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A few filler scenes for spirals project

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This commit is contained in:
Grant Sanderson
2019-09-30 12:06:35 -07:00
parent c94aef8845
commit 67f3d1584c
1 changed files with 483 additions and 65 deletions
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548
active_projects/spirals.py
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@ -8,6 +8,15 @@ INV_113_MOD_710 = 377 # Inverse of 113 mode 710
INV_7_MOD_44 = 19
def is_prime(n):
if n < 2:
return False
for k in range(2, int(np.sqrt(n)) + 1):
if n % k == 0:
return False
return True
def generate_prime_list(*args):
if len(args) == 1:
start, stop = 2, args[0]
@ -17,15 +26,10 @@ def generate_prime_list(*args):
else:
raise TypeError("generate_prime_list takes 1 or 2 arguments")
result = []
for n in range(start, stop):
include = True
for k in range(2, int(np.sqrt(n)) + 1):
if n % k == 0:
include = False
break
if include:
result.append(n)
result = [
n for n in range(start, stop)
if is_prime(n)
]
return result
@ -189,6 +193,77 @@ class SpiralScene(MovingCameraScene):
# Scenes
class MathExchange(ExternallyAnimatedScene):
pass
class PrimesAndPi(Scene):
def construct(self):
self.show_primes()
self.show_rational_approximations()
def show_primes(self):
n_rows = 10
n_cols = 10
matrix = IntegerMatrix([
[n_cols * x + y for y in range(n_cols)]
for x in range(n_rows)
])
numbers = matrix.get_entries()
primes = VGroup(*filter(
lambda m: is_prime(m.get_value()),
numbers,
))
non_primes = VGroup(*filter(
lambda m: not is_prime(m.get_value()),
numbers
))
self.add(numbers)
self.play(
LaggedStart(*[
ApplyFunction(
lambda m: m.set_color(YELLOW).scale(1.2),
prime
)
for prime in primes
]),
non_primes.set_opacity, 0.25,
run_time=2,
)
self.wait()
self.numbers = numbers
def show_rational_approximations(self):
numbers = self.numbers
approxs = TexMobject(
"{22 \\over 7} &=", "{:.12}\\dots\\\\".format(22 / 7),
"{355 \\over 113} &=", "{:.12}\\dots\\\\".format(355 / 113),
"\\pi &=", "{:.12}\\dots\\\\".format(PI),
)
approxs[:2].shift(MED_LARGE_BUFF * UP)
approxs[-2:].shift(MED_LARGE_BUFF * DOWN)
approxs[-2:].set_color(YELLOW)
approxs[1][:4].set_color(YELLOW)
approxs[3][:8].set_color(YELLOW)
approxs.scale(1.5)
randy = Randolph(color=YELLOW, height=1)
randy.move_to(approxs[-2][0], RIGHT)
approxs[-2][0].set_opacity(0)
self.play(
LaggedStartMap(FadeOutAndShiftDown, numbers),
LaggedStartMap(FadeIn, approxs),
FadeIn(randy)
)
self.play(Blink(randy))
self.play(randy.change, "pondering", UR)
self.wait()
class RefresherOnPolarCoordinates(MovingCameraScene):
CONFIG = {
@ -391,12 +466,15 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
r_line.add_updater(
lambda l: l.put_start_and_end_on(ORIGIN, dot.get_center())
)
theta_tracker = ValueTracker(0)
theta_tracker.add_updater(
lambda m: m.set_value(r_line.get_angle() % TAU)
)
self.add(theta_tracker)
arc.add_updater(
lambda m: m.become(Arc(
angle=(r_line.get_angle() % TAU),
color=theta_color,
radius=1,
))
lambda m: m.become(
self.get_arc(theta_tracker.get_value())
)
)
self.add(coord_label)
@ -408,16 +486,25 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
self.wait()
self.play(
FadeOut(coord_label),
FadeOut(r_line),
FadeOut(arc),
FadeOut(r_label),
FadeOut(theta_label),
FadeOut(radians_word),
FadeOut(r_line),
FadeOut(arc),
FadeOut(dot),
)
self.dot = dot
self.r_line = r_line
self.arc = arc
self.theta_tracker = theta_tracker
def show_all_nn_tuples(self):
self.remove(self.dot)
dot = self.dot
arc = self.arc
r_line = self.r_line
theta_tracker = self.theta_tracker
primes = generate_prime_list(20)
non_primes = list(range(1, 20))
for prime in primes:
@ -446,11 +533,15 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
# Talk about 1
one = np_points[0]
r_line = Line(ORIGIN, one.dot.get_center())
r_line.set_color(self.r_color)
dot.move_to(self.get_polar_point(1, 1))
self.add(dot)
theta_tracker.clear_updaters()
theta_tracker.set_value(1)
# r_line = Line(ORIGIN, one.dot.get_center())
# r_line.set_color(self.r_color)
# pre_arc = Line(RIGHT, UR, color=self.r_color)
theta_tracker = ValueTracker(1)
arc = always_redraw(lambda: self.get_arc(theta_tracker.get_value()))
# theta_tracker = ValueTracker(1)
# arc = always_redraw(lambda: self.get_arc(theta_tracker.get_value()))
one_rect = SurroundingRectangle(one)
one_r_rect = SurroundingRectangle(one.label[1])
@ -467,43 +558,37 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
# self.play(TransformFromCopy(r_line, pre_arc))
# self.add(pre_arc, one)
self.play(
TransformFromCopy(r_line, arc),
ReplacementTransform(
Line(*r_line.get_start_and_end()), arc
),
ReplacementTransform(one_r_rect, one_theta_rect)
)
self.add(arc, one, one_theta_rect)
self.play(FadeOut(one_theta_rect))
self.wait()
# Talk about 2
self.play(theta_tracker.set_value, 2)
self.wait()
self.play(Rotate(r_line, angle=1, about_point=ORIGIN))
self.play(r_line.scale, 2, {'about_point': ORIGIN})
self.wait()
# And now 3
self.play(
theta_tracker.set_value, 3,
Rotate(r_line, angle=1, about_point=ORIGIN),
)
self.wait()
self.play(
r_line.scale, 3 / 2, {"about_point": ORIGIN}
)
self.wait()
# Finally 4
self.play(
theta_tracker.set_value, 4,
Rotate(r_line, angle=1, about_point=ORIGIN),
)
self.wait()
self.play(
r_line.scale, 4 / 3, {"about_point": ORIGIN}
)
self.wait()
# Talk about 2, 3 then 4
for n in [2, 3, 4]:
self.play(
Rotate(dot, 1, about_point=ORIGIN),
theta_tracker.set_value, n,
)
self.wait()
self.play(dot.move_to, self.get_polar_point(n, n))
self.wait()
# Zoom out and show spiral
big_anim = Succession(*3 * [Animation(Mobject())], *it.chain(*[
[
AnimationGroup(
Rotate(dot, 1, about_point=ORIGIN),
ApplyMethod(theta_tracker.set_value, n),
),
ApplyMethod(dot.move_to, self.get_polar_point(n, n))
]
for n in [5, 6, 7, 8, 9]
]))
spiral = ParametricFunction(
lambda t: self.get_polar_point(t, t),
t_min=0,
@ -511,7 +596,7 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
stroke_width=1.5,
)
self.add(spiral, pp_points, np_points)
# self.add(spiral, pp_points, np_points)
self.polar_grid.generate_target()
for mob in self.polar_grid:
@ -519,18 +604,24 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
mob.set_stroke(width=1)
self.play(
ApplyMethod(
frame.scale, 3,
run_time=5,
rate_func=lambda t: smooth(t, 2)
),
frame.scale, 3,
big_anim,
run_time=10,
)
self.play(
# ApplyMethod(
# frame.scale, 1.5,
# run_time=2,
# rate_func=lambda t: smooth(t, 2)
# ),
ShowCreation(
spiral,
run_time=6,
run_time=4,
),
FadeOut(r_line),
FadeOut(arc),
MoveToTarget(self.polar_grid)
FadeOut(dot),
# MoveToTarget(self.polar_grid)
)
self.wait()
@ -621,11 +712,19 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
def get_arc(self, theta, r=1, color=None):
if color is None:
color = self.theta_color
return Arc(
angle=theta,
radius=r,
stroke_color=color,
return ParametricFunction(
lambda t: self.get_polar_point(1 + 0.025 * t, t),
t_min=0,
t_max=theta,
dt=0.25,
color=color,
stroke_width=3,
)
# return Arc(
# angle=theta,
# radius=r,
# stroke_color=color,
# )
class ReplacePolarCoordinatesWithPrimes(RefresherOnPolarCoordinates):
@ -887,7 +986,7 @@ class AskAboutRelationToPrimes(TeacherStudentsScene):
q_marks = TextMobject("???")
q_marks.set_color(YELLOW)
group = VGroup(numbers, arrow, primes)
group = VGroup(primes, arrow, numbers)
group.arrange(RIGHT)
q_marks.next_to(arrow, UP)
group.add(q_marks)
@ -958,6 +1057,94 @@ class ThickZoomOutOnPrimesWithNumbers(ZoomOutOnPrimesWithNumbers):
}
class HighlightGapsInSpirals(IntroducePrimePatterns):
def construct(self):
self.setup_spiral()
max_n_tracker = ValueTracker(0)
get_max_n = max_n_tracker.get_value
gaps = always_redraw(lambda: VGroup(*[
self.get_highlighted_gap(n - 1, n + 1, get_max_n())
for n in [11, 33]
]))
self.add(gaps)
self.play(max_n_tracker.set_value, 25000, run_time=5)
gaps.clear_updaters()
self.play(FadeOut(gaps))
def setup_spiral(self):
p_spiral = self.get_p_spiral(read_in_primes(self.small_n_primes))
self.add(p_spiral)
self.set_scale(
scale=self.spiral_scale,
spiral=p_spiral,
target_p_spiral_width=5,
run_time=0,
)
def get_highlighted_gap(self, n1, n2, max_n):
l1, l2 = [
[
self.get_polar_point(k, k)
for k in range(INV_7_MOD_44 * n, int(max_n), 5 * 44)
]
for n in (n1, n2)
]
if len(l1) == 0 or len(l2) == 0:
return VectorizedPoint()
result = VMobject()
result.set_points_as_corners(
[*l1, *reversed(l2)]
)
result.make_smooth()
result.set_stroke(GREY, width=0)
result.set_fill(DARK_GREY, 1)
return result
class ImagineYouFoundIt(TeacherStudentsScene):
def construct(self):
you = self.students[1]
others = VGroup(
self.students[0],
self.students[2],
self.teacher,
)
bubble = you.get_bubble(direction=LEFT)
you_label = TextMobject("You")
arrow = Vector(DOWN)
arrow.next_to(you, UP)
you_label.next_to(arrow, UP)
self.play(
you.change, "hesitant", you_label,
FadeInFromDown(you_label),
GrowArrow(arrow),
others.set_opacity, 0.25,
)
self.play(Blink(you))
self.play(
ShowCreation(bubble),
FadeOut(you_label),
FadeOut(arrow),
you.change, "pondering",
)
self.play(you.look_at, bubble.get_corner(UR))
self.play(Blink(you))
self.wait(3)
self.play(you.change, "hooray")
self.play(Blink(you))
self.wait(2)
self.play(you.change, "sassy", bubble.get_top())
self.wait(3)
class ShowSpiralsForWholeNumbers(CountSpirals):
CONFIG = {
"max_prime": 10000,
@ -1061,6 +1248,50 @@ class ShowSpiralsForWholeNumbers(CountSpirals):
)
class SeparateIntoTwoQuestions(Scene):
def construct(self):
top_q = TextMobject("Why do", " primes", " cause", " spirals", "?")
top_q.scale(2)
top_q.to_edge(UP)
q1 = TextMobject("Where do the\\\\", "spirals", " come from?")
q2 = TextMobject("What happens when\\\\", "filtering to", " primes", "?")
for q in q1, q2:
q.scale(1.3)
q.next_to(top_q, DOWN, LARGE_BUFF)
q1.to_edge(LEFT)
q1.set_color(YELLOW)
q2.to_edge(RIGHT)
q2.set_color(TEAL)
v_line = DashedLine(
top_q.get_bottom() + MED_SMALL_BUFF * DOWN,
FRAME_HEIGHT * DOWN / 2,
)
self.add(top_q)
self.wait()
for q, text in [(q1, "spirals"), (q2, "primes")]:
self.play(
top_q.get_part_by_tex(text).set_color, q.get_color(),
TransformFromCopy(
top_q.get_part_by_tex(text),
q.get_part_by_tex(text),
),
LaggedStartMap(
FadeIn,
filter(
lambda m: m is not q.get_part_by_tex(text),
q,
)
),
)
self.wait()
self.play(ShowCreation(v_line))
self.wait()
class ExplainSixSpirals(ShowSpiralsForWholeNumbers):
CONFIG = {
"max_N": 150,
@ -2510,13 +2741,200 @@ class CompareTauToApprox(Scene):
class ShowClassesOfPrimeRays(SpiralScene):
CONFIG = {
"max_N": int(1e6),
"scale": 1e5
}
def construct(self):
self.setup_rays()
self.show_classes()
def setup_rays(self):
pass
spiral = self.get_prime_p_spiral(self.max_N)
self.add(spiral)
self.set_scale(
scale=self.scale,
spiral=spiral,
target_p_spiral_width=3,
run_time=0
)
def show_classes(self):
pass
max_N = self.max_N
mod = 710
primes = read_in_primes(max_N)
rect = FullScreenFadeRectangle()
rect.set_opacity(0)
self.add(rect)
last_ray = PMobject()
last_label = VGroup(*[VectorizedPoint() for x in range(3)])
for i in range(40):
if get_gcd(i, mod) != 1:
continue
r = (INV_113_MOD_710 * i) % mod
sequence = filter(
lambda x: x in primes,
range(r, max_N, mod)
)
ray = self.get_v_spiral(sequence, box_width=0.03)
ray.set_color(GREEN)
ray.set_opacity(0.9)
label = VGroup(
*TexMobject("710k", "+"),
Integer(r)
)
label.arrange(RIGHT, buff=SMALL_BUFF)
label.next_to(ray[100], UL, SMALL_BUFF)
label[2].save_state()
label[2].set_opacity(0)
label[2].move_to(last_label, RIGHT)
self.play(
rect.set_opacity, 0.5,
ShowCreation(ray),
LaggedStartMap(FadeOut, last_ray),
ReplacementTransform(last_label[:2], label[:2]),
Restore(label[2]),
last_label[2].move_to, label[2].saved_state,
last_label[2].set_opacity, 0,
)
self.remove(last_label)
self.add(label)
self.wait()
last_ray = ray
last_label = label
class ShowFactorsOf710(Scene):
def construct(self):
equation = TexMobject(
"710", "=",
"71", "\\cdot",
"5", "\\cdot",
"2",
)
equation.scale(1.5)
equation.to_corner(UL)
ten = TexMobject("10")
ten.match_height(equation)
ten.move_to(equation.get_part_by_tex("5"), LEFT)
self.add(equation[0])
self.wait()
self.play(
TransformFromCopy(
equation[0][:2],
equation[2],
),
FadeIn(equation[1]),
FadeIn(equation[3]),
TransformFromCopy(
equation[0][2:],
ten,
)
)
self.wait()
self.remove(ten)
self.play(*[
TransformFromCopy(ten, mob)
for mob in equation[4:]
])
self.wait()
# Circle factors
colors = [RED, BLUE, PINK]
for factor, color in zip(equation[:-6:-2], colors):
rect = SurroundingRectangle(factor)
rect.set_color(color)
self.play(ShowCreation(rect))
self.wait()
self.play(FadeOut(rect))
class LookAtRemainderMod710(Scene):
def construct(self):
t2c = {
"n": YELLOW,
"r": GREEN
}
equation = TexMobject(
"n", "=", "710", "\\cdot", "q", "+", "r",
tex_to_color_map=t2c,
)
equation.scale(1.5)
n_arrow = Vector(UP).next_to(equation.get_part_by_tex("n"), DOWN)
r_arrow = Vector(UP).next_to(equation.get_part_by_tex("r"), DOWN)
n_arrow.set_color(t2c["n"])
r_arrow.set_color(t2c["r"])
n_label = TextMobject("Some\\\\number")
r_label = TextMobject("Remainder")
VGroup(n_label, r_label).scale(1.5)
n_label.next_to(n_arrow, DOWN)
n_label.match_color(n_arrow)
r_label.next_to(r_arrow, DOWN)
r_label.match_color(r_arrow)
self.add(equation)
self.play(
FadeInFrom(n_label, UP),
ShowCreation(n_arrow),
)
self.wait()
self.play(
FadeInFrom(r_label, DOWN),
ShowCreation(r_arrow),
)
self.wait()
class EliminateNonPrimative710Residues(ShowClassesOfPrimeRays):
def construct(self):
self.setup_rays()
self.eliminate_classes()
def setup_rays(self):
mod = 710
rays = PGroup(*[
self.get_p_spiral(range(r, self.max_N, mod))
for r in range(mod)
])
rays.set_color(YELLOW)
self.add(rays)
self.set_scale(
scale=self.scale,
spiral=rays,
target_p_spiral_width=1,
run_time=0,
)
self.wait()
self.rays = rays
def eliminate_classes(self):
rays = self.rays
rect = FullScreenFadeRectangle()
rect.set_opacity(0)
mod = 710
mult2 = PGroup(*[rays[i] for i in range(0, mod, 2)])
mult5 = PGroup(*[rays[i] for i in range(0, mod, 5) if i % 2 != 0])
mult71 = PGroup(*[rays[i] for i in range(0, mod, 71) if (i % 2 != 0 and i % 5 != 0)])
groups = Group(mult2, mult5, mult71)
colors = [RED, BLUE, PINK]
for group, color in zip(groups, colors):
group.set_stroke_width(2)
self.add(rect, group)
self.wait()
self.remove(rect, group)
self.wait()