove(*strand_groups)
        self.add(necklace)

    ########

    def get_necklace(self, **kwargs):
        colors = reduce(op.add, [
            num*[color]
            for num, color in zip(self.num_per_jewel, self.jewel_colors)
        ])
        self.necklace = Necklace(*colors, **kwargs)
        self.necklace.shift(self.necklace_center)
        return self.necklace

    def get_jewels_organized_by_type(self, jewels):
        return [
            VGroup(*filter(lambda m : m.get_color() == color, jewels))
            for color in map(Color, self.jewel_colors)
        ]

    def shuffle_jewels(self, jewels, run_time = 2, path_arc = np.pi/2, **kwargs):
        shuffled_indices = range(len(jewels))
        random.shuffle(shuffled_indices)
        target_group = VGroup(*[
            jewel.copy().move_to(jewels[shuffled_indices[i]])
            for i, jewel in enumerate(jewels)
        ])
        self.play(Transform(
            jewels, target_group,
            run_time = run_time,            
            path_arc = path_arc,
            **kwargs
        ))

    def find_slice_indices(self, jewels, jewel_types):

        def jewel_to_type_number(jewel):
            for i, jewel_type in enumerate(jewel_types):
                if jewel in jewel_type:
                    return i
            raise Exception("Not in any jewel_types")
        type_numbers = map(jewel_to_type_number, jewels)

        n_types = len(jewel_types)
        for slice_indices in it.combinations(range(1, len(jewels)), n_types):
            slice_indices = [0] + list(slice_indices) + [len(jewels)]
            if self.forced_binary_choices is not None:
                all_binary_choices = [self.forced_binary_choices]
            else:
                all_binary_choices = it.product(*[range(2)]*(n_types+1))
            for binary_choices in all_binary_choices:
                subsets = [
                    type_numbers[i1:i2]
                    for i1, i2 in zip(slice_indices, slice_indices[1:])
                ]
                left_sets, right_sets = [
                    [
                        subset
                        for subset, index in zip(subsets, binary_choices)
                        if index == target_index
                    ]
                    for target_index in range(2)
                ]
                flat_left_set = np.array(list(it.chain(*left_sets)))
                flat_right_set = np.array(list(it.chain(*right_sets)))


                match_array = [
                    sum(flat_left_set == n) == sum(flat_right_set == n)
                    for n in range(n_types)
                ]
                if np.all(match_array):
                    return slice_indices, binary_choices
        raise Exception("No fair division found")

class FiveJewelCase(IntroduceStolenNecklaceProblem):
    CONFIG = {
        "jewel_colors" : [BLUE, GREEN, WHITE, RED, YELLOW],
        "num_per_jewel" : [6, 4, 4, 2, 8],
        "forced_binary_choices" : (0, 1, 0, 1, 0, 1),
    }
    def construct(self):
        random.seed(self.random_seed)
        self.add(self.get_necklace())
        jewels = self.necklace.jewels
        self.shuffle_jewels(jewels, run_time = 0)
        self.jewel_types = self.get_jewels_organized_by_type(jewels)     
        self.add_title()
        self.add_thieves()
        for thief in self.thieves:
            ApplyMethod(thief.change_mode, "pondering").update(1)
            thief.look_at(self.necklace)
        self.divvy_with_n_cuts()

    def add_title(self):
        n_cuts = len(self.jewel_colors)
        title = TextMobject(
            "%d jewel types, %d cuts"%(n_cuts, n_cuts)
        )
        title.to_edge(UP)
        self.add(title)

class SixJewelCase(FiveJewelCase):
    CONFIG = {
        "jewel_colors" : [BLUE, GREEN, WHITE, RED, YELLOW, MAROON_B],
        "num_per_jewel" : [6, 4, 4, 2, 2, 6],
        "forced_binary_choices" : (0, 1, 0, 1, 0, 1, 0),
    }

class DiscussApplicability(TeacherStudentsScene):
    def construct(self):
        self.teacher_says("""
            Minize sharding,
            allocate resources evenly
        """)
        self.change_student_modes(*["pondering"]*3)
        self.dither(2)

class ThreeJewelCase(FiveJewelCase):
    CONFIG = {
        "jewel_colors" : [BLUE, GREEN, WHITE],
        "num_per_jewel" : [6, 4, 8],
        "forced_binary_choices" : (0, 1, 0, 1),
    }

class RepeatedShuffling(IntroduceStolenNecklaceProblem):
    CONFIG = {
        "num_shuffles" : 5,
        "random_seed" : 3,
        "show_matching_after_divvying" : False,
    }
    def construct(self):
        random.seed(self.random_seed)
        self.add(self.get_necklace())
        jewels = self.necklace.jewels
        self.jewel_types = self.get_jewels_organized_by_type(jewels)     
        self.add_thieves()
        for thief in self.thieves:
            ApplyMethod(thief.change_mode, "pondering").update(1)
            thief.look_at(self.necklace)

        for x in range(self.num_shuffles):
            self.shuffle_jewels(jewels)
            self.divvy_with_n_cuts(
                show_matching_after_divvying = False
            )

class NowForTheTopology(TeacherStudentsScene):
    def construct(self):
        self.teacher_says("Now for the \\\\ topology")
        self.change_student_modes(*["hooray"]*3)
        self.dither(3)

class ExternallyAnimatedScene(Scene):
    def construct(self):
        raise Exception("Don't actually run this class.")

class SphereOntoPlaneIn3D(ExternallyAnimatedScene):
    pass

class DiscontinuousSphereOntoPlaneIn3D(ExternallyAnimatedScene):
    pass

class WriteWords(Scene):
    CONFIG = {
        "words" : "",
        "color" : WHITE,
    }
    def construct(self):
        words = TextMobject(self.words)
        words.highlight(self.color)
        words.scale_to_fit_width(2*SPACE_WIDTH-1)
        words.to_edge(DOWN)
        self.play(Write(words))
        self.dither(2)

class WriteNotAllowed(WriteWords):
    CONFIG = {
        "words" : "Not allowed",
        "color" : RED,
    }

class NonAntipodalCollisionIn3D(ExternallyAnimatedScene):
    pass

class AntipodalCollisionIn3D(ExternallyAnimatedScene):
    pass

class WriteBorsukUlam(WriteWords):
    CONFIG = {
        "words" : "Borsuk-Ulam Theorem",
    }

class WriteAntipodal(WriteWords):
    CONFIG = {
        "words" : "``Antipodal''",
        "color" : MAROON_B,
    }

class ProjectOntoEquatorIn3D(ExternallyAnimatedScene):
    pass

class ProjectOntoEquatorWithPolesIn3D(ExternallyAnimatedScene):
    pass

class ProjectAntipodalNonCollisionIn3D(ExternallyAnimatedScene):
    pass

class ShearThenProjectnOntoEquatorPolesMissIn3D(ExternallyAnimatedScene):
    pass

