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Up to introduction of factorization in leibniz
This commit is contained in:
Grant Sanderson
parent
8d53f67f03
commit
0092ac9a2a
3 changed files with 576 additions and 19 deletions
588
leibniz.py
588
leibniz.py
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@ -81,7 +81,9 @@ class LatticePointScene(Scene):
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)
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dot.r_squared = r_squared
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self.lattice_points.add(dot)
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self.lattice_points.sort_submobjects(np.linalg.norm)
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self.lattice_points.sort_submobjects(
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lambda p : np.linalg.norm(p - self.plane_center)
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)
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def get_circle(self, radius = None, color = None):
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if radius is None:
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@ -109,7 +111,7 @@ class LatticePointScene(Scene):
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r_squared = int(np.round(radius**2))
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root_label = TexMobject("\\sqrt{%d}"%r_squared)
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root_label.add_background_rectangle()
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root_label.next_to(radial_line, DOWN, SMALL_BUFF)
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root_label.next_to(radial_line, UP, SMALL_BUFF)
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return radial_line, root_label
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@ -142,6 +144,54 @@ class LatticePointScene(Scene):
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)
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])
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def add_axis_labels(self, spacing = 2):
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x_max = int(self.plane.point_to_coords(SPACE_WIDTH*RIGHT)[0])
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y_max = int(self.plane.point_to_coords(SPACE_HEIGHT*UP)[1])
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x_range = range(spacing, x_max, spacing)
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y_range = range(spacing, y_max, spacing)
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for r in x_range, y_range:
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r += [-n for n in r]
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tick = Line(ORIGIN, MED_SMALL_BUFF*UP)
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x_ticks = VGroup(*[
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tick.copy().move_to(self.plane.coords_to_point(x, 0))
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for x in x_range
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])
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tick.rotate(-np.pi/2)
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y_ticks = VGroup(*[
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tick.copy().move_to(self.plane.coords_to_point(0, y))
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for y in y_range
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])
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x_labels = VGroup(*[
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TexMobject(str(x))
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for x in x_range
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])
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y_labels = VGroup(*[
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TexMobject(str(y) + "i")
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for y in y_range
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])
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for labels, ticks in (x_labels, x_ticks), (y_labels, y_ticks):
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labels.scale(0.6)
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for tex_mob, tick in zip(labels, ticks):
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tex_mob.add_background_rectangle()
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tex_mob.next_to(
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tick,
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tick.get_start() - tick.get_end(),
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SMALL_BUFF
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)
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self.add(x_ticks, y_ticks, x_labels, y_labels)
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digest_locals(self, [
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"x_ticks", "y_ticks",
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"x_labels", "y_labels",
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])
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def point_to_int_coords(self, point):
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x, y = self.plane.point_to_coords(point)[:2]
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return (int(np.round(x)), int(np.round(y)))
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def dot_to_int_coords(self, dot):
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return self.point_to_int_coords(dot.get_center())
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######
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@ -1319,14 +1369,13 @@ class IntroduceComplexConjugate(LatticePointScene):
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CONFIG = {
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"y_radius" : 20,
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"x_radius" : 30,
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"plane_scale_factor" : 2,
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"plane_scale_factor" : 1.7,
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"plane_center" : 2*LEFT,
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"example_coords" : (3, 4),
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"x_color" : GREEN,
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"y_color" : RED,
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}
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def construct(self):
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self.force_skipping()
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self.resize_plane()
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self.write_points_with_complex_coords()
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self.introduce_complex_conjugate()
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@ -1335,7 +1384,11 @@ class IntroduceComplexConjugate(LatticePointScene):
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self.show_geometrically()
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def resize_plane(self):
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self.plane.scale(self.plane_scale_factor)
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self.plane.scale(
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self.plane_scale_factor,
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about_point = self.plane_center
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)
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self.plane.set_stroke(width = 3)
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def write_points_with_complex_coords(self):
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x, y = self.example_coords
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@ -1370,7 +1423,7 @@ class IntroduceComplexConjugate(LatticePointScene):
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label.add_background_rectangle()
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label.next_to(dot, UP+RIGHT, buff = 0)
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y_range = range(-8, 10, 2)
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y_range = range(-9, 10, 3)
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ticks = VGroup(*[
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Line(
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ORIGIN, MED_SMALL_BUFF*RIGHT
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@ -1419,13 +1472,14 @@ class IntroduceComplexConjugate(LatticePointScene):
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def introduce_complex_conjugate(self):
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x, y = self.example_coords
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equation = VGroup(
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TexMobject(str(x), "^2", "+", str(y), "^2", "="),
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TexMobject("25 = ", str(x), "^2", "+", str(y), "^2", "="),
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TexMobject("(", str(x), "+", str(y), "i", ")"),
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TexMobject("(", str(x), "-", str(y), "i", ")"),
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)
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equation.arrange_submobjects(
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RIGHT, buff = SMALL_BUFF,
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)
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VGroup(*equation[-2:]).shift(0.5*SMALL_BUFF*DOWN)
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equation.scale(0.9)
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equation.to_corner(UP+RIGHT, buff = MED_SMALL_BUFF)
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for tex_mob in equation:
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@ -1449,9 +1503,11 @@ class IntroduceComplexConjugate(LatticePointScene):
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conjugate_words.add_background_rectangle()
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conjugate_words.next_to(brace, DOWN)
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self.revert_to_original_skipping_status()
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for part in equation:
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self.play(FadeIn(part))
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self.play(FadeIn(
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equation,
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run_time = 3,
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submobject_mode = "lagged_start"
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))
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self.dither(2)
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self.play(
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GrowFromCenter(brace),
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@ -1467,17 +1523,519 @@ class IntroduceComplexConjugate(LatticePointScene):
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])
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self.dither(2)
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self.conjugate_label = VGroup(brace, conjugate_words)
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self.equation = equation
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self.conjugate_dot = dot
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def show_confusion(self):
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pass
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randy = Randolph(color = BLUE_C).to_corner(DOWN+LEFT)
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morty = Mortimer().to_edge(DOWN)
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randy.make_eye_contact(morty)
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self.play(*map(FadeIn, [randy, morty]))
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self.play(PiCreatureSays(
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randy, "Wait \\dots why?",
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target_mode = "confused",
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))
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self.play(Blink(randy))
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self.dither(2)
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self.play(
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RemovePiCreatureBubble(
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randy, target_mode = "erm",
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),
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PiCreatureSays(
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morty, "Now it's a \\\\ factoring problem!",
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target_mode = "hooray",
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bubble_kwargs = {"width" : 5, "height" : 3}
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)
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)
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self.play(
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morty.look_at, self.equation,
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randy.look_at, self.equation,
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)
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self.play(Blink(morty))
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self.play(randy.change_mode, "pondering")
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self.play(RemovePiCreatureBubble(morty))
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self.play(*map(FadeOut, [randy, morty]))
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def expand_algebraically(self):
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pass
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x, y = self.example_coords
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expansion = VGroup(
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TexMobject(str(x), "^2"),
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TexMobject("-", "(", str(y), "i", ")^2")
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)
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expansion.arrange_submobjects(RIGHT, buff = SMALL_BUFF)
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expansion.next_to(
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VGroup(*self.equation[-2:]),
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DOWN, LARGE_BUFF
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)
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alt_y_term = TexMobject("+", str(y), "^2")
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alt_y_term.move_to(expansion[1], LEFT)
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for tex_mob in list(expansion) + [alt_y_term]:
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tex_mob.highlight_by_tex(str(x), self.x_color)
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tex_mob.highlight_by_tex(str(y), self.y_color)
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tex_mob.rect = BackgroundRectangle(tex_mob)
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x1 = self.equation[-2][1][1]
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x2 = self.equation[-1][1][1]
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y1 = VGroup(*self.equation[-2][1][3:5])
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y2 = VGroup(*self.equation[-1][1][2:5])
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vect = MED_LARGE_BUFF*UP
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self.play(FadeOut(self.conjugate_label))
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group = VGroup(x1, x2)
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self.play(group.shift, -vect)
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self.play(
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FadeIn(expansion[0].rect),
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ReplacementTransform(group.copy(), expansion[0]),
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)
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self.play(group.shift, vect)
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group = VGroup(x1, y2)
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self.play(group.shift, -vect)
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self.dither()
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self.play(group.shift, vect)
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group = VGroup(x2, y1)
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self.play(group.shift, -vect)
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self.dither()
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self.play(group.shift, vect)
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group = VGroup(*it.chain(y1, y2))
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self.play(group.shift, -vect)
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self.dither()
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self.play(
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FadeIn(expansion[1].rect),
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ReplacementTransform(group.copy(), expansion[1]),
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)
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self.play(group.shift, vect)
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self.dither(2)
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self.play(
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Transform(expansion[1].rect, alt_y_term.rect),
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Transform(expansion[1], alt_y_term),
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)
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self.dither()
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self.play(*map(FadeOut, [
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expansion[0].rect,
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expansion[1].rect,
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expansion
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]))
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def show_geometrically(self):
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pass
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dots = [self.example_dot, self.conjugate_dot]
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top_dot, low_dot = dots
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for dot in dots:
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dot.line = Line(
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self.plane_center, dot.get_center(),
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color = BLUE
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)
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dot.angle = dot.line.get_angle()
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dot.arc = Arc(
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dot.angle,
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radius = 0.75,
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color = YELLOW
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)
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dot.arc.shift(self.plane_center)
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dot.arc.add_tip(tip_length = 0.2)
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dot.rotate_word = TextMobject("Rotate")
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dot.rotate_word.scale(0.5)
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dot.rotate_word.next_to(dot.arc, RIGHT, SMALL_BUFF)
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dot.magnitude_word = TextMobject("Length 5")
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dot.magnitude_word.scale(0.6)
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dot.magnitude_word.next_to(
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ORIGIN,
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np.sign(dot.get_center()[1])*UP,
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buff = SMALL_BUFF
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)
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dot.magnitude_word.add_background_rectangle()
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dot.magnitude_word.rotate(dot.angle)
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dot.magnitude_word.shift(dot.line.get_center())
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twenty_five_label = TexMobject("25")
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twenty_five_label.add_background_rectangle()
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twenty_five_label.next_to(
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self.plane.coords_to_point(25, 0),
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DOWN
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)
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self.play(ShowCreation(top_dot.line))
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mover = VGroup(
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top_dot.line.copy().highlight(PINK),
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top_dot.copy()
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)
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self.play(FadeIn(
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top_dot.magnitude_word,
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submobject_mode = "lagged_start"
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))
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self.dither()
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self.play(ShowCreation(top_dot.arc))
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self.dither(2)
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self.play(ShowCreation(low_dot.line))
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self.play(
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ReplacementTransform(
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top_dot.arc,
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low_dot.arc
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),
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FadeIn(low_dot.rotate_word)
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)
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self.play(
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Rotate(
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mover, low_dot.angle,
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about_point = self.plane_center
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),
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run_time = 2
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)
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self.play(
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FadeOut(low_dot.arc),
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FadeOut(low_dot.rotate_word),
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FadeIn(low_dot.magnitude_word),
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)
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self.play(
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mover[0].scale_about_point, 5, self.plane_center,
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mover[1].move_to, self.plane.coords_to_point(25, 0),
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run_time = 2
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)
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self.dither()
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self.play(Write(twenty_five_label))
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self.dither(3)
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class NameGaussianIntegers(LatticePointScene):
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CONFIG = {
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"max_lattice_point_radius" : 15,
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"dot_radius" : 0.05,
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"plane_center" : 2*LEFT,
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"x_radius" : 15,
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}
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def construct(self):
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self.add_axis_labels()
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self.add_a_plus_bi()
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self.draw_lattice_points()
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self.add_name()
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self.restrict_to_one_circle()
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self.show_question_algebraically()
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def add_a_plus_bi(self):
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label = TexMobject(
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"a", "+", "b", "i"
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)
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a = label.get_part_by_tex("a")
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b = label.get_part_by_tex("b")
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a.highlight(GREEN)
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b.highlight(RED)
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label.add_background_rectangle()
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label.to_corner(UP+RIGHT)
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integers = TextMobject("Integers")
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integers.next_to(label, DOWN, LARGE_BUFF)
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integers.add_background_rectangle()
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arrows = VGroup(*[
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Arrow(integers.get_top(), mob, tip_length = 0.15)
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for mob in a, b
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])
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self.add_foreground_mobjects(label, integers, arrows)
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self.a_plus_bi = label
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self.integers_label = VGroup(integers, arrows)
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def add_name(self):
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gauss_name = TextMobject(
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"Carl Friedrich Gauss"
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)
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gauss_name.add_background_rectangle()
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gauss_name.next_to(ORIGIN, UP, MED_LARGE_BUFF)
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gauss_name.to_edge(LEFT)
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gaussian_integers = TextMobject("``Gaussian integers'': ")
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gaussian_integers.scale(0.9)
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gaussian_integers.next_to(self.a_plus_bi, LEFT)
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gaussian_integers.add_background_rectangle()
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self.play(FadeIn(gaussian_integers))
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self.add_foreground_mobject(gaussian_integers)
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self.play(FadeIn(
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gauss_name,
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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(3)
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self.play(FadeOut(gauss_name))
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self.gaussian_integers = gaussian_integers
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def restrict_to_one_circle(self):
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dots = self.get_lattice_points_on_r_squared_circle(25).copy()
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for dot in dots:
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dot.scale_in_place(2)
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circle = self.get_circle(5)
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radius, root_label = self.get_radial_line_with_label(5)
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self.play(
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FadeOut(self.lattice_points),
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ShowCreation(circle),
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Rotating(
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radius,
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run_time = 1, rate_func = smooth,
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about_point = self.plane_center
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),
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*map(GrowFromCenter, dots)
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)
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self.play(Write(root_label))
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self.dither()
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self.circle_dots = dots
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def show_question_algebraically(self):
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for i, dot in enumerate(self.circle_dots):
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x, y = self.dot_to_int_coords(dot)
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x_str = str(x)
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y_str = str(y) if y >= 0 else "(%d)"%y
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label = TexMobject(x_str, "+", y_str, "i")
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label.scale(0.8)
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label.next_to(
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dot,
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dot.get_center()-self.plane_center + SMALL_BUFF*(UP+RIGHT),
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buff = 0,
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)
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label.add_background_rectangle()
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dot.label = label
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equation = TexMobject(
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"25 = "
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"(", x_str, "+", y_str, "i", ")",
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"(", x_str, "-", y_str, "i", ")",
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)
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equation.scale(0.9)
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equation.add_background_rectangle()
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equation.to_corner(UP + RIGHT)
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dot.equation = equation
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for mob in label, equation:
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mob.highlight_by_tex(x_str, GREEN, substring = False)
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mob.highlight_by_tex(y_str, RED, substring = False)
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dot.line_pair = VGroup(*[
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Line(
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self.plane_center,
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self.plane.coords_to_point(x, u*y),
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color = PINK,
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)
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for u in 1, -1
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])
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dot.conjugate_dot = self.circle_dots[-i]
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self.play(*map(FadeOut, [
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self.a_plus_bi, self.integers_label,
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self.gaussian_integers,
|
||||
]))
|
||||
|
||||
last_dot = None
|
||||
for dot in self.circle_dots:
|
||||
anims = [
|
||||
dot.highlight, PINK,
|
||||
dot.conjugate_dot.highlight, PINK,
|
||||
]
|
||||
if last_dot is None:
|
||||
anims += [
|
||||
FadeIn(dot.equation),
|
||||
FadeIn(dot.label),
|
||||
]
|
||||
anims += map(ShowCreation, dot.line_pair)
|
||||
else:
|
||||
anims += [
|
||||
last_dot.highlight, self.dot_color,
|
||||
last_dot.conjugate_dot.highlight, self.dot_color,
|
||||
ReplacementTransform(last_dot.equation, dot.equation),
|
||||
ReplacementTransform(last_dot.label, dot.label),
|
||||
ReplacementTransform(last_dot.line_pair, dot.line_pair),
|
||||
]
|
||||
self.play(*anims)
|
||||
self.dither()
|
||||
last_dot = dot
|
||||
|
||||
class FactorOrdinaryNumber(TeacherStudentsScene):
|
||||
def construct(self):
|
||||
equation = TexMobject(
|
||||
"2{,}250", "=", "2 \\cdot 3^2 \\cdot 5^3"
|
||||
)
|
||||
equation.next_to(self.get_pi_creatures(), UP, LARGE_BUFF)
|
||||
number = equation[0]
|
||||
alt_rhs_list = list(it.starmap(TexMobject, [
|
||||
("\\ne", "2^2 \\cdot 563"),
|
||||
("\\ne", "2^2 \\cdot 3 \\cdot 11 \\cdot 17"),
|
||||
("\\ne", "2 \\cdot 7^2 \\cdot 23"),
|
||||
("=", "(-2) \\cdot (-3) \\cdot (3) \\cdot 5^3"),
|
||||
]))
|
||||
for alt_rhs in alt_rhs_list:
|
||||
if "\\ne" in alt_rhs.get_tex_string():
|
||||
alt_rhs.highlight(RED)
|
||||
else:
|
||||
alt_rhs.highlight(GREEN)
|
||||
alt_rhs.move_to(equation.get_right())
|
||||
number.save_state()
|
||||
number.next_to(self.teacher, UP+LEFT)
|
||||
|
||||
self.play(
|
||||
self.teacher.change_mode, "raise_right_hand",
|
||||
Write(number)
|
||||
)
|
||||
self.dither(2)
|
||||
self.play(
|
||||
number.restore,
|
||||
Write(VGroup(*equation[1:]))
|
||||
)
|
||||
self.change_student_modes(
|
||||
*["pondering"]*3,
|
||||
look_at_arg = equation,
|
||||
added_anims = [self.teacher.change_mode, "happy"]
|
||||
)
|
||||
self.dither()
|
||||
last_alt_rhs = None
|
||||
for alt_rhs in alt_rhs_list:
|
||||
equation.generate_target()
|
||||
equation.target.next_to(alt_rhs, LEFT)
|
||||
anims = [MoveToTarget(equation)]
|
||||
if last_alt_rhs:
|
||||
anims += [ReplacementTransform(last_alt_rhs, alt_rhs)]
|
||||
else:
|
||||
anims += [FadeIn(alt_rhs)]
|
||||
self.play(*anims)
|
||||
if alt_rhs is alt_rhs_list[-1]:
|
||||
self.change_student_modes(
|
||||
*["sassy"]*3,
|
||||
look_at_arg = alt_rhs
|
||||
)
|
||||
self.dither(2)
|
||||
last_alt_rhs = alt_rhs
|
||||
|
||||
self.play(
|
||||
FadeOut(VGroup(equation, alt_rhs)),
|
||||
PiCreatureSays(
|
||||
self.teacher,
|
||||
"It's similar for \\\\ Gaussian integers"
|
||||
)
|
||||
)
|
||||
self.change_student_modes(*["happy"]*3)
|
||||
self.dither(3)
|
||||
|
||||
class IntroduceGaussianPrimes(LatticePointScene):
|
||||
CONFIG = {
|
||||
"plane_center" : LEFT,
|
||||
"x_radius" : 13,
|
||||
}
|
||||
def construct(self):
|
||||
five_dot, p1_dot, p2_dot, p3_dot, p4_dot = dots = [
|
||||
Dot(self.plane.coords_to_point(*coords))
|
||||
for coords in [
|
||||
(5, 0),
|
||||
(2, 1), (2, -1),
|
||||
(-1, 2), (-1, -2),
|
||||
]
|
||||
]
|
||||
five_dot.highlight(YELLOW)
|
||||
p_dots = VGroup(*dots[1:])
|
||||
p_dots.highlight(PINK)
|
||||
|
||||
five_label, p1_label, p2_label, p3_label, p4_label = labels = [
|
||||
TexMobject(tex).add_background_rectangle()
|
||||
for tex in "5", "2+i", "2-i", "-1+2i", "-1-2i",
|
||||
]
|
||||
vects = [DOWN, UP+RIGHT, DOWN+RIGHT, UP+LEFT, DOWN+LEFT]
|
||||
for dot, label, vect in zip(dots, labels, vects):
|
||||
label.next_to(dot, vect, SMALL_BUFF)
|
||||
|
||||
arc_angle = 0.8*np.pi
|
||||
times_i_arc = Arrow(
|
||||
p1_dot.get_top(), p3_dot.get_top(),
|
||||
path_arc = arc_angle
|
||||
)
|
||||
times_neg_i_arc = Arrow(
|
||||
p2_dot.get_bottom(), p4_dot.get_bottom(),
|
||||
path_arc = -arc_angle
|
||||
)
|
||||
times_i = TexMobject("\\times i")
|
||||
times_i.add_background_rectangle()
|
||||
times_i.next_to(
|
||||
times_i_arc.point_from_proportion(0.5),
|
||||
UP
|
||||
)
|
||||
times_neg_i = TexMobject("\\times (-i)")
|
||||
times_neg_i.add_background_rectangle()
|
||||
times_neg_i.next_to(
|
||||
times_neg_i_arc.point_from_proportion(0.5),
|
||||
DOWN
|
||||
)
|
||||
VGroup(
|
||||
times_i, times_neg_i, times_i_arc, times_neg_i_arc
|
||||
).highlight(MAROON_B)
|
||||
|
||||
gaussian_prime = TextMobject("$\\Rightarrow$ ``Gaussian prime''")
|
||||
gaussian_prime.add_background_rectangle()
|
||||
gaussian_prime.scale(0.9)
|
||||
gaussian_prime.next_to(p1_label, RIGHT)
|
||||
|
||||
factorization = TexMobject(
|
||||
"5", "= (2+i)(2-i)"
|
||||
)
|
||||
factorization.to_corner(UP+RIGHT)
|
||||
factorization.shift(1.5*LEFT)
|
||||
factorization.add_background_rectangle()
|
||||
alt_factorization = TexMobject("=(-1+2i)(-1-2i)")
|
||||
alt_factorization.next_to(
|
||||
factorization.get_part_by_tex("="), DOWN,
|
||||
aligned_edge = LEFT
|
||||
)
|
||||
alt_factorization.add_background_rectangle()
|
||||
|
||||
|
||||
for dot in dots:
|
||||
dot.add(Line(
|
||||
self.plane_center,
|
||||
dot.get_center(),
|
||||
color = dot.get_color()
|
||||
))
|
||||
|
||||
self.add(factorization)
|
||||
self.play(
|
||||
DrawBorderThenFill(five_dot),
|
||||
FadeIn(five_label)
|
||||
)
|
||||
self.dither()
|
||||
self.play(
|
||||
ReplacementTransform(
|
||||
VGroup(five_dot).copy(),
|
||||
VGroup(p1_dot, p2_dot)
|
||||
)
|
||||
)
|
||||
self.play(*map(Write, [p1_label, p2_label]))
|
||||
self.dither()
|
||||
self.play(Write(gaussian_prime))
|
||||
self.dither(2)
|
||||
self.play(
|
||||
ShowCreation(times_i_arc),
|
||||
FadeIn(times_i),
|
||||
*[
|
||||
ReplacementTransform(
|
||||
mob1.copy(), mob2,
|
||||
path_arc = np.pi/2
|
||||
)
|
||||
for mob1, mob2 in [
|
||||
(p1_dot, p3_dot),
|
||||
(p1_label, p3_label),
|
||||
]
|
||||
]
|
||||
)
|
||||
self.dither()
|
||||
self.play(
|
||||
ShowCreation(times_neg_i_arc),
|
||||
FadeIn(times_neg_i),
|
||||
*[
|
||||
ReplacementTransform(
|
||||
mob1.copy(), mob2,
|
||||
path_arc = -np.pi/2
|
||||
)
|
||||
for mob1, mob2 in [
|
||||
(p2_dot, p4_dot),
|
||||
(p2_label, p4_label),
|
||||
]
|
||||
]
|
||||
)
|
||||
self.dither()
|
||||
self.play(Write(alt_factorization))
|
||||
self.dither(3)
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -101,7 +101,6 @@ class TexMobject(SVGMobject):
|
|||
tex = tex.replace("{", "", num_lefts - num_rights)
|
||||
return tex
|
||||
|
||||
|
||||
def get_tex_string(self):
|
||||
return self.tex_string
|
||||
|
||||
|
|
|
|||
|
|
@ -202,10 +202,10 @@ class NumberPlane(VMobject):
|
|||
return result
|
||||
|
||||
def point_to_coords(self, point):
|
||||
new_point = point-self.get_center()
|
||||
new_point = point-self.axes.get_center()
|
||||
center_x, center_y = self.coords_at_center
|
||||
x = center_x + point[0]/self.get_space_unit_to_x_unit()
|
||||
y = center_y + point[1]/self.get_space_unit_to_y_unit()
|
||||
x = center_x + new_point[0]/self.get_space_unit_to_x_unit()
|
||||
y = center_y + new_point[1]/self.get_space_unit_to_y_unit()
|
||||
return x, y
|
||||
|
||||
def get_space_unit_to_x_unit(self):
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue