mirror of
https://github.com/3b1b/manim.git
synced 2025-09-01 00:48:45 +00:00
FromGeometryToAlgebra
This commit is contained in:
Grant Sanderson
parent
8589515e82
commit
4ae12a6856
1 changed files with 394 additions and 9 deletions
|
|
@ -7,7 +7,7 @@ from once_useful_constructs.light import SwitchOn
|
|||
PRODUCT_COLOR = BLUE
|
||||
CHEAP_AMBIENT_LIGHT_CONFIG = {
|
||||
"num_levels": 5,
|
||||
"radius": 1,
|
||||
"radius": 0.5,
|
||||
}
|
||||
|
||||
|
||||
|
|
@ -608,30 +608,415 @@ class Lemma2With9Lighthouses(Lemma2):
|
|||
class FromGeometryToAlgebra(DistanceProductScene):
|
||||
CONFIG = {
|
||||
"num_lighthouses": 7,
|
||||
"ambient_light_config": CHEAP_AMBIENT_LIGHT_CONFIG,
|
||||
# "ambient_light_config": CHEAP_AMBIENT_LIGHT_CONFIG,
|
||||
}
|
||||
|
||||
def construct(self):
|
||||
self.setup_lights()
|
||||
self.point_out_evenly_spaced()
|
||||
self.transition_to_complex_plane()
|
||||
self.name_roots_of_unity()
|
||||
self.discuss_powers()
|
||||
self.raise_everything_to_the_nth()
|
||||
|
||||
def setup_lights(self):
|
||||
lights = self.get_lights()
|
||||
circle = self.circle
|
||||
circle.scale_to_fit_height(5, about_edge=DOWN)
|
||||
lights = self.get_lights()
|
||||
dots = VGroup(*[Dot(point) for point in self.get_lh_points()])
|
||||
for dot, light in zip(dots, lights):
|
||||
light.add_to_back(dot)
|
||||
|
||||
self.add(circle, lights)
|
||||
|
||||
def point_out_evenly_spaced(self):
|
||||
pass
|
||||
circle = self.circle
|
||||
step = 1.0 / self.num_lighthouses / 2
|
||||
alpha_range = np.arange(0, 1 + step, step)
|
||||
arcs = VGroup(*[
|
||||
VMobject().pointwise_become_partial(circle, a1, a2)
|
||||
for a1, a2 in zip(alpha_range, alpha_range[1:])
|
||||
])
|
||||
arcs.set_stroke(YELLOW, 5)
|
||||
|
||||
for arc in arcs[::2]:
|
||||
arc.reverse_points()
|
||||
|
||||
arcs_anim = ShowCreationThenDestruction(
|
||||
arcs, submobject_mode="all_at_once", run_time=2
|
||||
)
|
||||
|
||||
spacing_words = self.spacing_words = TextMobject("Evenly-spaced")
|
||||
spacing_words.scale_to_fit_width(self.get_radius())
|
||||
spacing_words.move_to(circle)
|
||||
|
||||
arrows = self.get_arrows()
|
||||
|
||||
geometric_words = self.geometric_words = TextMobject("Geometric property")
|
||||
geometric_words.to_edge(UP)
|
||||
geometric_words.add_background_rectangle()
|
||||
|
||||
self.add(geometric_words)
|
||||
self.play(
|
||||
FadeIn(spacing_words),
|
||||
arcs_anim,
|
||||
*map(GrowArrow, arrows)
|
||||
)
|
||||
self.play(FadeOut(arrows), arcs_anim)
|
||||
self.wait()
|
||||
|
||||
def transition_to_complex_plane(self):
|
||||
pass
|
||||
|
||||
def name_roots_of_unity(self):
|
||||
pass
|
||||
plane = self.complex_plane = ComplexPlane(
|
||||
unit_size=2, y_radius=6, x_radius=9,
|
||||
)
|
||||
plane.shift(1.5 * RIGHT)
|
||||
plane.add_coordinates()
|
||||
origin = plane.number_to_point(0)
|
||||
h_line = Line(plane.number_to_point(-1), plane.number_to_point(1))