class ShearThenProjectnOntoEquatorAntipodalCollisionIn3D(ExternallyAnimatedScene):
    pass

class ClassicExample(TeacherStudentsScene):
    def construct(self):
        self.teacher_says("The classic example...")
        self.change_student_modes(*["happy"]*3)
        self.dither(2)

class AntipodalEarthPoints(ExternallyAnimatedScene):
    pass

class RotatingEarth(ExternallyAnimatedScene):
    pass

class TemperaturePressurePlane(GraphScene):
    CONFIG = {
        "x_labeled_nums" : [],
        "y_labeled_nums" : [],
        "x_axis_label" : "Temperature",
        "y_axis_label" : "Pressure",
        "graph_origin" : 2.5*DOWN + 2*LEFT,
        "corner_square_width" : 4,
        "example_point_coords" : (2, 5),
    }
    def construct(self):
        self.setup_axes()
        self.draw_corner_square()
        self.add_example_coordinates()
        self.wander_continuously()

    def draw_corner_square(self):
        square = Square(
            side_length = self.corner_square_width,
            stroke_color = WHITE,
            stroke_width = 2
        )
        square.to_corner(UP+LEFT, buff = 0)

        arrow = Arrow(
            square.get_right(), 
            self.coords_to_point(*self.example_point_coords)
        )

        self.play(ShowCreation(square))
        self.play(ShowCreation(arrow))

    def add_example_coordinates(self):
        dot = Dot(self.coords_to_point(*self.example_point_coords))
        dot.highlight(YELLOW)
        tex = TexMobject("(25^\\circ\\text{C}, 101 \\text{ kPa})")
        tex.next_to(dot, UP+RIGHT, buff = SMALL_BUFF)

        self.play(ShowCreation(dot))
        self.play(Write(tex))
        self.dither()
        self.play(FadeOut(tex))

    def wander_continuously(self):
        path = VMobject().set_points_smoothly([
            ORIGIN, 2*UP+RIGHT, 2*DOWN+RIGHT,
            5*RIGHT, 4*RIGHT+UP, 3*RIGHT+2*DOWN,
            DOWN+LEFT, 2*RIGHT
        ])
        point = self.coords_to_point(*self.example_point_coords)
        path.shift(point)

        path.highlight(GREEN)

        self.play(ShowCreation(path, run_time = 10, rate_func = None))
        self.dither()

class AlternateSphereSquishing(ExternallyAnimatedScene):
    pass

class AlternateAntipodalCollision(ExternallyAnimatedScene):
    pass

class AskWhy(TeacherStudentsScene):
    def construct(self):
        self.student_says("But...why?")
        self.change_student_modes("pondering", None, "thinking")
        self.play(self.get_teacher().change_mode, "happy")
        self.dither(3)

class PointOutVSauce(CheckOutMathologer):
    CONFIG = {
        "channel_name" : "",
        "logo_file" : "Vsauce_logo",
        "logo_height" : 1,
        "logo_color" : GREY,
    }

class WalkEquatorPostTransform(GraphScene):
    CONFIG = {
        "x_labeled_nums" : [],
        "y_labeled_nums" : [],
        "graph_origin" : 2.5*DOWN + 2*LEFT,
        "curved_arrow_color" : WHITE,
        "curved_arrow_radius" : 3,
        "num_great_arcs" : 10,
    }
    def construct(self):
        self.setup_axes()
        self.add_curved_arrow()
        self.great_arc_images = self.get_great_arc_images()

        self.walk_equator()
        self.walk_tilted_equator()
        self.draw_transverse_curve()
        self.walk_transverse_curve()

    def add_curved_arrow(self):
        arc = Arc(
            start_angle = 2*np.pi/3, angle = -np.pi/2, 
            radius = self.curved_arrow_radius,
            color = self.curved_arrow_color
        )
        arc.add_tip()
        arc.move_to(self.coords_to_point(0, 7))

        self.add(arc)

    def walk_equator(self):
        equator = self.great_arc_images[0]
        dots = VGroup(Dot(), Dot())
        dots.highlight(MAROON_B)
        dot_movement = self.get_arc_walk_dot_movement(equator, dots)
        dot_movement.update(0)

        self.play(ShowCreation(equator, run_time = 3))
        self.play(FadeIn(dots[0]))
        dots[1].set_fill(opacity = 0)
        self.play(dot_movement)
        self.play(dots[1].set_fill, None, 1)
        self.play(dot_movement)
        self.play(dot_movement)

        proportion = equator.collision_point_proportion
        self.play(self.get_arc_walk_dot_movement(
            equator, dots,
            rate_func = lambda t : 2*proportion*smooth(t)
        ))
        v_line = DashedLine(SPACE_HEIGHT*UP, SPACE_HEIGHT*DOWN)
        v_line.shift(dots.get_center()[0]*RIGHT)
        self.play(ShowCreation(v_line))
        self.dither()
        self.play(FadeOut(v_line))

        dots.save_state()
        equator.save_state()
        self.play(
            equator.fade,
            dots.fade
        )

        self.first_dots = dots

    def walk_tilted_equator(self):
        equator = self.great_arc_images[0]
        tilted_eq = self.great_arc_images[1]

        dots = VGroup(Dot(), Dot())
        dots.highlight(MAROON_B)
        dot_movement = self.get_arc_walk_dot_movement(tilted_eq, dots)
        dot_movement.update(0)

        self.play(ReplacementTransform(equator.copy(), tilted_eq))
        self.dither()
        self.play(FadeIn(dots))
        self.play(dot_movement)

        proportion = tilted_eq.collision_point_proportion
        self.play(self.get_arc_walk_dot_movement(
            tilted_eq, dots,
            rate_func = lambda t : 2*proportion*smooth(t)
        ))
        v_line = DashedLine(SPACE_HEIGHT*UP, SPACE_HEIGHT*DOWN)
        v_line.shift(dots.get_center()[0]*RIGHT)
        self.play(ShowCreation(v_line))
        self.dither()
        self.play(FadeOut(v_line))
        self.play(*map(FadeOut, [tilted_eq, dots]))

    def draw_transverse_curve(self):
        transverse_curve = self.get_transverse_curve(self.great_arc_images)
        dots = self.first_dots
        equator = self.great_arc_images[0]

        self.play(dots.restore)
        equator.restore()
        self.great_arc_images.fade()

        target_arcs = list(self.great_arc_images[1:])
        target_dots = []
        for arc in target_arcs:
            new_dots = dots.copy()
            for dot, point in zip(new_dots, arc.x_collision_points):
                dot.move_to(point)
            target_dots.append(new_dots)

        alt_eq = equator.copy()
        alt_eq.points = np.array(list(reversed(alt_eq.points)))
        alt_dots = dots.copy()
        alt_dots.submobjects.reverse()
        target_arcs += [alt_eq, alt_eq.copy()]
        target_dots += [alt_dots, alt_dots.copy()]

        equator_transform = Succession(*[
            Transform(equator, arc, rate_func = None)
            for arc in target_arcs
        ])
        dots_transform = Succession(*[
            Transform(dots, target, rate_func = None)
            for target in target_dots
        ])

        self.play(
            ShowCreation(transverse_curve, submobject_mode = "all_at_once"),
            equator_transform,
            dots_transform,
            run_time = 10,
            rate_func = None,
        )
        self.dither(2)

    def walk_transverse_curve(self):
        transverse_curve = self.get_transverse_curve(self.great_arc_images)
        dots = self.first_dots

        def dot_update(dots, alpha):
            for dot, curve in zip(dots, transverse_curve):
                dot.move_to(curve.point_from_proportion(alpha))
            return dots

        for x in range(2):
            self.play(
                UpdateFromAlphaFunc(dots, dot_update),
                run_time = 4
            )
        self.play(
            UpdateFromAlphaFunc(dots, dot_update),
            run_time = 4,
            rate_func = lambda t : 0.455*smooth(t)
        )
        self.play(
            dots.highlight, YELLOW,
            dots.scale_in_place, 1.2,
            rate_func = there_and_back
        )
        self.dither()

    #######

    def get_arc_walk_dot_movement(self, arc, dots, **kwargs):
        def dot_update(dots, alpha):
            dots[0].move_to(arc.point_from_proportion(0.5*alpha))
            dots[1].move_to(arc.point_from_proportion(0.5+0.5*alpha))
            return dots
        if "run_time" not in kwargs:
            kwargs["run_time"] = 5
        return UpdateFromAlphaFunc(dots, dot_update, **kwargs)

    def sphere_to_plane(self, point):
        x, y, z = point
        return np.array([
            x - 2*x*z + y + 1,
            y+0.5*y*np.cos(z*np.pi),
            0
        ])

    def sphere_point(self, portion_around_equator, equator_tilt = 0):
        theta = portion_around_equator*2*np.pi
        point = np.cos(theta)*RIGHT + np.sin(theta)*UP
        phi = equator_tilt*np.pi
        return rotate_vector(point, phi, RIGHT)

    def get_great_arc_images(self):
        curves = VGroup(*[
            ParametricFunction(
                lambda t : self.sphere_point(t, s)
            ).apply_function(self.sphere_to_plane)
            for s in np.arange(0, 1, 1./self.num_great_arcs)
            # for s in [0]
        ])
        curves.highlight(YELLOW)
        curves[0].highlight(RED)
        for curve in curves:
            antipodal_x_diff = lambda x : \
                curve.point_from_proportion(x+0.5)[0]-\
                curve.point_from_proportion(x)[0]
            last_x = 0                
            last_sign = np.sign(antipodal_x_diff(last_x))
            for x in np.linspace(0, 0.5, 100):
                sign = np.sign(antipodal_x_diff(x))
                if sign != last_sign:
                    mean = np.mean([last_x, x])
                    curve.x_collision_points = [
                        curve.point_from_proportion(mean),
                        curve.point_from_proportion(mean+0.5),
                    ]
                    curve.collision_point_proportion = mean
                    break
                last_x = x
                last_sign = sign
        return curves

    def get_transverse_curve(self, gerat_arc_images):
        points = list(it.chain(*[
            [
                curve.x_collision_points[i]
                for curve in gerat_arc_images
            ]
            for i in 0, 1
        ]))
        full_curve = VMobject(close_new_points = True)
        full_curve.set_points_smoothly(points + [points[0]])
        full_curve.highlight(MAROON_B)
        first_half = full_curve.copy().pointwise_become_partial(
            full_curve, 0, 0.5
        )
        second_half = first_half.copy().rotate_in_place(np.pi, RIGHT)
        broken_curve = VGroup(first_half, second_half)
        return broken_curve

class WalkAroundEquatorPreimage(ExternallyAnimatedScene):
    pass

class WalkTiltedEquatorPreimage(ExternallyAnimatedScene):
    pass

class FormLoopTransverseToEquator(ExternallyAnimatedScene):
    pass

class AntipodalWalkAroundTransverseLoop(ExternallyAnimatedScene):
    pass

class MentionGenerality(TeacherStudentsScene):
    CONFIG = {
        "camera_class" : ShadingCamera,
    }
    def construct(self):
        necklace = Necklace(width = SPACE_WIDTH)
        necklace.shift(2*UP)
        necklace.to_edge(RIGHT)
        arrow = TexMobject("\\Leftrightarrow")
        arrow.scale(2)
        arrow.next_to(necklace, LEFT)
        q_marks = TexMobject("???")
        q_marks.next_to(arrow, UP)
        arrow.add(q_marks)

        formula = TexMobject("f(\\textbf{x}) = f(-\\textbf{x})")
        formula.next_to(self.get_students(), UP, buff = LARGE_BUFF)
        formula.to_edge(LEFT, buff = LARGE_BUFF)

        self.play(
            self.teacher.change_mode, "raise_right_hand",
            self.teacher.look_at, arrow
        )
        self.play(
            FadeIn(necklace, run_time = 2, submobject_mode = "lagged_start"),
            Write(arrow),
            *[
                ApplyMethod(pi.look_at, arrow)
                for pi in self.get_everyone()
            ]
        )
        self.change_student_modes("pondering", "erm", "confused")
        self.dither()
        self.play(*[
            ApplyMethod(pi.look_at, arrow)
            for pi in self.get_everyone()
        ])
        self.play(Write(formula))
        self.dither(3)

class SimpleSphere(ExternallyAnimatedScene):
    pass

class PointsIn3D(Scene):
    CONFIG = {
        # "colors" : [RED, GREEN, BLUE],
        "colors" : color_gradient([GREEN, BLUE], 3),
    }
    def construct(self):
        sphere_def = TextMobject(
            "\\doublespacing Sphere in 3D: All", "$(x_1, x_2, x_3)$\\\\", 
            "such that", "$x_1^2 + x_2^2 + x_3^2 = 1$",
            alignment = "",
        )
        sphere_def.next_to(ORIGIN, DOWN)
        for index, subindex_list in (1, [1, 2, 4, 5, 7, 8]), (3, [0, 2, 4, 6, 8, 10]):
            colors = np.repeat(self.colors, 2)
            for subindex, color in zip(subindex_list, colors):
                sphere_def[index][subindex].highlight(color)

        point_ex = TextMobject(
            "For example, ", 
            "(", "0.41", ", ", "-0.58", ", ", "0.71", ")",
            arg_separator = ""
        )
        for index, color in zip([2, 4, 6], self.colors):
            point_ex[index].highlight(color)
        point_ex.scale(0.8)
        point_ex.next_to(
            sphere_def[1], UP+RIGHT,
            buff = 1.5*LARGE_BUFF
        )
        point_ex.shift_onto_screen()
        arrow = Arrow(sphere_def[1].get_top(), point_ex.get_bottom())

        self.play(Write(sphere_def[1]))
        self.play(ShowCreation(arrow))
        self.play(Write(point_ex))
        self.dither()
        self.play(
            Animation(sphere_def[1].copy(), remover = True),
            Write(sphere_def),
        )
        self.dither()

class AntipodalPairToBeGivenCoordinates(ExternallyAnimatedScene):
    pass

class WritePointCoordinates(Scene):
    CONFIG = {
        "colors" : color_gradient([GREEN, BLUE], 3),
        "corner" : DOWN+RIGHT,
    }
    def construct(self):
        coords = self.get_coords()
        arrow = Arrow(
            -self.corner, self.corner, 
            stroke_width = 8,
            color = MAROON_B
        )
        x_component = self.corner[0]*RIGHT
        y_component = self.corner[1]*UP
        arrow.next_to(
            coords.get_edge_center(y_component), 
            y_component, 
            aligned_edge = -x_component,
            buff = MED_SMALL_BUFF
        )

        group = VGroup(coords, arrow)
        group.scale(2)        
        group.to_corner(self.corner)


        self.play(FadeIn(coords))
        self.play(ShowCreation(arrow))
        self.dither()

    def get_coords(self):
        coords = TexMobject(
            "(", "0.41", ", ", "-0.58", ", ", "0.71", ")",
            arg_separator = ""
        )
        for index, color in zip([1, 3, 5], self.colors):
            coords[index].highlight(color)
        return coords

class WriteAntipodalCoordinates(WritePointCoordinates):
    CONFIG = {
        "corner" : UP+LEFT,
        "sign_color" : RED,
    }

    def get_coords(self):
        coords = TexMobject(
            "(", "-", "0.41", ", ", "+", "0.58", ", ", "-", "0.71", ")",
            arg_separator = ""
        )
        for index, color in zip([2, 5, 8], self.colors):
            coords[index].highlight(color)
            coords[index-1].highlight(self.sign_color)
        return coords

class GeneralizeBorsukUlam(Scene):
    CONFIG = {
        "n_dims" : 3,
        "boundary_colors" : [GREEN_B, BLUE],
        "output_boundary_color" : [MAROON_B, YELLOW],
        "negative_color" : RED,
    }
    def setup(self):
        self.colors = color_gradient(self.boundary_colors, self.n_dims)

    def construct(self):
        sphere_set = self.get_sphere_set()
        arrow = Arrow(LEFT, RIGHT)
        f = TexMobject("f")
        output_space = self.get_output_space()
        equation = self.get_equation()

        sphere_set.to_corner(UP+LEFT)
        arrow.next_to(sphere_set, RIGHT)
        f.next_to(arrow, UP)
        output_space.next_to(arrow, RIGHT)
        equation.to_edge(RIGHT)
        lhs = VGroup(*equation[:2])
        eq = equation[2]
        rhs = VGroup(*equation[3:])

        self.play(FadeIn(sphere_set))
        self.dither()
        self.play(
            ShowCreation(arrow),
            Write(f)
        )
        self.play(Write(output_space))
        self.dither()
        self.play(FadeIn(lhs))
        self.play(
            ReplacementTransform(lhs.copy(), rhs),
            Write(eq)
        )
        self.dither()

    def get_condition(self):
        squares = map(TexMobject, [
            "x_%d^2"%d
            for d in range(1, 1+self.n_dims)
        ])
        for square, color in zip(squares, self.colors):
            square[0].highlight(color)
            square[-1].highlight(color)
        plusses = [TexMobject("+") for x in range(self.n_dims-1)]
        plusses += [TexMobject("=1")]
        condition = VGroup(*it.chain(*zip(squares, plusses)))
        condition.arrange_submobjects(RIGHT)

        return condition

    def get_tuple(self):
        mid_parts = list(it.chain(*[
            ["x_%d"%d, ","]
            for d in range(1, self.n_dims)
        ]))
        tup = TexMobject(*["("] + mid_parts + ["x_%d"%self.n_dims, ")"])
        for index, color in zip(it.count(1, 2), self.colors):
            tup[index].highlight(color)

        return tup

    def get_negative_tuple(self):
        mid_parts = list(it.chain(*[
            ["-", "x_%d"%d, ","]
            for d in range(1, self.n_dims)
        ]))
        tup = TexMobject(*["("] + mid_parts + ["-", "x_%d"%self.n_dims, ")"])
        for index, color in zip(it.count(1, 3), self.colors):
            tup[index].highlight(self.negative_color)
            tup[index+1].highlight(color)

        return tup

    def get_output_space(self):
        return TextMobject("%dD space"%(self.n_dims-1))
        # n_dims = self.n_dims-1
        # colors = color_gradient(self.output_boundary_color, n_dims)
        # mid_parts = list(it.chain(*[
        #     ["y_%d"%d, ","]
        #     for d in range(1, n_dims)
        # ]))
        # tup = TexMobject(*["("] + mid_parts + ["y_%d"%n_dims, ")"])
        # for index, color in zip(it.count(1, 2), colors):
        #     tup[index].highlight(color)

        # return tup

    def get_equation(self):
        tup = self.get_tuple()
        neg_tup = self.get_negative_tuple()
        f1, f2 = [TexMobject("f") for x in range(2)]
        equals = TexMobject("=")
        equation = VGroup(f1, tup, equals, f2, neg_tup)
        equation.arrange_submobjects(RIGHT, buff = SMALL_BUFF)

        return equation

    def get_sphere_set(self):
        tup = self.get_tuple()
        such_that = TextMobject("such that")
        such_that.next_to(tup, RIGHT)
        condition = self.get_condition()
        condition.next_to(
            tup, DOWN, 
            buff = MED_LARGE_BUFF,
            aligned_edge = LEFT
        )
        group = VGroup(tup, such_that, condition)
        l_brace = Brace(group, LEFT)
        r_brace = Brace(group, RIGHT)
        group.add(l_brace, r_brace)

