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zagier.py
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zagier.py
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from manimlib import *
from itertools import product
YELLOW_Z = "#e2e1a4"
A_AQUA = "#8dd3c7"
A_YELLOW = "#ffffb3"
A_LAVENDER = "#bebada"
A_RED = "#fb8072"
A_BLUE = "#80b1d3"
A_ORANGE = "#fdb462"
A_GREEN = "#b3de69"
A_PINK = "#fccde5"
A_GREY = "#d9d9d9"
A_VIOLET = "#bc80bd"
A_UNKA = "#ccebc5"
A_UNKB = "#ffed6f"
def get_windmills(n):
return [
(x, y, z)
for (x, y, z) in product(range(1, n), repeat=3)
if x**2 + 4 * y * z == n
]
class GridRectangle(VGroup):
def __init__(
self, height, width, freq=1, rect_kwargs={}, line_kwargs={}, *args, **kwargs
):
super().__init__(*args, **kwargs)
self.rect = Rectangle(height=height, width=width, **rect_kwargs)
self.lines = VGroup()
for i in np.arange(-height / 2, height / 2, freq):
self.lines.add(
Line(LEFT * width / 2, RIGHT * width / 2, **line_kwargs).shift(i * UP)
)
for i in np.arange(-width / 2, width / 2, freq):
self.lines.add(
Line(UP * height / 2, DOWN * height / 2, **line_kwargs).shift(i * RIGHT)
)
self.add(self.rect, self.lines)
class Windmill(VGroup):
def __init__(
self,
x,
y,
z,
freq=1,
stroke_width=2,
i_kwargs={"fill_color": "#7e6773", "fill_opacity": 1.0},
o_kwargs={"fill_color": "#476a64", "fill_opacity": 1.0},
l_kwargs={},
*args,
**kwargs,
):
super().__init__(*args, **kwargs)
self.square = GridRectangle(
x,
x,
freq=freq,
rect_kwargs={"stroke_width": stroke_width, **i_kwargs},
line_kwargs={"stroke_width": stroke_width, **l_kwargs},
)
self.rects = VGroup()
for i in range(4):
# curr_vec[0] = 1 if last two bits are 10 or 01 else -1
# curr_vec[1] = 1 if 2nd last bit is 1 else -1
curr_vec = np.array(
[(((i & 1) ^ (i >> 1 & 1)) * 2 - 1), ((i >> 1 & 1) * -2 + 1), 0]
)
r = GridRectangle(
z,
y,
freq=freq,
rect_kwargs={"stroke_width": stroke_width, **o_kwargs},
line_kwargs={"stroke_width": stroke_width, **l_kwargs},
)
r.rotate(i * 90 * DEGREES)
r.move_to(curr_vec * x / 2)
r.shift(curr_vec[0] * (i & 1) * z / 2 * RIGHT)
r.shift(curr_vec[1] * (~i & 1) * z / 2 * UP)
r.shift(curr_vec[0] * (i & 1) * y / 2 * DOWN)
r.shift(curr_vec[1] * (~i & 1) * y / 2 * RIGHT)
self.rects.add(r)
self.add(self.square, self.rects)
class PartScene(Scene):
CONFIG = {"n": 1, "title": "Placeholder", "title_color": RED}
def construct(self):
self.part = TexText(f"Part {self.n}")
self.part.scale(1.5)
self.part.shift(2 * UP)
self.title = TexText(self.title, color=self.title_color)
self.title.scale(1.5)
self.play(Write(self.part))
self.play(Write(self.title))
self.wait()
class TitleScene(Scene):
CONFIG = {"color": None, "text": None}
def construct(self):
if self.text is None:
raise NotImplementedError
brect = Rectangle(
height=FRAME_HEIGHT, width=FRAME_WIDTH, fill_opacity=1, color=self.color
)
title = TexText(self.text)
title.scale(1.5)
title.to_edge(UP)
rect = ScreenRectangle(height=6)
rect.next_to(title, DOWN)
self.add(brect)
self.play(FadeIn(rect, DOWN), Write(title), run_time=2)
self.wait()
class Recap(TitleScene):
CONFIG = {"color": "#252626", "text": "What You Just Saw"}
class PartOne(PartScene):
CONFIG = {"n": 1, "title": "The Involution", "title_color": A_RED}
class OneSentence(Scene):
def construct(self):
title = TexText("Zagier's One Sentence Proof", color=A_LAVENDER)
title.scale(1.5)
title.shift(3 * UP)
img = ImageMobject("img/zagier_proof.png")
img.scale(0.85)
rect = SurroundingRectangle(img, color=A_LAVENDER, buff=0, stroke_width=8)
self.play(Write(title))
self.play(FadeIn(img), Write(rect))
self.wait()
self.embed()
class ZagierIntro(Scene):
def construct(self):
vectors = [
FRAME_WIDTH / 4 * LEFT + FRAME_HEIGHT / 4 * UP,
FRAME_WIDTH / 4 * RIGHT + FRAME_HEIGHT / 4 * UP,
FRAME_HEIGHT / 4 * DOWN,
]
screens = [
ScreenRectangle(height=2.75, fill_opacity=1, fill_color=BLACK).move_to(
vectors[i]
)
for i in range(3)
]
texts = ["Simplicity", "Elegance", "Visualization"]
text_objs = [TexText(texts[i]).next_to(screens[i], UP) for i in range(3)]
VGroup(*screens, *text_objs).center()
for i in range(3):
self.play(FadeIn(screens[i], DOWN))
self.play(Write(text_objs[i]))
self.wait(0.25)
self.embed()
class ZagierIntroVertical(Scene):
def construct(self):
b_rect = Rectangle(
height=FRAME_HEIGHT,
width=9 / 16 * FRAME_HEIGHT,
fill_opacity=1,
fill_color="#252626",
stroke_width=0,
)
self.add(b_rect)
vectors = [
((FRAME_HEIGHT / 2) - (FRAME_HEIGHT / 3.25)) * UP,
((FRAME_HEIGHT / 2) - (2 * FRAME_HEIGHT / 3.25)) * UP,
((FRAME_HEIGHT / 2) - (3 * FRAME_HEIGHT / 3.25)) * UP,
]
screens = [
ScreenRectangle(height=1.75, fill_opacity=1, fill_color=BLACK).move_to(
vectors[i]
)
for i in range(3)
]
texts = ["Simplicity", "Elegance", "Visualization"]
text_objs = [
TexText(texts[i]).move_to(screens[i]).scale(0.75).shift(1.15 * UP)
for i in range(3)
]
VGroup(*screens, *text_objs).center()
for i in range(3):
self.play(FadeIn(screens[i], DOWN))
self.play(Write(text_objs[i]))
self.wait(0.25)
self.embed()
class Involution(Scene):
def construct(self):
# Sinusodal offset to an ellipse to make it less uniform-looking
def set_bound_curve(t):
return [5 * np.cos(t), 3 * np.sin(t) + np.sin(0.5 * t), 0]
set_bound = ParametricCurve(
set_bound_curve, t_range=(0, 2 * PI), color=A_YELLOW, fill_opacity=0.35
)
set_bound.center()
s_text = Tex("S")
s_text.scale(3)
s_text.move_to(set_bound, DL)
s_text.shift(0.5 * LEFT)
self.play(Write(set_bound), Write(s_text))
circles = VGroup(
Circle(fill_color=RED_E, stroke_color=RED_A, fill_opacity=0.75),
Circle(fill_color=GREEN_E, stroke_color=GREEN_A, fill_opacity=0.75),
Circle(fill_color=BLUE_E, stroke_color=BLUE_A, fill_opacity=0.75),
)
# Arbritrarly chosen coordinates
circles[0].move_to(1.5 * LEFT + UP)
circles[1].move_to(RIGHT + 0.75 * UP)
circles[2].move_to(0.25 * LEFT + 1.5 * DOWN)
self.play(Write(circles))
self.wait()
circle_move_anims = [
ApplyMethod(circles[i].move_to, (3.5 * (i - 1) + 1.75) * RIGHT + 2 * UP)
for i in range(3)
]
self.play(Uncreate(set_bound))
self.play(*circle_move_anims, ApplyMethod(s_text.move_to, 5 * LEFT + 2 * UP))
self.wait()
f_circles = circles.deepcopy()
f_circles.shift(4 * DOWN)
f_circles[1].set_fill(color=BLUE_E)
f_circles[1].set_stroke(color=BLUE_A)
f_circles[2].set_fill(color=GREEN_E)
f_circles[2].set_stroke(color=GREEN_A)
transform_anims = [
TransformFromCopy(circles[i], f_circles[(i != 0) * (3 - i)])
for i in range(3)
]
fs_text = Tex(
"f(S)", tex_to_color_map={"f": A_YELLOW, ")": A_UNKA, "(": A_UNKA}
)
fs_text.scale(3)
fs_text.move_to(5 * LEFT + 2 * DOWN)
arrows = VGroup()
for i in range(3):
start = circles[i].get_center() + 1.25 * DOWN
end = f_circles[(i != 0) * (3 - i)].get_center() + 1.25 * UP
arrows.add(
Arrow(
start,
end,
stroke_width=10,
tip_width_ratio=6,
stroke_color=A_YELLOW,
buff=0,
)
)
self.play(*transform_anims, run_time=6)
self.play(Write(fs_text), Write(arrows))
self.bring_to_front(circles, f_circles)
self.wait()
inv_title = Tex(
r"\text{Involution} \leftrightarrow f(f(x)) = x",
tex_to_color_map={
r"\leftrightarrow": A_RED,
"f": A_YELLOW,
"(": A_UNKA,
")": A_UNKA,
},
)
inv_title[1].set_color(A_RED)
inv_title.scale(1.5)
inv_title.shift(3.25 * UP)
all_curr_obj = VGroup(arrows, s_text, fs_text, circles, f_circles)
self.play(all_curr_obj.shift, 0.5 * DOWN)
self.play(Write(inv_title))
self.wait()
inv_transform_anims = [
TransformFromCopy(f_circles[i], circles[(i != 0) * (3 - i)])
for i in range(3)
]
self.play(FadeOut(circles, UP))
circles.shift(4 * DOWN)
self.play(f_circles.shift, 4 * UP)
self.play(*inv_transform_anims, run_time=3)
self.wait()
self.play(
FadeOut(f_circles, UP),
FadeOut(arrows, UP),
ApplyMethod(circles.shift, 4 * UP),
)
self.wait()
arrows2 = VGroup()
for i in range(3):
start = circles[i].get_center() + 1.25 * DOWN
end = (
circles[(i != 2) * (1 - i) + 2 * (i == 2)].get_center()
+ 1.25 * UP
+ 4 * DOWN
)
arrows2.add(
Arrow(
start,
end,
stroke_width=10,
tip_width_ratio=6,
stroke_color=A_YELLOW,
buff=0,
)
)
f_circles2 = circles.deepcopy()
f_circles2.shift(4 * DOWN)
f_circles2[1].set_fill(color=RED_E)
f_circles2[1].set_stroke(color=RED_A)
f_circles2[0].set_fill(color=GREEN_E)
f_circles2[0].set_stroke(color=GREEN_A)
self.play(Write(arrows2[2]), Write(f_circles2[2]))
b_rect = SurroundingRectangle(
VGroup(f_circles2[2], arrows2[2], f_circles[2]),
color=BLUE_E,
buff=0.15,
stroke_width=8,
)
fixed_text = TexText("Fixed Element", color=BLUE_E)
fixed_text.scale(1.5)
fixed_text.move_to(b_rect, LEFT)
fixed_text.shift(5 * LEFT)
self.play(Write(b_rect), Write(fixed_text))
self.wait()
self.play(Uncreate(b_rect), Uncreate(fixed_text))
self.wait()
grp = VGroup(f_circles2[2], circles, fs_text, s_text, arrows2[2])
self.play(ApplyMethod(grp.scale, 0.5))
self.play(ApplyMethod(grp.shift, UP))
ff_text = Tex(
"f(f(S))", tex_to_color_map={"f": A_YELLOW, ")": A_UNKA, "(": A_UNKA}
)
ff_text.scale(1.5)
ff_text.move_to(fs_text.get_center() + 2 * DOWN)
arr = arrows2[-1].copy()
arr.shift(1.75 * LEFT)
arr2 = arr.copy()
arr2.shift(2 * DOWN)
circ = Circle(stroke_color=WHITE)
circ.scale(0.5)
circ.move_to(f_circles2[2].get_center() + 1.75 * LEFT)
q = TexText("?")
q.scale(1.5)
q.move_to(circ)
f_circ = f_circles2[0].copy()
f_circ.scale(0.5)
f_circ.move_to(circ.get_center() + 2 * DOWN)
self.play(Write(ff_text))
self.play(Write(arr), Write(VGroup(q, circ)))
self.play(Write(arr2), Write(f_circ))
self.wait()
self.play(Uncreate(VGroup(circ, q, ff_text, arr, arr2, f_circ)))
self.play(grp.shift, DOWN)
self.play(grp.scale, 2)
self.play(Write(f_circles2[:-1]), Write(arrows2[:-1]))
self.wait()
c = color_gradient([RED_E, GREEN_E, RED_E], 15)
p_rect = SurroundingRectangle(
VGroup(*f_circles2[:-1], *arrows2[:-1], *f_circles[:-1]),
buff=0.15,
color=c,
stroke_width=8,
)
p_text = Text("Paired Element", gradient=color_gradient([RED_E, GREEN_E], 10))
p_text.scale(1.5)
p_text.move_to(p_rect)
p_text.add_background_rectangle(buff=0.15)
self.play(ShowCreation(p_rect), Write(p_text))
self.wait()
self.embed()
class CommonInvolution(Scene):
def construct(self):
input_line = NumberLine()
input_line.shift(1 * UP)
input_line.add_numbers(font_size=36)
x_label = Tex("x", color=A_YELLOW)
x_label.move_to([-6.5, 1.5, 0])
output_line = NumberLine()
output_line.shift(2 * DOWN)
output_line.add_numbers(font_size=36)
y_label = Tex("f(x)", tex_to_color_map={"x": A_YELLOW, "f": A_BLUE})
y_label.move_to([-6.1902, -1.5, 0])
f_label = Tex("f(x) = -x", tex_to_color_map={"x": A_YELLOW, "f": A_BLUE})
f_label.scale(1.5)
f_label.shift(3 * UP)
self.play(Write(input_line), Write(output_line))
self.play(Write(x_label), Write(y_label), Write(f_label))
x_vals, y_vals = [], []
for x in np.linspace(-8, 8, 129):
x_vals.append(input_line.n2p(x))
y_vals.append(output_line.n2p(-x))
grad = color_gradient([A_PINK, A_GREEN], len(x_vals))
input_c = DotCloud(x_vals, color=grad)
output_c = DotCloud(y_vals, color=grad)
lines = VGroup()
for i in range(129):
lines.add(Line(x_vals[i], y_vals[i], color=grad[i], stroke_opacity=0.3))
self.play(ShowCreation(input_c))
self.play(TransformFromCopy(input_c, output_c), run_time=5)
self.play(ShowCreation(lines))
self.wait()
lines2 = VGroup()
for i in range(129):
lines2.add(
Line(
x_vals[i],
y_vals[i],
color=grad[len(grad) - i - 1],
stroke_opacity=0.3,
)
)
self.play(FadeOut(input_c, UP), FadeOut(lines))
self.play(ApplyMethod(output_c.shift, 3 * UP))
input_c.shift(3 * DOWN)
self.play(TransformFromCopy(output_c, input_c), run_time=3)
self.play(ShowCreation(lines2))
self.wait()
self.embed()
class MoreInvolution(Scene):
def construct(self):
color_map = {
r"\text{paired}": A_PINK,
"{f}": A_YELLOW,
"0": A_UNKA,
"2": A_UNKA,
"S": A_GREEN,
r"\text{fixed}": A_PINK,
}
eq1 = Tex(
r"| \text{paired} \ {f} \ S | \equiv 0 \mod 2", tex_to_color_map=color_map
)
eq1.scale(2)
eq1.shift(1.5 * UP)
eq2 = Tex(
r"|S| = | \text{fixed} \ {f} \ S | + | \text{paired} \ {f} \ S |",
tex_to_color_map=color_map,
)
eq2.scale(2)
eq2.shift(1.5 * DOWN)
eq3 = Tex(
r"|S| \equiv | \text{fixed} \ {f} \ S | \mod 2", tex_to_color_map=color_map
)
eq3.scale(2)
eq3.shift(1.5 * DOWN)
self.play(Write(eq1))
self.wait()
self.play(Write(eq2))
self.wait()
self.play(TransformMatchingTex(eq2, eq3))
self.play(Indicate(eq3))
self.wait()
self.embed()
class PartTwo(PartScene):
CONFIG = {"n": 2, "title": "The Windmill", "title_color": A_AQUA}
class WindmillIntro(Scene):
def construct(self):
head_t = Tex(
r"\text{For all } p = 4k + 1",
tex_to_color_map={"p": A_GREEN, "k": A_ORANGE, "4": A_UNKA, "1": A_UNKA},
)
head_t.scale(2)
head_t.shift(2.5 * UP)
bot_t = Tex(
r"p = a^2 + b^2",
tex_to_color_map={"p": A_GREEN, "a": A_PINK, "b": A_PINK, "2": A_UNKA},
)
bot_t.scale(2)
self.play(Write(head_t))
self.play(Write(bot_t))
self.wait()
self.play(FadeOut(head_t, UP))
self.play(bot_t.shift, 3 * UP + FRAME_WIDTH / 4 * LEFT)
s1 = Square(side_length=4, fill_opacity=0.5, fill_color=A_GREEN)
s1.shift(FRAME_WIDTH / 4 * LEFT + 0.5 * DOWN)
s1_lbl = Tex("p").scale(2)
s1_lbl.move_to(s1)
self.play(Write(s1), Write(s1_lbl))
self.wait(1)
sr1 = Square(side_length=2, fill_color=A_PINK, fill_opacity=0.5)
r1_lbl = Tex("a^2").scale(2)
r1_lbl.move_to(sr1)
r1 = VGroup(sr1, r1_lbl)
r1.shift(2 * LEFT)
sr2 = Square(side_length=np.sqrt(12), fill_color=A_PINK, fill_opacity=0.5)
r2_lbl = Tex("b^2").scale(2)
r2_lbl.move_to(sr2)
r2 = VGroup(sr2, r2_lbl)
r2.shift(RIGHT)
r_left = VGroup(r1, r2)
r_left.move_to(s1)
s1 = VGroup(s1, s1_lbl)
self.play(Transform(s1, r_left))
other_t = Tex(
r"p = x^2 + 4yz",
tex_to_color_map={
"p": A_GREEN,
"x": A_PINK,
"y": A_AQUA,
"z": A_AQUA,
"4": A_UNKA,
"2": A_UNKA,
},
)
other_t.scale(2)
other_t.shift(FRAME_WIDTH / 4 * RIGHT + 3 * UP)
s2 = Square(side_length=4, fill_opacity=0.5, fill_color=A_GREEN)
s2.shift(FRAME_WIDTH / 4 * RIGHT + 0.5 * DOWN)
s2_lbl = Tex("p").scale(2)
s2_lbl.move_to(s2)
self.play(Write(other_t))
self.play(Write(s2), Write(s2_lbl))
self.wait(1)
wind_f = Windmill(2, 1, 3, freq=float("inf"))
# Positions calculated by moving rects and square relative as desired,
# then, moving all to FRAME_WIDTH / 4
wind_f.square.move_to(FRAME_WIDTH / 4 * RIGHT + 1.125 * UP)
for i in range(4):
wind_f.rects[i].move_to(ORIGIN + 1.25 * i * RIGHT)
if i & 1:
wind_f.rects[i].rotate(90 * DEGREES)
wind_f.rects.move_to(FRAME_WIDTH / 4 * RIGHT + 1.625 * DOWN)
sq_lbl = Tex("x^2")
sq_lbl.scale(2)
sq_lbl.move_to(wind_f.square)
wr_lbls = VGroup()
for i in range(4):
wr_lbls.add(Tex("yz").move_to(wind_f.rects[i]))
s2 = VGroup(s2, s2_lbl)
self.play(Transform(s2, wind_f))
self.remove(s2)
self.add(wind_f)
self.play(Write(sq_lbl), Write(wr_lbls))
self.wait(1)
self.play(FadeOut(VGroup(bot_t, s1), UP))
self.play(other_t.scale, 1 / 1.5)
self.play(
ApplyMethod(other_t.shift, FRAME_WIDTH / 4 * LEFT + 0.25 * UP),
FadeOut(VGroup(sq_lbl, wr_lbls)),
)
wind = Windmill(2, 1, 3, freq=float("inf"))
wind.scale(0.75)
wind.shift(0.5 * DOWN)
self.play(Transform(wind_f.square, wind.square))
self.play(*[Transform(wind_f.rects[i], wind.rects[i]) for i in range(4)])
self.wait()
x_label = Tex("x")
x_label.scale(1.5)
y_label = Tex("y", color=A_AQUA)
y_label.scale(1.5)
z_label = Tex("z", color=A_AQUA)
z_label.scale(1.5)
b1 = Brace(wind_f.square, LEFT)
b1.rotate(180 * DEGREES)
b1.shift(0.5 * RIGHT)
b1.put_at_tip(x_label)
b2 = Brace(wind_f.rects[1], RIGHT)
b2.put_at_tip(y_label)
b3 = Brace(wind_f.rects[1], UP)
b3.put_at_tip(z_label)
self.play(GrowFromCenter(b1), TransformFromCopy(other_t[2], x_label))
self.play(GrowFromCenter(b2), TransformFromCopy(other_t[-2], y_label))
self.play(GrowFromCenter(b3), TransformFromCopy(other_t[-1], z_label))
self.wait()
self.embed()
class WindmillDef(Scene):
def construct(self):
eq1 = Tex(
r"W_{n} = \lbrace (x, y, z) \in \mathbb{{N}}^3 \ | \ x^2 + 4yz = {n} \rbrace",
tex_to_color_map={
"W": A_YELLOW,
"{n}": A_GREEN,
"x": A_PINK,
"y": A_AQUA,
"z": A_AQUA,
"2": A_UNKA,
"3": A_UNKA,
"4": A_UNKA,
"{N}": A_LAVENDER,
},
)
eq1.scale(1.5)
eq1.shift(3 * UP)
self.play(Write(eq1[:2]))
self.wait()
self.play(Write(eq1[2:12]))
self.wait()
self.play(Write(eq1[12:]))
self.wait()
w_29 = get_windmills(29)
windmill_stroke_width = 2
windmills = VGroup()
# Arbritrarily chosen relative positions
positions = [-4.825, -1.25, 2, 4.5, 6.375]
for n, i in enumerate(w_29):
windmill = Windmill(
*i,
l_kwargs={"stroke_width": windmill_stroke_width},
i_kwargs={
"fill_color": A_PINK,
"fill_opacity": 0.5,
"stroke_width": windmill_stroke_width,
},
o_kwargs={
"fill_color": A_AQUA,
"fill_opacity": 0.5,
"stroke_width": windmill_stroke_width,
},
)
windmill.scale(0.25)
windmill.move_to(positions[n] * RIGHT)
windmills.add(windmill)
windmills.center()
windmills.shift(0.75 * DOWN)
labels = VGroup(
*[
Tex(f"({x}, {y}, {z})").move_to(windmills[i]).shift(2.5 * DOWN)
for i, (x, y, z) in enumerate(w_29)
]
)
self.play(Write(windmills[2]), Write(labels[2]))
for i in range(5):
if i != 2:
self.play(Write(windmills[i]), Write(labels[i]))
b = Brace(windmills, UP)
t = Tex(r"W_{29}", tex_to_color_map={"W": A_YELLOW, "{29}": A_UNKA})
b.put_at_tip(t)
self.play(Write(b), Write(t))
self.wait()
self.play(Uncreate(VGroup(b, t, windmills, labels)))
head_t = Tex(
r"\text{For all } p = 4k + 1",
tex_to_color_map={"p": A_GREEN, "k": A_ORANGE, "4": A_UNKA, "1": A_UNKA},
)
head_t.scale(1.5)
head_t.shift(3 * UP)
eq2 = Tex(
r"W_{p} = \lbrace (x, y, z) \in \mathbb{{N}}^3 \ | \ x^2 + 4yz = {p} \rbrace",
tex_to_color_map={
"W": A_YELLOW,
"{p}": A_GREEN,
"x": A_PINK,
"y": A_AQUA,
"z": A_AQUA,
"2": A_UNKA,
"3": A_UNKA,
"4": A_UNKA,
"{N}": A_LAVENDER,
},
)
eq2.scale(1.5)
eq2.shift(1.5 * UP)
eq3 = Tex(
r"{p} = x^2 + 4yz",
tex_to_color_map={
"{p}": A_GREEN,
"x": A_PINK,
"y": A_AQUA,
"z": A_AQUA,
"2": A_UNKA,
"4": A_UNKA,
},
)
eq3.scale(1.5)
self.play(Transform(eq1, eq2))
self.play(Write(head_t))
self.wait()
self.play(TransformMatchingTex(eq2[12:-1], eq3))
self.wait()
eq4 = Tex(
r"{p} = x^2 + 4y^2",
tex_to_color_map={
"{p}": A_GREEN,
"x": A_PINK,
"y": A_AQUA,
"z": A_AQUA,
"2": A_UNKA,
"4": A_UNKA,
},
)
eq4.scale(1.5)
eq4.shift(3 * DOWN)
arrow = Arrow(eq3, eq4, stroke_color=A_YELLOW, buff=0.35, stroke_width=16)
arrow_tex = Tex("y=z", tex_to_color_map={"y": A_AQUA, "z": A_AQUA})
arrow_tex.scale(1.5)
arrow_tex.move_to(arrow)
arrow_tex.shift(1.5 * RIGHT)
cp = eq3.copy()
self.play(Write(arrow), Write(arrow_tex))
self.wait()
self.play(Transform(cp, eq4))
self.wait()
eq5 = Tex(
r"{p} = x^2 + (2y)^2",
tex_to_color_map={
"{p}": A_GREEN,
"x": A_PINK,
"y": A_AQUA,
"z": A_AQUA,
"2": A_UNKA,
"4": A_UNKA,
},
)
eq5.scale(1.5)
eq5.shift(3 * DOWN)
self.play(TransformMatchingTex(cp, eq5))
self.wait()
self.embed()
class WindmillOne(Scene):
def construct(self):
w_13 = get_windmills(13)
windmill_stroke_width = 2
windmills = VGroup()
# Arbritrarily chosen relative positions
positions = [-4.825, -1.25, 2]
for n, i in enumerate(w_13):
windmill = Windmill(
*i,
l_kwargs={"stroke_width": windmill_stroke_width},
i_kwargs={
"fill_color": A_PINK,
"fill_opacity": 0.5,
"stroke_width": windmill_stroke_width,
},
o_kwargs={
"fill_color": A_AQUA,
"fill_opacity": 0.5,
"stroke_width": windmill_stroke_width,
},
)
windmill.scale(0.5)
windmill.move_to(positions[n] * RIGHT)
windmills.add(windmill)
windmills.center()
labels = VGroup(
*[
Tex(f"({x}, {y}, {z})").move_to(windmills[i]).shift(2.5 * DOWN)
for i, (x, y, z) in enumerate(w_13)
]
)
title = Tex("W_{13}", tex_to_color_map={"W": A_YELLOW, "{13}": A_GREEN})
title.scale(1.5)
b = Brace(windmills, UP)
b.put_at_tip(title)
grp = VGroup(windmills, b, labels, title)
grp.center()
self.play(Write(title), Write(b))
for i in range(3):
self.play(Write(windmills[i]), Write(labels[i]))
self.wait()
self.play(Indicate(VGroup(windmills[-1], labels[-1])))
self.wait()
eq = Tex(
r"(1, 1, k)",
r"\in W_{4k+1}",
tex_to_color_map={"W": A_YELLOW, "4": A_UNKA, "1": A_UNKA, "k": A_ORANGE},
)
eq.scale(1.5)
eq.shift(3.25 * DOWN)
self.play(
ApplyMethod(VGroup(windmills, labels, b, title).shift, 0.5 * UP), Write(eq)
)
self.play(Indicate(eq[:7]), Indicate(VGroup(windmills[0], labels[0])))
self.wait()
self.embed()
class WindmillProof(Scene):
def construct(self):
head_t = Tex(
r"\text{In }",
r"{W_{p = 4k}}",
r"{}_{+}",
r"{}_{1}",
r"\text{, (1, 1, k) is the ONLY windmill with } {x} = {y}",
tex_to_color_map={
"p": A_GREEN,
"k": A_ORANGE,
"4": A_UNKA,
"1": A_UNKA,
"{x}": A_PINK,
"{y}": A_AQUA,
"{z}": A_AQUA,
"W": A_YELLOW,
"{p}": A_GREEN,
},
)
head_t.shift(3.25 * UP)
self.play(Write(head_t))
w = Windmill(1, 1, 3)
w.scale(0.75)
w.shift(0.8 * DOWN)
self.play(Write(w))
self.play(w.rects[0].rotate, 90 * DEGREES)
self.play(w.rects[0].shift, 2 * LEFT + 0.2 * DOWN)
b1 = Brace(w.rects[0], RIGHT)
b2 = Brace(w.square, UP)
x_label = Tex("x", color=A_PINK)
x_label.scale(1.5)
x_label.shift(0.5 * UP)
self.play(Write(b1), Write(b2), Write(x_label))
self.wait()
self.play(
Uncreate(VGroup(b1, b2, x_label)),
ApplyMethod(w.rects[0].rotate, -90 * DEGREES),