|
||||
|
||||
circle = self.circle
|
||||
circle_group = VGroup(circle, self.lights)
|
||||
circle_group.generate_target()
|
||||
circle_group.target.scale(h_line.get_width() / circle.get_width())
|
||||
circle_group.target.shift(
|
||||
origin - circle_group.target[0].get_center()
|
||||
)
|
||||
circle_group.target[0].set_stroke(RED)
|
||||
|
||||
geometric_words = self.geometric_words
|
||||
geometric_words.generate_target()
|
||||
arrow = TexMobject("\\rightarrow")
|
||||
arrow.add_background_rectangle()
|
||||
algebraic_words = TextMobject("Algebraic property")
|
||||
algebraic_words.add_background_rectangle()
|
||||
word_group = VGroup(geometric_words.target, arrow, algebraic_words)
|
||||
word_group.arrange_submobjects(RIGHT)
|
||||
word_group.move_to(origin)
|
||||
word_group.to_edge(UP)
|
||||
|
||||
unit_circle_words = TextMobject("Unit circle", "")
|
||||
unit_circle_words.match_color(circle_group.target[0])
|
||||
for part in unit_circle_words:
|
||||
part.add_background_rectangle()
|
||||
unit_circle_words.next_to(origin, UP)
|
||||
|
||||
complex_plane_words = TextMobject("Complex Plane")
|
||||
self.complex_plane_words = complex_plane_words
|
||||
complex_plane_words.move_to(word_group)
|
||||
complex_plane_words.add_background_rectangle()
|
||||
|
||||
roots_of_unity_words = TextMobject("Roots of\\\\", "unity")
|
||||
roots_of_unity_words.move_to(origin)
|
||||
roots_of_unity_words.set_color(YELLOW)
|
||||
for part in roots_of_unity_words:
|
||||
part.add_background_rectangle()
|
||||
|
||||
self.play(
|
||||
Write(plane),
|
||||
MoveToTarget(circle_group),
|
||||
FadeOut(self.spacing_words),
|
||||
MoveToTarget(geometric_words),
|
||||
FadeIn(arrow),
|
||||
FadeIn(algebraic_words),
|
||||
)
|
||||
word_group.submobjects[0] = geometric_words
|
||||
self.play(Write(unit_circle_words, run_time=1))
|
||||
|
||||
# Show complex values
|
||||
outer_arrows = self.outer_arrows = self.get_arrows()
|
||||
for arrow, point in zip(outer_arrows, self.get_lh_points()):
|
||||
arrow.rotate(np.pi, about_point=point)
|
||||
outer_arrow = self.outer_arrow = outer_arrows[3].copy()
|
||||
|
||||
values = map(plane.point_to_number, self.get_lh_points())
|
||||
complex_decimal = self.complex_decimal = DecimalNumber(
|
||||
values[3],
|
||||
num_decimal_points=3,
|
||||
include_background_rectangle=True
|
||||
)
|
||||
complex_decimal.next_to(outer_arrow.get_start(), LEFT, SMALL_BUFF)
|
||||
complex_decimal_rect = SurroundingRectangle(complex_decimal)
|
||||
complex_decimal_rect.fade(1)
|
||||
|
||||
self.play(
|
||||
FadeIn(complex_plane_words),
|
||||
FadeOut(word_group),
|
||||
FadeIn(complex_decimal),
|
||||
FadeIn(outer_arrow)
|
||||
)
|
||||
self.wait(2)
|
||||
self.play(
|
||||
ChangeDecimalToValue(
|
||||
complex_decimal, values[1],
|
||||
tracked_mobject=complex_decimal_rect
|
||||
),
|
||||
complex_decimal_rect.next_to, outer_arrows[1].get_start(), UP, SMALL_BUFF,
|
||||
Transform(outer_arrow, outer_arrows[1]),
|
||||
run_time=1.5
|
||||
)
|
||||
self.wait()
|
||||
|
||||
arrows = self.get_arrows()
|
||||
arrows.set_color(YELLOW)
|
||||
self.play(
|
||||
ReplacementTransform(unit_circle_words, roots_of_unity_words),
|
||||
LaggedStart(GrowArrow, arrows)
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
complex_plane_words.move_to, word_group,
|
||||
LaggedStart(FadeOut, VGroup(*it.chain(
|
||||
arrows, roots_of_unity_words
|
||||
)))
|
||||
)
|
||||
|
||||
# Turn decimal into z
|
||||
x_term = self.x_term = TexMobject("x")
|
||||
x_term.add_background_rectangle()
|
||||
x_term.move_to(complex_decimal, DOWN)
|
||||
x_term.shift(0.5 * SMALL_BUFF * (DR))
|
||||
self.play(ReplacementTransform(complex_decimal, x_term))
|
||||
|
||||
def discuss_powers(self):
|
||||
x_term = self.x_term
|
||||
outer_arrows = self.outer_arrows
|
||||
outer_arrows.add(outer_arrows[0].copy())
|
||||
plane = self.complex_plane
|
||||
origin = plane.number_to_point(0)
|
||||
|
||||
question = TextMobject("What is $x^2$")
|
||||
question.next_to(x_term, RIGHT, LARGE_BUFF)
|
||||
question.set_color(YELLOW)
|
||||
|
||||
lh_points = list(self.get_lh_points())
|
||||
lh_points.append(lh_points[0])
|
||||
lines = VGroup(*[
|
||||
Line(origin, point)
|
||||
for point in lh_points
|
||||
])
|
||||
lines.set_color(GREEN)
|
||||
step = 1.0 / self.num_lighthouses
|
||||
angle_arcs = VGroup(*[
|
||||
Arc(angle=alpha * TAU, radius=0.35).shift(origin)
|
||||
for alpha in np.arange(0, 1 + step, step)
|
||||
])
|
||||
angle_labels = VGroup()
|
||||
for i, arc in enumerate(angle_arcs):
|
||||
label = TexMobject("(%d / %d)\\tau" % (i, self.num_lighthouses))
|
||||
label.scale(0.5)
|
||||
label.add_background_rectangle()
|
||||
point = arc.point_from_proportion(0.5)
|
||||
point += 1.2 * (point - origin)
|
||||
label.move_to(point)
|
||||
angle_labels.add(label)
|
||||
|
||||
line = self.angle_line = lines[1].copy()
|
||||
line_ghost = DashedLine(line.get_start(), line.get_end())
|
||||
self.ghost_angle_line = line_ghost
|
||||
line_ghost.set_stroke(line.get_color(), 2)
|
||||
angle_arc = angle_arcs[1].copy()
|
||||
angle_label = angle_labels[1].copy()
|
||||
angle_label.shift(0.25 * SMALL_BUFF * DR)
|
||||
|
||||
magnitude_label = TexMobject("1")
|
||||
magnitude_label.next_to(line.get_center(), UL, buff=SMALL_BUFF)
|
||||
|
||||
power_labels = VGroup()
|
||||
for i, arrow in enumerate(outer_arrows):
|
||||
label = TexMobject("x^%d" % i)
|
||||
label.next_to(
|
||||
arrow.get_start(), -arrow.get_vector(),
|
||||
submobject_to_align=label[0]
|
||||
)
|
||||
label.add_background_rectangle()
|
||||
power_labels.add(label)
|
||||
power_labels[-1].next_to(outer_arrows[-1].get_start(), UR, SMALL_BUFF)
|
||||
power_labels.submobjects[1] = x_term
|
||||
|
||||
L_labels = self.L_labels = VGroup(*[
|
||||
TexMobject("L_%d" % i).move_to(power_label, DOWN).add_background_rectangle()
|
||||
for i, power_label in enumerate(power_labels)
|
||||
])
|
||||
|
||||
# Ask about squaring
|
||||
self.play(Write(question))
|
||||
self.wait()
|
||||
self.play(
|
||||
ShowCreation(line),
|
||||
Write(magnitude_label)
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
ShowCreation(angle_arc),
|
||||
Write(angle_label)
|
||||
)
|
||||
self.wait()
|
||||
self.add(line_ghost)
|
||||
for i in range(2, self.num_lighthouses + 1):
|
||||
anims = [
|
||||
Transform(angle_arc, angle_arcs[i]),
|
||||
Transform(angle_label, angle_labels[i]),
|
||||
Transform(line, lines[i], path_arc=TAU / self.num_lighthouses),
|
||||
]
|
||||
if i == 2:
|
||||
anims.append(FadeOut(magnitude_label))
|
||||
if i == 3:
|
||||
anims.append(FadeOut(question))
|
||||
self.play(*anims)
|
||||
new_anims = [
|
||||
GrowArrow(outer_arrows[i]),
|
||||
Write(power_labels[i]),
|
||||
]
|
||||
if i == 2:
|
||||
new_anims.append(FadeOut(self.complex_plane_words))
|
||||
self.play(*new_anims)
|
||||
self.wait()
|
||||
self.play(ReplacementTransform(power_labels[1:], L_labels[1:]))
|
||||
self.wait()
|
||||
self.play(
|
||||
Rotate(self.lights, TAU / self.num_lighthouses / 2),
|
||||
rate_func=wiggle
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
FadeOut(angle_arc),
|
||||
FadeOut(angle_label),
|
||||
*map(ShowCreationThenDestruction, lines)
|
||||
)
|
||||
self.wait()
|
||||
|
||||
def raise_everything_to_the_nth(self):
|
||||
func_label = TexMobject("L \\rightarrow L^7")
|
||||
func_label.set_color(YELLOW)
|
||||
func_label.to_corner(UL, buff=LARGE_BUFF)
|
||||
func_label.add_background_rectangle()
|
||||
|
||||
polynomial_scale_factor = 0.8
|
||||
|
||||
polynomial = TexMobject("x^%d - 1" % self.num_lighthouses, "=", "0")
|
||||
polynomial.scale(polynomial_scale_factor)
|
||||
polynomial.next_to(func_label, UP)
|
||||
polynomial.to_edge(LEFT)
|
||||
|
||||
factored_polynomial = TexMobject(
|
||||
"(x-L_1)(x-L_2)\\cdots(x-L_%d)" % self.num_lighthouses, "=", "0"
|
||||
)
|
||||
factored_polynomial.scale(polynomial_scale_factor)
|
||||
factored_polynomial.next_to(polynomial, DOWN, aligned_edge=LEFT)
|
||||
for group in polynomial, factored_polynomial:
|
||||
for part in group:
|
||||
part.add_background_rectangle()
|
||||
|
||||
origin = self.complex_plane.number_to_point(0)
|
||||
|
||||
lights = self.lights
|
||||
lights.save_state()
|
||||
rotations = []
|
||||
for i, light in enumerate(lights):
|
||||
rotations.append(Rotating(
|
||||
light,
|
||||
radians=(i * TAU - i * TAU / self.num_lighthouses),
|
||||
about_point=origin,
|
||||
rate_func=bezier([0, 0, 1, 1]),
|
||||
))
|
||||
|
||||
self.play(Write(func_label, run_time=1))
|
||||
for i, rotation in enumerate(rotations[:4]):
|
||||
anims = [rotation]
|
||||
if i == 3:
|
||||
rect = SurroundingRectangle(polynomial)
|
||||
rect.set_color(YELLOW)
|
||||
anims += [
|
||||
FadeIn(polynomial),
|
||||
ShowCreationThenDestruction(rect)
|
||||
]
|
||||
self.play(*anims, run_time=np.sqrt(i + 1))
|
||||
self.play(*rotations[4:], run_time=3)
|
||||
self.wait()
|
||||
|
||||
self.play(lights.restore)
|
||||
self.play(
|
||||
FadeOut(func_label),
|
||||
FadeIn(factored_polynomial)
|
||||
)
|
||||
self.wait(3)
|
||||
self.play(
|
||||
factored_polynomial[0].next_to, polynomial[1], RIGHT, 1.5 * SMALL_BUFF,
|
||||
FadeOut(polynomial[2]),
|
||||
FadeOut(factored_polynomial[1:]),
|
||||
)
|
||||
|
||||
# Comment on formula
|
||||
formula = VGroup(polynomial[0], polynomial[1], factored_polynomial[0])
|
||||
rect = SurroundingRectangle(formula)
|
||||
|
||||
brace = Brace(factored_polynomial[0], DOWN)
|
||||
brace2 = Brace(polynomial[0], DOWN)
|
||||
|
||||
morty = PiCreature(color=GREY_BROWN)
|
||||
morty.scale(0.5)
|
||||
morty.next_to(brace.get_center(), DL, buff=LARGE_BUFF)
|
||||
|
||||
L1_rhs = TexMobject("= \\cos(\\tau / 7) + \\\\", "\\sin(\\tau / 7)i")
|
||||
L1_rhs.next_to(self.L_labels[1], RIGHT, aligned_edge=UP)
|
||||
for part in L1_rhs:
|
||||
part.add_background_rectangle()
|
||||
|
||||
self.play(ShowCreation(rect))
|
||||
self.play(FadeOut(rect))
|
||||
self.wait()
|
||||
self.play(GrowFromCenter(brace))
|
||||
self.play(FadeIn(morty))
|
||||
self.play(morty.change, "horrified", brace)
|
||||
self.play(Blink(morty))
|
||||
self.wait()
|
||||
self.play(
|
||||
Write(L1_rhs),
|
||||
morty.change, "confused", L1_rhs
|
||||
)
|
||||
self.play(Blink(morty))
|
||||
self.wait()
|
||||
self.play(
|
||||
Transform(brace, brace2),
|
||||
morty.change, "hooray", brace2
|
||||
)
|
||||
self.play(Blink(morty))
|
||||
self.wait()
|
||||
|
||||
# Nothing special about 7
|
||||
new_lights = self.lights.copy()
|
||||
new_lights.rotate(
|
||||
TAU / self.num_lighthouses / 2,
|
||||
about_point=origin
|
||||
)
|
||||
sevens = VGroup(polynomial[0][1][1], factored_polynomial[0][1][-2])
|
||||
n_terms = VGroup()
|
||||
for seven in sevens:
|
||||
n_term = TexMobject("N")
|
||||
n_term.replace(seven, dim_to_match=1)
|
||||
n_term.scale(0.9)
|
||||
n_term.shift(0.25 * SMALL_BUFF * DR)
|
||||
n_terms.add(n_term)
|
||||
|
||||
self.play(LaggedStart(FadeOut, VGroup(*it.chain(
|
||||
L1_rhs, self.outer_arrows, self.L_labels[1:], self.outer_arrow,
|
||||
self.angle_line, self.ghost_angle_line
|
||||
))))
|
||||
self.play(LaggedStart(SwitchOn, new_lights), morty.look_at, new_lights)
|
||||
self.play(Transform(sevens, n_terms))
|
||||
self.wait()
|
||||
self.play(Blink(morty))
|
||||
self.wait()
|
||||
#
|
||||
|
||||
def get_arrows(self):
|
||||
return VGroup(*[
|
||||
Arrow(
|
||||
interpolate(self.circle.get_center(), point, 0.6),
|
||||
interpolate(self.circle.get_center(), point, 0.9),
|
||||
buff=0
|
||||
)
|
||||
for point in self.get_lh_points()
|
||||
])
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue