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Showed sphere/necklace-division association
This commit is contained in:
Grant Sanderson
parent
44dc2f6968
commit
fd5dc23839
3 changed files with 453 additions and 9 deletions
457
borsuk.py
457
borsuk.py
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@ -1079,13 +1079,14 @@ class WriteAntipodalCoordinates(WritePointCoordinates):
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class GeneralizeBorsukUlam(Scene):
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CONFIG = {
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"n_dims" : 3,
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"boundary_colors" : [GREEN, BLUE],
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"boundary_colors" : [GREEN_B, BLUE],
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"output_boundary_color" : [MAROON_B, YELLOW],
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"negative_color" : RED,
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}
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def construct(self):
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def setup(self):
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self.colors = color_gradient(self.boundary_colors, self.n_dims)
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def construct(self):
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sphere_set = self.get_sphere_set()
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arrow = Arrow(LEFT, RIGHT)
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f = TexMobject("f")
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@ -1556,13 +1557,18 @@ class ThinkAboutTheChoices(TeacherStudentsScene):
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Think about the choices
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behind a division...
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""")
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self.change_student_modes(*["pondering"]*3)
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self.change_student_modes(
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*["pondering"]*3,
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look_at_arg = SPACE_WIDTH*RIGHT+SPACE_HEIGHT*DOWN
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)
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self.dither(3)
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class ChoicesInNecklaceCutting(Scene):
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CONFIG = {
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"num_continuous_division_searches" : 4,
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"final_num_pair" : [1./6, 1./3],
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"final_num_pair" : [1./6, 1./2],
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"necklace_center" : DOWN,
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"thief_box_offset" : 1.2,
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}
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def construct(self):
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self.add_necklace()
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@ -1589,6 +1595,7 @@ class ChoicesInNecklaceCutting(Scene):
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numbers_with_elongated_ticks = [],
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)
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interval.stretch_to_fit_width(width)
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interval.shift(self.necklace_center)
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tick_marks = interval.tick_marks
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tick_marks.set_stroke(WHITE, width = 2)
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@ -1602,11 +1609,12 @@ class ChoicesInNecklaceCutting(Scene):
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segments.add(segment)
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self.necklace = VGroup(segments, tick_marks)
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self.interval = interval
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self.add(self.necklace)
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self.interval = interval
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def choose_places_to_cut(self):
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v_lines = [DashedLine(UP, DOWN), for x in range(2)]
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v_lines = VGroup(*[DashedLine(UP, DOWN) for x in range(2)])
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num_pairs = [
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sorted([random.random(), random.random()])
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for x in range(self.num_continuous_division_searches)
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@ -1614,23 +1622,443 @@ class ChoicesInNecklaceCutting(Scene):
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point_pairs = [
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map(self.interval.number_to_point, num_pair)
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for num_pair in num_pair
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for num_pair in num_pairs
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]
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for line, point in zip(v_lines, point_pairs[0]):
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line.move_to(point)
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self.play(ShowCreation(v_lines))
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for point_pair in point_pairs[1:]:
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self.dither()
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self.play(*[
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ApplyMethod(line.move_to, point)
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for line, point in zip(v_lines, point_pair)
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])
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self.dither()
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self.division_points = list(it.chain(
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[self.interval.get_left()],
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point_pairs[-1],
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[self.interval.get_right()]
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))
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self.v_lines = v_lines
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def show_three_numbers_adding_to_one(self):
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pass
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points = self.division_points
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braces = [
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Brace(Line(p1+SMALL_BUFF*RIGHT/2, p2+SMALL_BUFF*LEFT/2))
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for p1, p2 in zip(points, points[1:])
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]
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for char, denom, brace in zip("abc", [6, 3, 2], braces):
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brace.label = brace.get_text("$%s$"%char)
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brace.concrete_label = brace.get_text("$\\frac{1}{%d}$"%denom)
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VGroup(
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brace.label,
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brace.concrete_label
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).highlight(YELLOW)
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words = TextMobject(
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"1) Choose", "$a$, $b$, $c$", "so that", "$a+b+c = 1$"
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)
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words[1].highlight(YELLOW)
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words[3].highlight(YELLOW)
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words.to_corner(UP+LEFT)
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self.play(*it.chain(*[
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[GrowFromCenter(brace), Write(brace.label)]
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for brace in braces
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]))
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self.play(Write(words))
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self.dither(2)
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self.play(*[
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ReplacementTransform(brace.label, brace.concrete_label)
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for brace in braces
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])
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self.dither()
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self.wiggle_v_lines()
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self.dither()
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self.play(*map(FadeOut, list(braces) + [
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brace.concrete_label for brace in braces
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]))
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self.choice_one_words = words
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def make_binary_choice(self):
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groups = self.get_groups()
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boxes, labels = self.get_boxes_and_labels()
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arrow_pairs, curr_arrows = self.get_choice_arrow_pairs(groups)
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words = TextMobject("2) Make a binary choice for each segment")
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words.next_to(
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self.choice_one_words, DOWN,
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buff = MED_LARGE_BUFF,
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aligned_edge = LEFT
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)
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self.play(Write(words))
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self.play(*map(FadeIn, [boxes, labels]))
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for binary_choices in it.product(*[[0, 1]]*3):
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self.play(*[
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ApplyMethod(group.move_to, group.target_points[choice])
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for group, choice in zip(groups, binary_choices)
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] + [
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Transform(
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curr_arrow, arrow_pair[choice],
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path_arc = np.pi
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)
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for curr_arrow, arrow_pair, choice in zip(
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curr_arrows, arrow_pairs, binary_choices
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)
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])
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self.dither()
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######
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def get_groups(self):
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segments, tick_marks = self.necklace
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n_segments = len(segments)
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indices = [0, n_segments/6, n_segments/2, n_segments]
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groups = [
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VGroup(
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VGroup(*segments[i1:i2]),
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VGroup(*tick_marks[i1:i2]),
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)
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for i1, i2 in zip(indices, indices[1:])
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]
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for group, index in zip(groups, indices[1:]):
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group.add(tick_marks[index].copy())
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groups[-1][-1].add(tick_marks[-1])
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for group in groups:
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group.target_points = [
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group.get_center() + self.thief_box_offset*vect
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for vect in UP, DOWN
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]
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return groups
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def get_boxes_and_labels(self):
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box = Rectangle(
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height = self.necklace.get_height()+SMALL_BUFF,
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width = self.necklace.get_width()+2*SMALL_BUFF,
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stroke_width = 0,
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fill_color = WHITE,
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fill_opacity = 0.25
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)
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box.move_to(self.necklace)
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boxes = VGroup(*[
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box.copy().shift(self.thief_box_offset*vect)
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for vect in UP, DOWN
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])
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labels = VGroup(*[
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TextMobject(
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"Thief %d"%(i+1)
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).next_to(box, UP, aligned_edge = RIGHT)
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for i, box in enumerate(boxes)
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])
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return boxes, labels
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def get_choice_arrow_pairs(self, groups):
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arrow = TexMobject("\\uparrow")
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arrow_pairs = [
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[arrow.copy(), arrow.copy().rotate(np.pi)]
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for group in groups
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]
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pre_arrow_points = [
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VectorizedPoint(group.get_center())
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for group in groups
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]
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for point, arrow_pair in zip(pre_arrow_points, arrow_pairs):
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for arrow, color in zip(arrow_pair, [GREEN, RED]):
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arrow.highlight(color)
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arrow.move_to(point.get_center())
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return arrow_pairs, pre_arrow_points
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def wiggle_v_lines(self):
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self.play(
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*it.chain(*[
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[
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line.rotate_in_place, np.pi/12, vect,
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line.highlight, RED
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]
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for line, vect in zip(self.v_lines, [OUT, IN])
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]),
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rate_func = wiggle
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)
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class CompareThisToSphereChoice(TeacherStudentsScene):
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def construct(self):
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self.teacher_says("""
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Compare this to choosing
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a point on the sphere.
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""")
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self.change_student_modes(
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*["pondering"]*3,
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look_at_arg = SPACE_WIDTH*RIGHT+SPACE_HEIGHT*DOWN
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)
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self.dither(3)
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class ChoicesForSpherePoint(GeneralizeBorsukUlam):
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def construct(self):
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self.add_sphere_set()
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self.initialize_words()
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self.play(Write(self.choice_one_words))
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self.dither()
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self.show_example_choices()
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self.show_binary_choices()
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def get_tuple(self):
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tup = TexMobject("(x, y, z)")
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for i, color in zip([1, 3, 5], self.colors):
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tup[i].highlight(color)
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return tup
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def get_condition(self):
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condition = TexMobject("x^2+y^2+z^2 = 1")
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for i, color in zip([0, 3, 6], self.colors):
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VGroup(*condition[i:i+2]).highlight(color)
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return condition
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def add_sphere_set(self):
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sphere_set = self.get_sphere_set()
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sphere_set.scale(0.7)
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sphere_set.to_edge(RIGHT)
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sphere_set.shift(UP)
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self.add(sphere_set)
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def initialize_words(self):
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choice_one_words = TextMobject(
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"1) Choose", "$x^2$, $y^2$, $z^2$",
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"so that", "$x^2+y^2+z^2 = 1$"
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)
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for i in 1, 3:
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for j, color in zip([0, 3, 6], self.colors):
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VGroup(*choice_one_words[i][j:j+2]).highlight(color)
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choice_one_words.to_corner(UP+LEFT)
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choice_two_words = TextMobject(
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"2) Make a binary choice for each one"
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)
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choice_two_words.next_to(
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choice_one_words, DOWN,
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buff = MED_LARGE_BUFF,
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aligned_edge = LEFT
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)
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self.choice_one_words = choice_one_words
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self.choice_two_words = choice_two_words
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def show_example_choices(self):
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choices = VGroup(*[
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TexMobject(*tex).highlight(color)
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for color, tex in zip(self.colors, [
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("x", "^2 = ", "1/6"),
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("y", "^2 = ", "1/3"),
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("z", "^2 = ", "1/2"),
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])
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])
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choices.arrange_submobjects(
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DOWN,
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buff = LARGE_BUFF,
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aligned_edge = LEFT
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)
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choices.scale_to_fit_height(SPACE_HEIGHT)
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choices.to_edge(LEFT)
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choices.shift(DOWN)
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self.play(FadeIn(
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choices,
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run_time = 2,
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submobject_mode = "lagged_start"
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))
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self.dither()
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self.choices = choices
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def show_binary_choices(self):
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for choice in self.choices:
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var_tex = choice.expression_parts[0]
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frac_tex = choice.expression_parts[2]
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sqrts = VGroup(*[
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TexMobject(
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var_tex + "=" + sign + \
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"\\sqrt{%s}"%frac_tex)
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for sign in ["+", "-"]
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])
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for sqrt in sqrts:
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sqrt.scale(0.6)
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sqrts.arrange_submobjects(DOWN)
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sqrts.next_to(choice, RIGHT, buff = LARGE_BUFF)
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sqrts.highlight(choice.get_color())
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arrows = VGroup(*[
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Arrow(
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choice.get_right(), sqrt.get_left(),
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color = WHITE,
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tip_length = 0.1,
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buff = SMALL_BUFF
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)
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for sqrt in sqrts
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])
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self.play(ShowCreation(arrows))
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self.play(FadeIn(sqrts, submobject_mode = "lagged_start"))
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self.play(Write(self.choice_two_words))
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self.dither()
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class NecklaceDivisionSphereAssociation(ChoicesInNecklaceCutting):
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CONFIG = {
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"xyz_colors" : color_gradient([GREEN_B, BLUE], 3),
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"necklace_center" : DOWN,
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"thief_box_offset" : 1.6,
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"denoms" : [6, 3, 2],
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}
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def construct(self):
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self.add_necklace()
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self.add_sphere_point_label()
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self.choose_places_to_cut()
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self.add_braces()
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self.add_boxes_and_labels()
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self.show_binary_choice_association()
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self.ask_about_antipodal_pairs()
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def add_sphere_point_label(self):
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label = TextMobject(
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"$(x, y, z)$",
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"such that",
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"$x^2 + y^2 + z^2 = 1$"
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)
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for i, j_list in (0, [1, 3, 5]), (2, [0, 3, 6]):
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for j, color in zip(j_list, self.xyz_colors):
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label[i][j].highlight(color)
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label.to_corner(UP+RIGHT)
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ghost_sphere_point = VectorizedPoint()
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ghost_sphere_point.to_corner(UP+LEFT, buff = LARGE_BUFF)
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ghost_sphere_point.shift(2*RIGHT)
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arrow = Arrow(
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label.get_left(), ghost_sphere_point,
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color = WHITE
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)
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self.add(label, arrow)
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self.sphere_point_label = label
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def add_braces(self):
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points = self.division_points
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braces = [
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Brace(
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Line(p1+SMALL_BUFF*RIGHT/2, p2+SMALL_BUFF*LEFT/2),
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UP
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)
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for p1, p2 in zip(points, points[1:])
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]
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for char, brace, color, denom in zip("xyz", braces, self.xyz_colors, self.denoms):
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brace.label = brace.get_text(
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"$%s^2$"%char, "$= 1/%d$"%denom,
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buff = SMALL_BUFF
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)
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brace.label.highlight(color)
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if brace.label.get_right()[0] > brace.get_right()[0]:
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brace.label.next_to(
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brace, UP, buff = SMALL_BUFF,
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aligned_edge = RIGHT
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)
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self.play(*it.chain(
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map(GrowFromCenter, braces),
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[Write(brace.label) for brace in braces]
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))
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self.dither()
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self.braces = braces
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def add_boxes_and_labels(self):
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boxes, labels = self.get_boxes_and_labels()
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self.play(*map(FadeIn, [boxes, labels]))
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self.dither()
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def show_binary_choice_association(self):
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groups = self.get_groups()
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self.swapping_anims = []
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final_choices = [1, 0, 1]
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quads = zip(self.braces, self.denoms, groups, final_choices)
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for brace, denom, group, final_choice in quads:
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char = brace.label.args[0][1]
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choices = [
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TexMobject(
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char, "=", sign, "\\sqrt{\\frac{1}{%d}}"%denom
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)
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for sign in "+", "-"
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]
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for choice, color in zip(choices, [GREEN, RED]):
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# choice[0].highlight(brace.label.get_color())
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choice[2].highlight(color)
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choice.scale(0.8)
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choice.move_to(group)
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if choice.get_width() > 0.8*group.get_width():
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choice.next_to(group.get_right(), LEFT, buff = MED_SMALL_BUFF)
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original_choices = [m.copy() for m in choices]
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self.play(
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ReplacementTransform(
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VGroup(brace.label[0], brace, brace.label[1]),
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choices[0]
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),
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group.move_to, group.target_points[0]
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)
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self.dither()
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self.play(
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Transform(*choices),
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group.move_to, group.target_points[1]
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)
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self.dither()
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if final_choice == 0:
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self.play(
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Transform(choices[0], original_choices[0]),
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group.move_to, group.target_points[0]
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)
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self.swapping_anims += [
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Transform(choices[0], original_choices[1-final_choice]),
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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
|
||||
|
||||
|
||||
|
|
@ -1645,6 +2073,19 @@ class ChoicesInNecklaceCutting(Scene):
|
|||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -470,6 +470,9 @@ class TeacherStudentsScene(Scene):
|
|||
pairs = [(s, m) for s, m in pairs if m is not None]
|
||||
start = VGroup(*[s for s, m in pairs])
|
||||
target = VGroup(*[s.copy().change_mode(m) for s, m in pairs])
|
||||
if "look_at_arg" in kwargs:
|
||||
for pi in target:
|
||||
pi.look_at(kwargs["look_at_arg"])
|
||||
self.play(
|
||||
Transform(
|
||||
start, target,
|
||||
|
|
|
|||
|
|
@ -169,7 +169,7 @@ class Arrow(Line):
|
|||
"color" : YELLOW_C,
|
||||
"tip_length" : 0.25,
|
||||
"tip_angle" : np.pi/6,
|
||||
"buff" : 0.3,
|
||||
"buff" : MED_SMALL_BUFF,
|
||||
"propogate_style_to_family" : False,
|
||||
"preserve_tip_size_when_scaling" : True,
|
||||
}
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue