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Animations for the introduction of the prime spiral video
This commit is contained in:
Grant Sanderson
@ -1,7 +1,10 @@
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from manimlib.imports import *
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import json
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OUTPUT_DIRECTORY = "spirals"
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INV_113_MOD_710 = 377 # Inverse of 113 mode 710
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INV_7_MOD_44 = 19
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def generate_prime_list(*args):
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@ -25,58 +28,133 @@ def generate_prime_list(*args):
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return result
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class SpiralScene(Scene):
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def get_gcd(x, y):
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while y > 0:
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x, y = y, x % y
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return x
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def read_in_primes(max_N=None):
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with open(os.path.join("assets", "primes.json")) as fp:
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primes = np.array(json.load(fp))
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if max_N:
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return primes[primes <= max_N]
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return primes
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class SpiralScene(MovingCameraScene):
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CONFIG = {
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"axes_config": {
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"number_line_config": {
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"stroke_width": 1.5,
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}
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}
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},
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"default_dot_color": TEAL,
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"p_spiral_width": 6,
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}
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def setup(self):
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super().setup()
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self.axes = Axes(**self.axes_config)
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self.add(self.axes)
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def get_dots_in_spiral(self, sequence, axes=None, dot_radius=None):
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def get_v_spiral(self, sequence, axes=None, data_point_width=None):
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if axes is None:
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axes = self.axes
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if dot_radius is None:
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if data_point_width is None:
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unit = get_norm(axes.c2p(1, 0) - axes.c2p(0, 0)),
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dot_radius = max(
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0.1 / (-np.log10(unit) + 1),
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0.01,
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data_point_width = max(
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0.2 / (-np.log10(unit) + 1),
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0.02,
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)
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return VGroup(*[
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Dot(
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self.get_polar_point(n, n, axes),
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radius=dot_radius,
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fill_color=TEAL,
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)
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Square(
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side_length=data_point_width,
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fill_color=self.default_dot_color,
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fill_opacity=1,
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stroke_width=0,
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).move_to(self.get_polar_point(n, n, axes))
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for n in sequence
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])
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def get_polar_point(self, r, theta, axes):
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def get_p_spiral(self, sequence, axes=None):
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if axes is None:
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axes = self.axes
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result = PMobject(
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color=self.default_dot_color,
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stroke_width=self.p_spiral_width,
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)
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result.add_points([
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self.get_polar_point(n, n, axes)
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for n in sequence
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])
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return result
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def get_prime_v_spiral(self, max_N, **kwargs):
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primes = read_in_primes(max_N)
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return self.get_v_spiral(primes, **kwargs)
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def get_prime_p_spiral(self, max_N, **kwargs):
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primes = read_in_primes(max_N)
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return self.get_p_spiral(primes, **kwargs)
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def get_polar_point(self, r, theta, axes=None):
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if axes is None:
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axes = self.axes
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return axes.c2p(r * np.cos(theta), r * np.sin(theta))
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def set_scale(self, scale, exceptions=None, run_time=3):
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axes = self.axes
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if exceptions is None:
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exceptions = []
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def set_scale(self, scale,
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axes=None,
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spiral=None,
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to_shrink=None,
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min_data_point_width=0.05,
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target_p_spiral_width=None,
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run_time=3):
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if axes is None:
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axes = self.axes
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sf = self.get_scale_factor(scale, axes)
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unit = get_norm(axes.c2p(1, 0) - axes.c2p(0, 0))
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sf = unit / scale
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anims = []
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for mob in [axes, spiral, to_shrink]:
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if mob is None:
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continue
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mob.generate_target()
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mob.target.scale(sf, about_point=ORIGIN)
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if mob is spiral:
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if isinstance(mob, VMobject):
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old_width = mob[0].get_width()
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for submob in mob.target:
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submob.set_width(max(
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old_width * sf,
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min_data_point_width,
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))
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elif isinstance(mob, PMobject):
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if target_p_spiral_width is not None:
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mob.target.set_stroke_width(target_p_spiral_width)
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anims.append(MoveToTarget(mob))
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self.remove(*exceptions)
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self.play(*[
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ApplyMethod(
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mob.scale, sf,
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about_point=ORIGIN,
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if run_time == 0:
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for anim in anims:
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anim.begin()
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anim.update(1)
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anim.finish()
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else:
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self.play(
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*anims,
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run_time=run_time,
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rate_func=lambda t: interpolate(
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smooth(t),
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smooth(t)**(sf**(0.5)),
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t,
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)
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)
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for mob in self.get_top_level_mobjects()
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])
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def get_scale_factor(self, target_scale, axes=None):
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if axes is None:
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axes = self.axes
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unit = get_norm(axes.c2p(1, 0) - axes.c2p(0, 0))
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return 1 / (target_scale * unit)
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# Scenes
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@ -101,7 +179,9 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
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plane = NumberPlane()
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plane.add_coordinates()
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x, y = 3, 2
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x = 3 * np.cos(PI / 6)
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y = 3 * np.sin(PI / 6)
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point = plane.c2p(x, y)
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xp = plane.c2p(x, 0)
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origin = plane.c2p(0, 0)
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@ -114,7 +194,7 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
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dot = Dot(point)
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coord_label = self.get_coord_label(x, y, x_color, y_color)
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coord_label = self.get_coord_label(0, 0, x_color, y_color)
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x_coord = coord_label.x_coord
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y_coord = coord_label.y_coord
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@ -251,6 +331,7 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
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ChangeDecimalToValue(degree_label, theta_value / DEGREES),
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)
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self.play(
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degree_label.scale, 0.9,
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degree_label.move_to, theta_coord,
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FadeInFromDown(theta_label),
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)
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@ -371,9 +452,8 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
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self.wait()
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# And now 3
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self.play(theta_tracker.set_value, 3)
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self.play(
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Transform(arc, self.get_arc(3)),
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theta_tracker.set_value, 3,
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Rotate(r_line, angle=1, about_point=ORIGIN),
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)
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self.wait()
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@ -382,6 +462,17 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
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)
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self.wait()
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# Finally 4
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self.play(
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theta_tracker.set_value, 4,
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Rotate(r_line, angle=1, about_point=ORIGIN),
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)
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self.wait()
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self.play(
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r_line.scale, 4 / 3, {"about_point": ORIGIN}
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)
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self.wait()
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# Zoom out and show spiral
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spiral = ParametricFunction(
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lambda t: self.get_polar_point(t, t),
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@ -466,7 +557,9 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
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include_background_rectangle=True,
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**decimal_kwargs):
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x_coord = DecimalNumber(x, **decimal_kwargs)
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x_coord.set_color(x_color)
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y_coord = DecimalNumber(y, **decimal_kwargs)
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y_coord.set_color(y_color)
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coord_label = VGroup(
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TexMobject("("), x_coord,
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@ -496,33 +589,253 @@ class RefresherOnPolarCoordinates(MovingCameraScene):
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)
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class SpiralSceneTest(SpiralScene):
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class IntroducePrimePatterns(SpiralScene):
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CONFIG = {
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"small_n_primes": 25000,
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"big_n_primes": 1000000,
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"axes_config": {
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"x_min": -25,
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"x_max": 25,
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"y_min": -25,
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"y_max": 25,
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},
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"spiral_scale": 3e3,
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"ray_scale": 1e5,
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}
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def construct(self):
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N = 1 * 10**4
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axes = self.axes
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self.slowly_zoom_out()
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self.show_clumps_of_four()
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primes = generate_prime_list(N)
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prime_dots = self.get_dots_in_spiral(primes)
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def slowly_zoom_out(self):
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zoom_time = 8
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scale_factor = TAU / N
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prime_spiral = self.get_prime_p_spiral(self.small_n_primes)
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prime_spiral.set_stroke_width(25)
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self.add(prime_spiral)
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axes.generate_target(use_deepcopy=True)
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axes.target.scale(scale_factor, about_point=ORIGIN)
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prime_dots.target = self.get_dots_in_spiral(primes, axes.target)
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self.add(prime_dots)
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self.set_scale(3, spiral=prime_spiral)
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self.wait()
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self.set_scale(0.1)
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# self.add(ParametricFunction(
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# lambda t: axes.c2p(t * np.cos(t), t * np.sin(t)),
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# t_min=0,
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# t_max=30,
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# ))
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return
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ap_dots = self.get_dots_in_spiral(range(0, N, 666))
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ap_dots.set_color(YELLOW)
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self.play(ShowIncreasingSubsets(ap_dots))
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self.set_scale(
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self.spiral_scale,
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spiral=prime_spiral,
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target_p_spiral_width=8,
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run_time=zoom_time,
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)
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self.wait()
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self.remove(prime_spiral)
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prime_spiral = self.get_prime_p_spiral(self.big_n_primes)
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prime_spiral.set_stroke_width(8)
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self.set_scale(
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self.ray_scale,
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spiral=prime_spiral,
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target_p_spiral_width=4,
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run_time=zoom_time,
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)
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self.wait()
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def show_clumps_of_four(self):
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line_groups = VGroup()
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for n in range(71):
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group = VGroup()
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for k in [-3, -1, 1, 3]:
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r = ((10 * n + k) * INV_113_MOD_710) % 710
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group.add(self.get_arithmetic_sequence_line(
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710, r, self.big_n_primes
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))
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line_groups.add(group)
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line_groups.set_stroke(YELLOW, 2, opacity=0.5)
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self.play(ShowCreation(line_groups[0]))
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for g1, g2 in zip(line_groups, line_groups[1:5]):
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self.play(
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FadeOut(g1),
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ShowCreation(g2)
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)
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self.play(
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FadeOut(line_groups[4]),
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LaggedStartMap(
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VFadeInThenOut,
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line_groups[4:],
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lag_ratio=0.5,
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run_time=5,
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)
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)
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self.wait()
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def get_arithmetic_sequence_line(self, N, r, max_val, skip_factor=5):
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line = VMobject()
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line.set_points_smoothly([
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self.get_polar_point(x, x)
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for x in range(r, max_val, skip_factor * N)
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])
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return line
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class AskWhat(TeacherStudentsScene):
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def construct(self):
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screen = self.screen
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self.student_says(
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"I'm sory,\\\\what?!?",
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target_mode="angry",
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look_at_arg=screen,
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student_index=2,
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added_anims=[
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self.teacher.change, "happy", screen,
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self.students[0].change, "confused", screen,
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self.students[1].change, "confused", screen,
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]
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)
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self.wait(3)
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class CountSpirals(IntroducePrimePatterns):
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CONFIG = {
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"count_sound": "pen_click.wav",
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}
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def construct(self):
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prime_spiral = self.get_prime_p_spiral(self.small_n_primes)
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self.add(prime_spiral)
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self.set_scale(
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self.spiral_scale,
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spiral=prime_spiral,
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run_time=0,
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)
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spiral_lines = self.get_all_primative_arithmetic_lines(
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44, self.small_n_primes, INV_7_MOD_44,
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)
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spiral_lines.set_stroke(YELLOW, 2, opacity=0.5)
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counts = VGroup()
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for n, spiral in zip(it.count(1), spiral_lines):
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count = Integer(n)
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count.move_to(spiral.point_from_proportion(0.25))
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counts.add(count)
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run_time = 3
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self.play(
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ShowIncreasingSubsets(spiral_lines),
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ShowSubmobjectsOneByOne(counts),
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run_time=run_time,
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rate_func=linear,
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)
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self.add_count_clicks(len(spiral_lines), run_time)
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self.play(
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counts[-1].scale, 3,
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counts[-1].set_stroke, BLACK, 5, {"background": True},
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)
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self.wait()
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def get_all_primative_arithmetic_lines(self, N, max_val, mult_factor):
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lines = VGroup()
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for r in range(1, N):
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if get_gcd(N, r) == 1:
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lines.add(
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self.get_arithmetic_sequence_line(N, (mult_factor * r) % N, max_val)
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)
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return lines
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def add_count_clicks(self, N, time, rate_func=linear):
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alphas = np.arange(0, 1, 1 / N)
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if rate_func is linear:
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delays = time * alphas
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else:
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delays = time * np.array([
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binary_search(
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rate_func,
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alpha,
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0,
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1,
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)
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for alpha in alphas
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])
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for delay in delays:
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self.add_sound(
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self.count_sound,
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time_offset=-delay,
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gain=-15,
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)
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class CountRays(CountSpirals):
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def construct(self):
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prime_spiral = self.get_prime_p_spiral(self.big_n_primes)
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self.add(prime_spiral)
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self.set_scale(
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self.ray_scale,
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spiral=prime_spiral,
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run_time=0,
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)
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spiral_lines = self.get_all_primative_arithmetic_lines(
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710, self.big_n_primes, INV_113_MOD_710,
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)
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spiral_lines.set_stroke(YELLOW, 2, opacity=0.5)
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counts = VGroup()
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for n, spiral in zip(it.count(1), spiral_lines):
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count = Integer(n)
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count.move_to(spiral.point_from_proportion(0.25))
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counts.add(count)
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run_time = 6
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self.play(
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ShowIncreasingSubsets(spiral_lines),
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ShowSubmobjectsOneByOne(counts),
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run_time=run_time,
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rate_func=smooth,
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)
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self.add_count_clicks(len(spiral_lines), run_time, rate_func=smooth)
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self.play(
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counts[-1].scale, 3,
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counts[-1].set_stroke, BLACK, 5, {"background": True},
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)
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self.wait()
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self.play(FadeOut(spiral_lines))
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self.wait()
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class AskAboutRelationToPrimes(TeacherStudentsScene):
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def construct(self):
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numbers = TextMobject("20, 280")
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arrow = Arrow(LEFT, RIGHT)
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primes = TextMobject("2, 3, 5, 7, 11, \\dots")
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q_marks = TextMobject("???")
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q_marks.set_color(YELLOW)
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|
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group = VGroup(numbers, arrow, primes)
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group.arrange(RIGHT)
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q_marks.next_to(arrow, UP)
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group.add(q_marks)
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group.scale(1.5)
|
||||
group.next_to(self.pi_creatures, UP, LARGE_BUFF)
|
||||
|
||||
self.play(
|
||||
self.get_student_changes(
|
||||
*3 * ["maybe"],
|
||||
look_at_arg=numbers,
|
||||
),
|
||||
self.teacher.change, "maybe", numbers,
|
||||
ShowCreation(arrow),
|
||||
FadeInFrom(numbers, RIGHT)
|
||||
)
|
||||
self.play(
|
||||
FadeInFrom(primes, LEFT),
|
||||
)
|
||||
self.play(
|
||||
LaggedStartMap(FadeInFromDown, q_marks[0]),
|
||||
Blink(self.teacher)
|
||||
)
|
||||
self.wait(3)
|
||||
|
||||
|
||||
class ZoomOutOnPrimesWithNumbers(IntroducePrimePatterns):
|
||||
def construct(self):
|
||||
pass
|
||||
|
||||
Reference in New Issue
Block a user