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More inventing math progress

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This commit is contained in:
Grant Sanderson
2015-08-03 22:23:00 -07:00
parent 6af2e9c6c2
commit 9cf8e7b75e
9 changed files with 290 additions and 15 deletions
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274
scripts/inventing_math.py
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@ -4,6 +4,7 @@ import numpy as np
import itertools as it
from copy import deepcopy
import sys
import operator as op
from animation import *
@ -18,7 +19,8 @@ DIVERGENT_SUM_TEXT = [
"1",
"+2",
"+4",
"+8", "+\\cdots",
"+8",
"+\\cdots",
"+2^n",
"+\\cdots",
"= -1",
@ -29,15 +31,47 @@ CONVERGENT_SUM_TEXT = [
"+\\frac{1}{4}",
"+\\frac{1}{8}",
"+\\frac{1}{16}",
"+\\cdots+",
"\\frac{1}{2^n}",
"+\\cdots",
"+\\frac{1}{2^n}",
"+\\cdots",
"=1",
]
CONVERGENT_SUM_TERMS = [
"\\frac{1}{2}",
"\\frac{1}{4}",
"\\frac{1}{8}",
"\\frac{1}{16}",
]
PARTIAL_CONVERGENT_SUMS_TEXT = [
"\\frac{1}{2}",
"", "", ",\\quad",
"\\frac{1}{2} + \\frac{1}{4}",
"=", "\\frac{3}{4}", ",\\quad",
"\\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8}",
"=", "\\frac{7}{8}", ",\\quad",
"\\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + \\frac{1}{16}",
"=", "\\frac{15}{16}", ",\\dots"
]
def partial_sum(n):
return sum([1.0/2**(k+1) for k in range(n)])
ALT_PARTIAL_SUM_TEXT = reduce(op.add, [
[str(partial_sum(n)), "&=", "+".join(CONVERGENT_SUM_TERMS[:n])+"\\\\"]
for n in range(1, len(CONVERGENT_SUM_TERMS)+1)
])+ [
"\\vdots", "&", "\\\\",
"1.0", "&=", "+".join(CONVERGENT_SUM_TERMS)+"+\\cdots+\\frac{1}{2^n}+\\cdots"
]
NUM_WRITTEN_TERMS = 4
INTERVAL_RADIUS = 5
NUM_INTERVAL_TICKS = 16
def divergent_sum():
return tex_mobject(DIVERGENT_SUM_TEXT, size = "\\large").scale(2)
@ -216,6 +250,26 @@ class ReasonsForMakingVideo(Scene):
self.add(line_two)
self.dither()
class DiscoverAndDefine(Scene):
def construct(self):
sum_mob = tex_mobject("\\sum_{n = 1}^\\infty a_n")
discover = text_mobject("What does it feel like to discover these?")
define = text_mobject([
"What does it feel like to",
"\\emph{define} ",
"them?"
])
sum_mob.shift(2*UP)
define.shift(2*DOWN)
define_parts = define.split()
define_parts[1].highlight("skyblue")
self.add(sum_mob)
self.animate(FadeIn(discover))
self.dither()
self.animate(FadeIn(CompoundMobject(*define_parts)))
self.dither()
class YouAsMathematician(Scene):
def construct(self):
you = draw_you()
@ -323,13 +377,17 @@ class ZoomInOnInterval(Scene):
class DanceDotOnInterval(Scene):
def construct(self):
num_height = 1.3*DOWN
interval = zero_to_one_interval()
dots = [
Dot(radius = 3*Dot.DEFAULT_RADIUS).shift(INTERVAL_RADIUS*x+UP)
for x in LEFT, RIGHT
]
color_range = Color("green").range_to("grey", NUM_WRITTEN_TERMS)
color_range = Color("green").range_to("yellow", NUM_WRITTEN_TERMS)
conv_sum = convergent_sum().split()
partial_sums = tex_mobject(PARTIAL_CONVERGENT_SUMS_TEXT, size = "\\small")
partial_sums.scale(1.5).to_edge(UP)
partial_sum_parts = partial_sums.split()
self.add(interval)
self.animate(*[
@ -346,17 +404,219 @@ class DanceDotOnInterval(Scene):
ApplyMethod(dots[0].shift, shift_val),
ShowCreation(line)
)
num = conv_sum[count]
num = conv_sum[count].scale(0.75)
num.shift(RIGHT*(line.get_center()[0]-num.get_center()[0]))
if num.get_width() > line.get_length():
num.stretch_to_fit_width(line.get_length())
num.shift(3*DOWN)
num.shift(num_height)
self.animate(
ApplyMethod(line.shift, 2*DOWN),
FadeIn(num)
)
self.dither()
self.animate(FadeIn(conv_sum[NUM_WRITTEN_TERMS]))
partial_sum_parts[0].highlight("yellow")
for x in range(0, len(partial_sum_parts), 4):
partial_sum_parts[x+2].highlight("yellow")
self.animate(*[
FadeIn(partial_sum_parts[y])
for y in range(x, x+4)
])
self.dither(2)
class OrganizePartialSums(Scene):
def construct(self):
partial_sums = tex_mobject(PARTIAL_CONVERGENT_SUMS_TEXT, size = "\\small")
partial_sums.scale(1.5).to_edge(UP)
partial_sum_parts = partial_sums.split()
for x in [0] + range(2, len(partial_sum_parts), 4):
partial_sum_parts[x].highlight("yellow")
pure_sums = [
partial_sum_parts[x]
for x in range(0, len(partial_sum_parts), 4)
]
new_pure_sums = deepcopy(pure_sums)
for pure_sum, count in zip(new_pure_sums, it.count(3, -1.2)):
pure_sum.center().scale(1/1.25).highlight("white")
pure_sum.to_edge(LEFT).shift(2*RIGHT+count*UP)
self.add(*partial_sum_parts)
self.dither()
self.animate(*[
SemiCircleTransform(*pair, counterclockwise=False)
for pair in zip(pure_sums, new_pure_sums)
]+[
FadeOut(mob)
for mob in partial_sum_parts
if mob not in pure_sums
])
self.dither()
down_arrow = tex_mobject("\\downarrow")
down_arrow.to_edge(LEFT).shift(2*RIGHT+2*DOWN)
infinite_sum = tex_mobject("".join(CONVERGENT_SUM_TEXT[:-1]), size = "\\samll")
infinite_sum.scale(1.5/1.25)
infinite_sum.to_corner(DOWN+LEFT).shift(2*RIGHT)
self.animate(FadeIn(CompoundMobject(down_arrow, infinite_sum)))
self.dither()
class SeeNumbersApproachOne(Scene):
def construct(self):
interval = zero_to_one_interval()
arrow = Arrow(INTERVAL_RADIUS*RIGHT, tail=ORIGIN).nudge()
arrow.shift(DOWN).highlight("yellow")
num_dots = 6
colors = Color("green").range_to("yellow", num_dots)
dots = CompoundMobject(*[
Dot(
density = 2*DEFAULT_POINT_DENSITY_1D
).scale(1+1.0/2.0**x).shift(
INTERVAL_RADIUS*RIGHT +\
(INTERVAL_RADIUS/2.0**x)*LEFT
).highlight(colors.next())
for x in range(num_dots)
])
self.add(interval)
self.animate(
ShowCreation(arrow),
ShowCreation(dots),
run_time = 2.0
)
self.dither()
class HowDoYouDefineInfiniteSums(Scene):
def construct(self):
you = draw_you().center().rewire_part_attributes()
text = text_mobject(
["How", " do", " you,\\\\", "\\emph{define}"],
size = "\\Huge"
).shift(UP).split()
text[-1].shift(3*DOWN).highlight("skyblue")
sum_mob = tex_mobject("\\sum_{n=0}^\\infty{a_n}")
text[-1].shift(LEFT)
sum_mob.shift(text[-1].get_center()+2*RIGHT)
self.add(you)
self.dither()
for mob in text[:-1]:
self.add(mob)
self.dither(0.1)
self.animate(BlinkPiCreature(you))
self.dither()
self.add(text[-1])
self.dither()
self.add(sum_mob)
self.dither()
class LessAboutNewThoughts(Scene):
def construct(self):
words = generating, new, thoughts, to, definitions = text_mobject([
"Generating", " new", " thoughts", "$\\rightarrow$",
"useful definitions"
], size = "\\large").split()
gen_cross = ImageMobject("cross").highlight("red")
new_cross = deepcopy(gen_cross)
for cross, mob in [(gen_cross, generating), (new_cross, new)]:
cross.replace(mob, stretch = True)
disecting = text_mobject("Disecting")
disecting.shift(generating.get_center() + 0.7*UP)
old = text_mobject("old")
old.shift(new.get_center()+0.7*DOWN)
kwargs = {"run_time" : 0.25}
self.add(*words)
self.dither()
self.animate(ShowCreation(gen_cross, **kwargs))
self.animate(ShowCreation(new_cross, **kwargs))
self.dither()
self.add(disecting)
self.dither(0.25)
self.add(old)
self.dither()
class ListOfPartialSums(Scene):
def construct(self):
all_terms = np.array(tex_mobject(
ALT_PARTIAL_SUM_TEXT,
size = "\\large"
).split())
numbers, equals, sums = [
all_terms[range(k, 12, 3)]
for k in 0, 1, 2
]
dots = all_terms[12]
one = all_terms[-3]
last_equal = all_terms[-2]
infinite_sum = all_terms[-1]
self.count(
numbers,
mode = "show",
display_numbers = False,
run_time = 1.0
)
self.animate(ShowCreation(dots))
self.dither()
self.animate(
FadeIn(CompoundMobject(*equals)),
*[
Transform(deepcopy(number), finite_sum)
for number, finite_sum in zip(numbers, sums)
]
)
self.animate(*[
ApplyMethod(s.highlight, "yellow", alpha_func = there_and_back)
for s in sums
])
self.dither()
self.add(one.highlight("green"))
self.dither()
class ShowDecreasingDistance(Scene):
args_list = [(1,), (2,)]
@staticmethod
def args_to_string(num):
return str(num)
def construct(self, num):
number_line = NumberLine(interval_size = 1).add_numbers()
vert_line0 = Line(0.5*UP, UP)
vert_line1 = Line(0.5*UP+num*RIGHT, UP+num*RIGHT)
horiz_line = Line(UP, UP+num*RIGHT)
lines = [vert_line0, vert_line1, horiz_line]
for line in lines:
line.highlight("green")
self.add(number_line, *lines)
self.dither()
self.animate(
ApplyMethod(vert_line0.shift, RIGHT),
Transform(
horiz_line,
Line(vert_line0.end+RIGHT, vert_line1.end).highlight("green")
),
run_time = 2.5
)
self.dither()