mirror of
https://github.com/3b1b/videos.git
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298 lines
9.3 KiB
Python
298 lines
9.3 KiB
Python
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from manim_imports_ext import *
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def get_integer_matrix_exponential(matrix):
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pass
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def get_vector_field_and_stream_lines(func, coordinate_system,
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magnitude_range=(0.5, 4),
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vector_opacity=0.75,
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vector_thickness=0.03,
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color_by_magnitude=True,
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line_color=GREY_A,
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line_width=3,
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sample_freq=3,
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n_samples_per_line=8,
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arc_len=3,
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time_width=0.5,
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):
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vector_field = VectorField(
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func, coordinate_system,
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magnitude_range=magnitude_range,
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vector_config={
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"fill_opacity": vector_opacity,
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"thickness": vector_thickness,
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}
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)
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stream_lines = StreamLines(
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func, coordinate_system,
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step_multiple=1.0 / sample_freq,
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n_samples_per_line=n_samples_per_line,
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arc_len=arc_len,
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magnitude_range=magnitude_range,
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color_by_magnitude=color_by_magnitude,
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stroke_color=line_color,
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stroke_width=line_width,
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)
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animated_lines = AnimatedStreamLines(
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stream_lines,
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line_anim_config={
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"time_width": time_width,
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},
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)
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return vector_field, animated_lines
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# Scenes
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class ShowConfusionAtMatrixExponenent(Scene):
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def construct(self):
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# Sticking a matrix in an exponent like this might strike you as...
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base = Tex("e")
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base.set_height(1.0)
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matrix = IntegerMatrix(
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[[3, 1, 4],
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[1, 5, 9],
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[2, 6, 5]],
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)
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matrix.move_to(base.get_corner(UR), DL)
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matrix_exp = VGroup(base, matrix)
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matrix_exp.set_height(2)
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matrix_exp.to_corner(UL)
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matrix_exp.shift(3 * RIGHT)
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randy = Randolph()
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randy.set_height(2)
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randy.to_corner(DL)
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matrix.save_state()
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matrix.center()
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matrix.set_height(2.5)
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self.add(randy)
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self.play(
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randy.animate.change("pondering", matrix),
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Write(matrix.get_brackets()),
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ShowIncreasingSubsets(matrix.get_entries()),
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)
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self.play(Blink(randy))
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self.play(
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matrix.animate.restore(),
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Write(base),
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randy.animate.change("erm", base),
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)
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self.play(Blink(randy))
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# ...odd, to say the least.
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rhs = Tex("= e \\cdot e \\dots e \\cdot e")
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rhs.set_height(0.75 * base.get_height())
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rhs.next_to(matrix_exp, RIGHT)
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rhs.align_to(base, DOWN)
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brace = Brace(rhs[0][1:], DOWN)
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matrix_copy = matrix.copy()
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matrix_copy.scale(0.5)
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brace_label = VGroup(
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matrix.copy().scale(0.5),
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Text("times?")
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)
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brace_label.arrange(RIGHT)
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brace_label.next_to(brace, DOWN, SMALL_BUFF)
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bubble = randy.get_bubble(
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TexText("I'm sorry,\\\\what?!").scale(0.75),
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height=2,
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width=3,
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bubble_class=SpeechBubble,
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)
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self.play(
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TransformMatchingParts(
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base.copy(), rhs,
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path_arc=10 * DEGREES,
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lag_ratio=0.01,
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),
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GrowFromCenter(brace),
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ReplacementTransform(
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matrix.copy(), brace_label[0],
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path_arc=30 * DEGREES,
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run_time=2,
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rate_func=squish_rate_func(smooth, 0.3, 1),
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),
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Write(
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brace_label[1],
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run_time=2,
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rate_func=squish_rate_func(smooth, 0.5, 1),
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),
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)
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self.play(
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randy.animate.change("angry", rhs),
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ShowCreation(bubble),
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Write(bubble.content),
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)
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self.wait()
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false_equation = VGroup(
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matrix_exp, rhs, brace, brace_label
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)
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# The short version of this video would be to say the notation is misleading,
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# having little to do with the number e being multiplying it by itself.
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morty = Mortimer()
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morty.match_height(randy)
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morty.to_corner(DR)
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false_equation.generate_target()
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false_equation.target.scale(0.5)
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false_equation.target.next_to(morty, UL)
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fe_rect = SurroundingRectangle(false_equation.target)
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fe_rect.set_color(GREY_BROWN)
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cross = Cross(false_equation.target[1])
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cross.set_stroke(RED, 5)
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nonsense = Text("This would be nonsense")
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nonsense.match_width(fe_rect)
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nonsense.next_to(fe_rect, UP)
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nonsense.set_color(RED)
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randy.bubble = bubble
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self.play(
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MoveToTarget(false_equation),
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RemovePiCreatureBubble(randy, target_mode="hesitant"),
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morty.animate.change("raise_right_hand"),
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ShowCreation(fe_rect),
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VFadeIn(morty),
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)
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self.play(
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ShowCreation(cross),
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FadeIn(nonsense),
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)
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self.play(Blink(morty))
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self.wait()
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# Writing e with a matrix up top is shorthand for plugging in the matrix to a certain infinite polynomial.
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real_equation = Tex(
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"e^X := X^0 + X^1 + \\frac{1}{2} X^2 + \\frac{1}{6} X^3 + \\cdots + \\frac{1}{n!} X^n + \\cdots",
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isolate=["X"]
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)
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xs = real_equation.get_parts_by_tex("X")
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xs.set_color(TEAL)
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real_equation.set_width(FRAME_WIDTH - 2.0)
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real_equation.to_edge(UP)
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by_definition = Text("(by definition)", font_size=24)
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by_definition.set_fill(GREY_B)
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bd_arrow = Vector(0.5 * UP)
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bd_arrow.match_color(by_definition)
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bd_arrow.next_to(real_equation.get_part_by_tex("="), DOWN, SMALL_BUFF)
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by_definition.next_to(bd_arrow, DOWN)
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self.play(
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TransformFromCopy(base, real_equation[0]),
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FadeTransform(matrix.copy(), real_equation[1]),
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)
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self.play(
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GrowArrow(bd_arrow),
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FadeIn(by_definition, DOWN),
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Write(real_equation[2]),
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FadeTransformPieces(xs[:1].copy(), xs[1:], path_arc=20 * DEGREES),
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LaggedStart(*(
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FadeIn(part)
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for part in real_equation[4:]
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if part not in xs
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))
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)
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self.add(real_equation)
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self.play(
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randy.animate.change("pondering", real_equation),
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morty.animate.change("pondering", real_equation),
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)
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self.wait()
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# Still, that’s a complicated thing to do, which takes some explaining in its own right,
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# and without context this definition does very little to explain how to think about this operation.
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rhs_tex = "X^0 + X^1 + \\frac{1}{2} X^2 + \\frac{1}{6} X^3 + \\cdots + \\frac{1}{n!} X^n + \\cdots"
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mat_tex = "\\left[ \\begin{array}{ccc} 3 & 1 & 4 \\\\ 1 & 5 & 9 \\\\ 2 & 6 & 5 \\end{array} \\right]"
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ex_rhs = Tex(
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rhs_tex.replace("X", mat_tex),
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tex_to_color_map={mat_tex: TEAL},
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)
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ex_rhs.scale(0.5)
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ex_eq = VGroup(matrix_exp.copy(), Tex("="), ex_rhs)
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ex_eq[0][1].set_color(TEAL)
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ex_eq.arrange(RIGHT)
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ex_rhs.align_to(ex_eq[0], DOWN)
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ex_eq = VGroup(ex_eq[0], ex_eq[1], *ex_rhs)
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ex_eq[1:].shift(0.1 * DOWN)
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ex_eq.next_to(real_equation, DOWN, buff=2, aligned_edge=DOWN)
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false_group = VGroup(false_equation, fe_rect, cross, nonsense)
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complicated = Text("Rather complicated!")
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complicated.set_color(YELLOW)
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complicated.next_to(ex_eq, DOWN, LARGE_BUFF)
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self.play(
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TransformFromCopy(matrix_exp, ex_eq[0]),
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FadeOut(false_group, DOWN),
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FadeTransformPieces(
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real_equation.copy()[2:], ex_eq[1:], run_time=2,
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),
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randy.animate.change("hesitant"),
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morty.animate.change("raise_right_hand"),
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)
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self.play(Blink(randy))
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self.play(
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FadeIn(complicated, shift=DOWN),
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randy.animate.change("confused"),
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morty.animate.change("hesitant"),
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)
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self.wait()
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class CircularPhaseFlow(Scene):
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def construct(self):
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plane = NumberPlane(
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x_range=[-4, 4],
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y_range=[-2, 2],
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height=8,
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width=16,
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)
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plane.add_coordinate_labels()
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vector_field, animated_lines = get_vector_field_and_stream_lines(
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self.func, plane
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)
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self.add(plane)
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self.add(vector_field)
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self.add(animated_lines)
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self.wait(10)
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#
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self.embed()
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def func(self, x, y):
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return (-y, x)
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def get_label(self):
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pass
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class HyperbolicPhaseFlow(CircularPhaseFlow):
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def func(self, x, y):
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return (x, -y)
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class WhyWedCare(Scene):
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def construct(self):
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pass
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# ...how to think about this operation, or more importantly why we’d care.
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# So if it’s alright with you I think we should hold off explaining this definition, or justifying the
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# abuse of notation, until it’s a bit clearer what problems it helps us to solve.
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#
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self.embed()
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