        return group

# class FourDBorsukUlam(GeneralizeBorsukUlam):
#     CONFIG = {
#         "n_dims" : 4,
#     }

# class FiveDBorsukUlam(GeneralizeBorsukUlam):
#     CONFIG = {
#         "n_dims" : 5,
#     }

class MakeTwoJewelCaseContinuous(IntroduceStolenNecklaceProblem):
    CONFIG = {
        "camera_class" : ShadingCamera,
        "jewel_colors" : [BLUE, GREEN],
        "num_per_jewel" : [8, 10],
        "random_seed" : 2,
        "forced_binary_choices" : (0, 1, 0),
        "show_matching_after_divvying" : True,
        "necklace_center" : ORIGIN,
        "necklace_width" : 2*SPACE_WIDTH - 3,
        "random_seed" : 0,
        "num_continuous_division_searches" : 4,
    }
    def construct(self):
        random.seed(self.random_seed)
        self.introduce_necklace()
        self.divvy_with_n_cuts()
        self.identify_necklace_with_unit_interval()
        self.color_necklace()
        self.find_continuous_fair_division()
        self.show_continuous_fair_division()
        self.highlight_continuous_groups()
        self.mention_equivalence_to_discrete_case()
        self.shift_divide_off_tick_marks()

    def introduce_necklace(self):
        self.get_necklace(
            width = self.necklace_width,
        )
        self.play(FadeIn(
            self.necklace,
            submobject_mode = "lagged_start"
        ))
        self.shuffle_jewels(self.necklace.jewels)
        jewel_types = self.get_jewels_organized_by_type(
            self.necklace.jewels
        )
        self.dither()
        self.count_jewel_types(jewel_types)
        self.dither()

        self.jewel_types = jewel_types

    def count_jewel_types(self, jewel_types):
        enumeration_labels = VGroup()
        for jewel_type in jewel_types:
            num_mob = TexMobject(str(len(jewel_type)))
            jewel_copy = jewel_type[0].copy()
            # jewel_copy.scale_to_fit_height(num_mob.get_height())
            jewel_copy.next_to(num_mob)
            label = VGroup(num_mob, jewel_copy)
            enumeration_labels.add(label)
        enumeration_labels.arrange_submobjects(RIGHT, buff = LARGE_BUFF)
        enumeration_labels.to_edge(UP)

        for jewel_type, label in zip(jewel_types, enumeration_labels):
            jewel_type.submobjects.sort(lambda m1, m2: cmp(m1.get_center()[0], m2.get_center()[0]))
            jewel_type.save_state()
            jewel_type.generate_target()
            jewel_type.target.arrange_submobjects()
            jewel_type.target.move_to(2*UP)
            self.play(
                MoveToTarget(jewel_type), 
                Write(label)
            )
            self.play(jewel_type.restore)

    def divvy_with_n_cuts(self):
        IntroduceStolenNecklaceProblem.divvy_with_n_cuts(
            self, 
            with_thieves = False, 
            highlight_groups = False,
            show_matching_after_divvying = True,
        )

    def identify_necklace_with_unit_interval(self):
        interval = UnitInterval(
            tick_frequency = 1./sum(self.num_per_jewel),
            tick_size = 0.2,
            numbers_with_elongated_ticks = [],
        )
        interval.stretch_to_fit_width(self.necklace.get_width())
        interval.move_to(self.necklace)
        tick_marks = interval.tick_marks
        tick_marks.set_stroke(WHITE, width = 2)

        brace = Brace(interval)
        brace_text = brace.get_text("Length = 1")

        self.play(
            GrowFromCenter(brace),
            Write(brace_text)
        )
        self.dither()
        self.play(
            ShowCreation(interval.tick_marks),
        )
        self.dither()

        self.tick_marks = interval.tick_marks
        self.length_brace = VGroup(brace, brace_text)

    def color_necklace(self):
        example_index = len(self.necklace.jewels)/2
        jewels = self.necklace.jewels
        chain = self.necklace.chain
        self.remove(self.necklace)
        self.add(chain, jewels)

        jewels.submobjects.sort(
            lambda m1, m2 : cmp(m1.get_center()[0], m2.get_center()[0])
        )
        remaining_indices = range(len(jewels))
        remaining_indices.remove(example_index)

        example_segment = self.color_necklace_by_indices(example_index)
        remaining_segments = self.color_necklace_by_indices(*remaining_indices)
        self.remove(chain)
        segments = VGroup(example_segment[0], *remaining_segments)
        segments.submobjects.sort(
            lambda m1, m2 : cmp(m1.get_center()[0], m2.get_center()[0])
        )
        segment_types = VGroup(*[
            VGroup(*filter(
                lambda m : m.get_color() == Color(color), 
                segments
            ))
            for color in self.jewel_colors
        ])

        for group in segment_types:
            length_tex = TexMobject("\\frac{%d}{%d}"%(
                len(group),
                len(jewels)
             ))
            length_tex.next_to(group, UP)
            length_tex.shift(UP)
            self.play(
                group.shift, UP,
                Write(length_tex, run_time = 1),
            )
            self.dither()
            self.play(
                group.shift, DOWN,
                FadeOut(length_tex)
            )
        self.play(FadeOut(self.length_brace))

        self.segments = segments

    def color_necklace_by_indices(self, *indices):
        chain = self.necklace.chain
        jewels = VGroup(*[
            self.necklace.jewels[i]
            for i in indices
        ])
        n_jewels = len(self.necklace.jewels)

        segments = VGroup(*[
            Line(
                chain.point_from_proportion(index/float(n_jewels)),
                chain.point_from_proportion((index+1)/float(n_jewels)),
                color = jewel.get_color()
            )
            for index, jewel in zip(indices, jewels)
        ])
        for jewel in jewels:
            jewel.save_state()

        self.play(jewels.shift, jewels.get_height()*UP)
        self.play(ReplacementTransform(
            jewels, segments,
            submobject_mode = "lagged_start",
            run_time = 2
        ))
        self.dither()
        return segments

    def find_continuous_fair_division(self):
        chain = self.necklace.chain
        n_jewels = len(self.necklace.jewels)

        slice_indices, ignore = self.find_slice_indices(
            self.necklace.jewels,
            self.jewel_types
        )
        cut_proportions = [
            sorted([random.random(), random.random()])
            for x in range(self.num_continuous_division_searches)
        ]
        cut_proportions.append([
            float(i)/n_jewels
            for i in slice_indices[1:-1]
        ])
        cut_points = [
            map(chain.point_from_proportion, pair)
            for pair in cut_proportions
        ]
        v_lines = VGroup(*[DashedLine(UP, DOWN) for x in range(2)])

        for line, point in zip(v_lines, cut_points[0]):
            line.move_to(point)

        self.play(ShowCreation(v_lines))
        self.dither()
        for target_points in cut_points[1:]:
            self.play(*[
                ApplyMethod(line.move_to, point)
                for line, point in zip(v_lines, target_points)
            ])
            self.dither()

        self.slice_indices = slice_indices
        self.v_lines = v_lines

    def show_continuous_fair_division(self):
        slice_indices = self.slice_indices

        groups = [
            VGroup(
                VGroup(*self.segments[i1:i2]),
                VGroup(*self.tick_marks[i1:i2]),
            )
            for i1, i2 in zip(slice_indices, slice_indices[1:])
        ]
        groups[-1].add(self.tick_marks[-1])
        vects = [[UP, DOWN][i] for i in self.forced_binary_choices]

        self.play(*[
            ApplyMethod(group.shift, 0.5*vect)
            for group, vect in zip(groups, vects)
        ])
        self.dither()

        self.groups = groups

    def highlight_continuous_groups(self):
        top_group = VGroup(self.groups[0], self.groups[2])
        bottom_group = self.groups[1]
        boxes = VGroup()
        for group in top_group, bottom_group:
            box = Rectangle(
                width = group.get_width()+2*SMALL_BUFF,
                height = group.get_height()+SMALL_BUFF,
                stroke_width = 0,
                fill_color = WHITE,
                fill_opacity = 0.25,
            )
            box.move_to(group)
            boxes.add(box)

        weight_description = VGroup(*[
            VGroup(
                TexMobject("\\frac{%d}{%d}"%(
                    len(jewel_type)/2, len(self.segments)
                )),
                Jewel(color = jewel_type[0].get_color())
            ).arrange_submobjects()
            for jewel_type in self.jewel_types
        ])
        weight_description.arrange_submobjects(buff = LARGE_BUFF)
        weight_description.next_to(boxes, UP, aligned_edge = LEFT)

        self.play(FadeIn(boxes))
        self.play(Write(weight_description))
        self.dither()

        self.highlight_box = boxes
        self.weight_description = weight_description

    def mention_equivalence_to_discrete_case(self):
        morty = Mortimer()
        morty.flip()
        morty.to_edge(DOWN)
        morty.shift(LEFT)
        self.play(FadeIn(morty))
        self.play(PiCreatureSays(
            morty, 
            """This is equivalent to
            the discrete case. """,
            bubble_kwargs = {
                "height" : 3, 
                "direction" : LEFT,
            }
        ))
        self.play(Blink(morty))
        self.dither()
        self.play(*map(FadeOut, [
            morty, morty.bubble, morty.bubble.content
        ]))
        
    def shift_divide_off_tick_marks(self):
        groups = self.groups
        slice_indices = self.slice_indices
        v_lines = self.v_lines

        left_segment = groups[1][0][0]
        left_tick = groups[1][1][0]
        right_segment = groups[-1][0][0]
        right_tick = groups[-1][1][0]

        segment_width = left_segment.get_width()

        for mob in left_segment, right_segment:
            mob.parts = VGroup(
                mob.copy().pointwise_become_partial(mob, 0, 0.5),
                mob.copy().pointwise_become_partial(mob, 0.5, 1),
            )
            self.remove(mob)
            self.add(mob.parts)
        restorers = [left_segment.parts, left_tick, right_segment.parts, right_tick]
        for mob in restorers:
            mob.save_state()

        self.play(v_lines.shift, segment_width*RIGHT/2)
        self.play(*[
            ApplyMethod(mob.shift, vect)
            for mob, vect in [
                (left_segment.parts[0], UP),
                (left_tick, UP),
                (right_segment.parts[0], DOWN),
                (right_tick, DOWN),
            ]
        ])
        self.dither()

        words = TextMobject("Cut part way through segment")
        words.to_edge(RIGHT)
        words.shift(2*UP)
        arrow1 = Arrow(words.get_bottom(), left_segment.parts[0].get_right())
        arrow2 = Arrow(words.get_bottom(), right_segment.parts[1].get_left())
        VGroup(words, arrow1, arrow2).highlight(RED)

        self.play(Write(words), ShowCreation(arrow1))
        self.dither()
        self.play(ShowCreation(arrow2))
        self.dither()
        self.play(*map(FadeOut, [words, arrow1, arrow2]))

        for line in v_lines:
            self.play(line.shift, segment_width*LEFT/2)
        self.play(*[mob.restore for mob in restorers])
        self.remove(left_segment.parts, right_segment.parts)
        self.add(left_segment, right_segment)

class ThinkAboutTheChoices(TeacherStudentsScene):
    def construct(self):
        self.teacher_says("""
            Think about the choices
            behind a division...
        """)
        self.change_student_modes(
            *["pondering"]*3,
            look_at_arg = SPACE_WIDTH*RIGHT+SPACE_HEIGHT*DOWN
        )
        self.dither(3)

class ChoicesInNecklaceCutting(Scene):
    CONFIG = {
        "num_continuous_division_searches" : 4,
        "final_num_pair" : [1./6, 1./2],
        "necklace_center" : DOWN,
        "thief_box_offset" : 1.2,
    }
    def construct(self):
        self.add_necklace()
        self.choose_places_to_cut()
        self.show_three_numbers_adding_to_one()
        self.make_binary_choice()

    def add_necklace(self):
        width, colors, num_per_color = [
            MakeTwoJewelCaseContinuous.CONFIG[key]
            for key in [
                "necklace_width", "jewel_colors", "num_per_jewel"
            ]
        ]
        color_list = list(it.chain(*[
            num*[color] 
            for num, color in zip(num_per_color, colors)
        ]))
        random.shuffle(color_list)

        interval = UnitInterval(
            tick_frequency = 1./sum(num_per_color),
            tick_size = 0.2,
            numbers_with_elongated_ticks = [],
        )
        interval.stretch_to_fit_width(width)
        interval.shift(self.necklace_center)
        tick_marks = interval.tick_marks
        tick_marks.set_stroke(WHITE, width = 2)

        segments = VGroup()
        for l_tick, r_tick, color in zip(tick_marks, tick_marks[1:], color_list):
            segment = Line(
                l_tick.get_center(),
                r_tick.get_center(),
                color = color
            )
            segments.add(segment)

        self.necklace = VGroup(segments, tick_marks)
        self.add(self.necklace)

        self.interval = interval        

    def choose_places_to_cut(self):
        v_lines = VGroup(*[DashedLine(UP, DOWN) for x in range(2)])
        num_pairs = [
            sorted([random.random(), random.random()])
            for x in range(self.num_continuous_division_searches)
        ] + [self.final_num_pair]

        point_pairs = [
            map(self.interval.number_to_point, num_pair)
            for num_pair in num_pairs
        ]
        
        for line, point in zip(v_lines, point_pairs[0]):
            line.move_to(point)
        self.play(ShowCreation(v_lines))
        for point_pair in point_pairs[1:]:
            self.dither()
            self.play(*[
                ApplyMethod(line.move_to, point)
                for line, point in zip(v_lines, point_pair)
            ])
        self.dither()

        self.division_points = list(it.chain(
            [self.interval.get_left()],
            point_pairs[-1],
            [self.interval.get_right()]
        ))

        self.v_lines = v_lines

    def show_three_numbers_adding_to_one(self):
        points = self.division_points
        braces = [
            Brace(Line(p1+SMALL_BUFF*RIGHT/2, p2+SMALL_BUFF*LEFT/2))
            for p1, p2 in zip(points, points[1:])
        ]
        for char, denom, brace in zip("abc", [6, 3, 2], braces):
            brace.label = brace.get_text("$%s$"%char)
            brace.concrete_label = brace.get_text("$\\frac{1}{%d}$"%denom)
            VGroup(
                brace.label,
                brace.concrete_label
            ).highlight(YELLOW)

        words = TextMobject(
            "1) Choose", "$a$, $b$, $c$", "so that", "$a+b+c = 1$"
        )
        words[1].highlight(YELLOW)
        words[3].highlight(YELLOW)
        words.to_corner(UP+LEFT)

        self.play(*it.chain(*[
            [GrowFromCenter(brace), Write(brace.label)]
            for brace in braces
        ]))
        self.play(Write(words))
        self.dither(2)
        self.play(*[
            ReplacementTransform(brace.label, brace.concrete_label)
            for brace in braces
        ])
        self.dither()
        self.wiggle_v_lines()
        self.dither()
        self.play(*map(FadeOut, list(braces) + [
            brace.concrete_label for brace in braces
        ]))

        self.choice_one_words = words

    def make_binary_choice(self):
        groups = self.get_groups()
        boxes, labels = self.get_boxes_and_labels()
        arrow_pairs, curr_arrows = self.get_choice_arrow_pairs(groups)
        words = TextMobject("2) Make a binary choice for each segment")
        words.next_to(
            self.choice_one_words, DOWN, 
            buff = MED_LARGE_BUFF, 
            aligned_edge = LEFT
        )

        self.play(Write(words))
        self.play(*map(FadeIn, [boxes, labels]))
        for binary_choices in it.product(*[[0, 1]]*3):
            self.play(*[
                ApplyMethod(group.move_to, group.target_points[choice])
                for group, choice in zip(groups, binary_choices)
            ] + [
                Transform(
                    curr_arrow, arrow_pair[choice],
                    path_arc = np.pi
                )
                for curr_arrow, arrow_pair, choice in zip(
                    curr_arrows, arrow_pairs, binary_choices
                )
            ])
            self.dither()

    ######

    def get_groups(self):
        segments, tick_marks = self.necklace
        n_segments = len(segments)
        indices = [0, n_segments/6, n_segments/2, n_segments]

        groups = [
            VGroup(
                VGroup(*segments[i1:i2]),
                VGroup(*tick_marks[i1:i2]),
            )
            for i1, i2 in zip(indices, indices[1:])
        ]
        for group, index in zip(groups, indices[1:]):
            group.add(tick_marks[index].copy())
        groups[-1][-1].add(tick_marks[-1])

        for group in groups:
            group.target_points = [
                group.get_center() + self.thief_box_offset*vect
                for vect in UP, DOWN
            ]

        return groups

    def get_boxes_and_labels(self):
        box = Rectangle(
            height = self.necklace.get_height()+SMALL_BUFF,
            width = self.necklace.get_width()+2*SMALL_BUFF,
            stroke_width = 0,
            fill_color = WHITE,
            fill_opacity = 0.25
        )
        box.move_to(self.necklace)

        boxes = VGroup(*[
            box.copy().shift(self.thief_box_offset*vect)
            for vect in UP, DOWN
        ])
        labels = VGroup(*[
            TextMobject(
                "Thief %d"%(i+1)
            ).next_to(box, UP, aligned_edge = RIGHT)
            for i, box in enumerate(boxes)
        ])
        return boxes, labels

    def get_choice_arrow_pairs(self, groups):
        arrow = TexMobject("\\uparrow")
        arrow_pairs = [
            [arrow.copy(), arrow.copy().rotate(np.pi)]
            for group in groups
        ]
        pre_arrow_points = [
            VectorizedPoint(group.get_center())
            for group in groups
        ]
        for point, arrow_pair in zip(pre_arrow_points, arrow_pairs):
            for arrow, color in zip(arrow_pair, [GREEN, RED]):
                arrow.highlight(color)
                arrow.move_to(point.get_center())
        return arrow_pairs, pre_arrow_points

    def wiggle_v_lines(self):
        self.play(
            *it.chain(*[
                [
                    line.rotate_in_place, np.pi/12, vect,
                    line.highlight, RED
                ]
                for line, vect in zip(self.v_lines, [OUT, IN])
            ]),
            rate_func = wiggle
        )

class CompareThisToSphereChoice(TeacherStudentsScene):
    def construct(self):
        self.teacher_says("""
            Compare this to choosing
            a point on the sphere.
        """)
        self.change_student_modes(
            *["pondering"]*3,
            look_at_arg = SPACE_WIDTH*RIGHT+SPACE_HEIGHT*DOWN
        )
        self.dither(3)

class ChoicesForSpherePoint(GeneralizeBorsukUlam):
    def construct(self):
        self.add_sphere_set()
        self.initialize_words()
        self.play(Write(self.choice_one_words))
        self.dither()
        self.show_example_choices()
        self.show_binary_choices()

    def get_tuple(self):
        tup = TexMobject("(x, y, z)")
        for i, color in zip([1, 3, 5], self.colors):
            tup[i].highlight(color)
        return tup

    def get_condition(self):
        condition = TexMobject("x^2+y^2+z^2 = 1")
        for i, color in zip([0, 3, 6], self.colors):
            VGroup(*condition[i:i+2]).highlight(color)
        return condition

    def add_sphere_set(self):
        sphere_set = self.get_sphere_set()
        sphere_set.scale(0.7)
        sphere_set.to_edge(RIGHT)
        sphere_set.shift(UP)

        self.add(sphere_set)

    def initialize_words(self):
        choice_one_words = TextMobject(
            "1) Choose", "$x^2$, $y^2$, $z^2$",
            "so that", "$x^2+y^2+z^2 = 1$"
        )
        for i in 1, 3:
            for j, color in zip([0, 3, 6], self.colors):
                VGroup(*choice_one_words[i][j:j+2]).highlight(color)
        choice_one_words.to_corner(UP+LEFT)

        choice_two_words = TextMobject(
            "2) Make a binary choice for each one"
        )
        choice_two_words.next_to(
            choice_one_words, DOWN, 
            buff = MED_LARGE_BUFF,
            aligned_edge = LEFT
        )

        self.choice_one_words = choice_one_words
        self.choice_two_words = choice_two_words

    def show_example_choices(self):
        choices = VGroup(*[
            TexMobject(*tex).highlight(color)
            for color, tex in zip(self.colors, [
                ("x", "^2 = ", "1/6"),
                ("y", "^2 = ", "1/3"),
                ("z", "^2 = ", "1/2"),
            ])
        ])
        choices.arrange_submobjects(
            DOWN, 
            buff = LARGE_BUFF,
            aligned_edge = LEFT
        )
        choices.scale_to_fit_height(SPACE_HEIGHT)
        choices.to_edge(LEFT)
        choices.shift(DOWN)

        self.play(FadeIn(
            choices,
            run_time = 2,
            submobject_mode = "lagged_start"
        ))
        self.dither()

        self.choices = choices

    def show_binary_choices(self):
        for choice in self.choices:
            var_tex = choice.expression_parts[0]
            frac_tex = choice.expression_parts[2]
            sqrts = VGroup(*[
                TexMobject(
                    var_tex + "=" + sign + \
                    "\\sqrt{%s}"%frac_tex)
                for sign in ["+", "-"]
            ])
            for sqrt in sqrts:
                sqrt.scale(0.6)
            sqrts.arrange_submobjects(DOWN)
            sqrts.next_to(choice, RIGHT, buff = LARGE_BUFF)
            sqrts.highlight(choice.get_color())

            arrows = VGroup(*[
                Arrow(
                    choice.get_right(), sqrt.get_left(), 
                    color = WHITE,
                    tip_length = 0.1,
                    buff = SMALL_BUFF
                )
                for sqrt in sqrts
            ])

            self.play(ShowCreation(arrows))
            self.play(FadeIn(sqrts, submobject_mode = "lagged_start"))
        self.play(Write(self.choice_two_words))
        self.dither()

class NecklaceDivisionSphereAssociation(ChoicesInNecklaceCutting):
    CONFIG = {
        "xyz_colors" : color_gradient([GREEN_B, BLUE], 3),
        "necklace_center" : DOWN,
        "thief_box_offset" : 1.6,
        "denoms" : [6, 3, 2],
    }
    def construct(self):
        self.add_necklace()
        self.add_sphere_point_label()
        self.choose_places_to_cut()
        self.add_braces()
        self.add_boxes_and_labels()
        self.show_binary_choice_association()
        self.ask_about_antipodal_pairs()

    def add_sphere_point_label(self):
        label = TextMobject(
            "$(x, y, z)$",
            "such that",
            "$x^2 + y^2 + z^2 = 1$"
        )
        for i, j_list in (0, [1, 3, 5]), (2, [0, 3, 6]):
            for j, color in zip(j_list, self.xyz_colors):
                label[i][j].highlight(color)
        label.to_corner(UP+RIGHT)

        ghost_sphere_point = VectorizedPoint()
        ghost_sphere_point.to_corner(UP+LEFT, buff = LARGE_BUFF)
        ghost_sphere_point.shift(2*RIGHT)

        arrow = Arrow(
            label.get_left(), ghost_sphere_point,
            color = WHITE
        )

        self.add(label, arrow)

        self.sphere_point_label = label

    def add_braces(self):
        points = self.division_points
        braces = [
            Brace(
                Line(p1+SMALL_BUFF*RIGHT/2, p2+SMALL_BUFF*LEFT/2),
                UP
            )
            for p1, p2 in zip(points, points[1:])
        ]
        for char, brace, color, denom in zip("xyz", braces, self.xyz_colors, self.denoms):
            brace.label = brace.get_text(
                "$%s^2$"%char, "$= 1/%d$"%denom,
                buff = SMALL_BUFF
            )
            brace.label.highlight(color)
            if brace.label.get_right()[0] > brace.get_right()[0]:
                brace.label.next_to(
                    brace, UP, buff = SMALL_BUFF,
                    aligned_edge = RIGHT
                )

        self.play(*it.chain(
            map(GrowFromCenter, braces),
            [Write(brace.label) for brace in braces]
        ))
        self.dither()

        self.braces = braces

    def add_boxes_and_labels(self):
        boxes, labels = self.get_boxes_and_labels()
        self.play(*map(FadeIn, [boxes, labels]))
        self.dither()

    def show_binary_choice_association(self):
        groups = self.get_groups()
        self.swapping_anims = []
        final_choices = [1, 0, 1]
        quads = zip(self.braces, self.denoms, groups, final_choices)
        for brace, denom, group, final_choice in quads:
            char = brace.label.args[0][1]
            choices = [
                TexMobject(
                    char, "=", sign, "\\sqrt{\\frac{1}{%d}}"%denom
                )
                for sign in "+", "-"
            ]
            for choice, color in zip(choices, [GREEN, RED]):
                # choice[0].highlight(brace.label.get_color())
                choice[2].highlight(color)
                choice.scale(0.8)
                choice.move_to(group)
                if choice.get_width() > 0.8*group.get_width():
                    choice.next_to(group.get_right(), LEFT, buff = MED_SMALL_BUFF)
            original_choices = [m.copy() for m in choices]

            self.play(
                ReplacementTransform(
                    VGroup(brace.label[0], brace, brace.label[1]), 
                    choices[0]
                ),
                group.move_to, group.target_points[0]
            )
            self.dither()
            self.play(
                Transform(*choices),
                group.move_to, group.target_points[1]
            )
            self.dither()
            if final_choice == 0:
                self.play(
                    Transform(choices[0], original_choices[0]),
                    group.move_to, group.target_points[0]
                )
            self.swapping_anims += [
                Transform(choices[0], original_choices[1-final_choice]),
                group.move_to, group.target_points[1-final_choice]
            ]

    def ask_about_antipodal_pairs(self):
        question = TextMobject("What do antipodal points signify?")
        question.move_to(self.sphere_point_label, LEFT)
        question.highlight(MAROON_B)
        antipodal_tex = TexMobject(
            "(x, y, z) \\rightarrow (-x, -y, -z)"
        )
        antipodal_tex.next_to(question, DOWN, aligned_edge = LEFT)

        self.play(FadeOut(self.sphere_point_label))
        self.play(FadeIn(question))
        self.dither()
        self.play(Write(antipodal_tex))
        self.dither()
        self.wiggle_v_lines()
        self.dither()
        self.play(*self.swapping_anims)
        self.dither()

class TotalLengthOfEachJewelEquals(NecklaceDivisionSphereAssociation):
    CONFIG = {
        "random_seed" : 2,
        "hard_coded_fair_division_indices" : [],
    }
    def construct(self):
        random.seed(self.random_seed)
        self.add_necklace()
        self.add_boxes_and_labels()
        self.find_fair_division()

    def find_fair_division(self):
        pass