Kind: captions
Language: en
a single sentence in one of the oldest
math books held the key to understanding
our universe ucl's elements has been
published in more editions than any
other book except the Bible it was the
go-to math text for over 2,000 years but
for all that time mathematicians were
skeptical of a single line which seemed
like a
mistake ultimately some of the greatest
math Minds realized that uclid wasn't
wrong after all
but there was more to the story slight
tweaks to this line opened up strange
new universes out of
nothing surprisingly 80 years later we
found out those strange new universes
are core to understanding our own
[Music]
Universe around 300 BC the Greek
mathematician uclid takes on a massive
project to summarize all mathematics
known at the the time to essentially
create the one book that contains
everything that everyone knows about
mathematics but that's no easy task see
before euklid there was a bit of a
problem with math people would prove
things but they would just be going
around in circles why does a triangle
have 180° cuz if you take two parallel L
yeah but why did parallel lines exist oh
that's that's because you can make a
square why does a square exist you have
this infinite recursion of what the
fundamental reason why something is true
is it's kind of like in the dictionary
every word is defined in terms of other
words so how do you get to ground
truth uid used a solution that was
pioneered by the Greeks let's just
accept a few of the most simple basic
things as being true these are our
postulates then based upon these
postulates we can prove theorems one at
a time building up our math using logic
so that as long as those first
statements are true then everything else
that follows from them must definitely
be true he had perfected the gold
standard for rigorous mathematical proof
that all modern math relies
on uid used this method when he
published his 13 book series called the
elements in which he proved 465 theorems
covering almost all of mathematics known
at the time including geometry and
number
Theory and all these theorems depended
on were some definitions a few common
Notions and five Poss
poates we go right to book one and book
one starts with definitions you got to
start somewhere and the definition is a
point is that which has no part a line
is a breadless length and the ends of
lines are points by line he really means
a curve and then it has some ends a
straight line is a line that lies evenly
within the points on itself and then so
on and so forth he's got 23 definitions
then he's got the five postulates
the first four are simple one if you
have two points you can draw a straight
line between them two if you have a
straight line you can extend it
indefinitely three given a center and a
radius you can draw a circle and the
fourth is that all right angles are
equal to each other but postul five gets
the big guns which is that if a straight
line falling on two straight lines makes
the interior angles on the same side
less than two right Ang angles the two
straight lines if produced indefinitely
meet on that side on which are the
angles less than the two right
angles what the hell is he talking about
that's a postulate like all of these are
you know half of a sentence and they're
all blatantly obvious and then comes
five at a left field and it's like an
entire paragraph what is he
doing this made mathematicians
suspicious it seemed like uclid made a
mistake Greek philosopher procus thought
postulate 5 ought even to be struck out
of the postulates altogether for it is a
theorem but if it's a theorem we should
be able to prove it from the first four
postulates so that's what many people
tried some including tmy and procus
believed they had succeeded but they
hadn't in fact all they managed to do
was just restate postulate 5 in
different
words here's one formulation if you have
a line and a point that is not on that
line then there is a single unique line
which will be parallel to the first line
for this reason the fifth postulate is
often called The Parallel
postulate when the method of direct
proof failed other mathematicians
including alham and Omar cam tried a
different approach proof by
contradiction the idea is simple you
keep the first four postulates the same
but assume that the fifth postulate is
false then you use those new postulates
to prove theorems and if that leads to
to a contradiction for example true
equals false well then it means your new
fifth postulate must be wrong therefore
the only remaining option would be that
ucl's version of the fifth postulate is
correct and you would have proven the
fifth postulate so what would it look
like if the fifth postulate were false
well according to uclid through a point
not on a line there could only be one
line that is parallel to the first one
alternative is that there are no
parallel lines that you could draw
through that point well people tried
that and they realized that then lines
had to be finite in length like well
that can't be so this option was ruled
out it contradicted the second postulate
which states that lines can be extended
indefinitely the other alternative is
that you can draw more than one parallel
line through a point not on the first
line so that's what they would do they
would assume the postulate 5 fails like
this has got to be wrong where's the
cont they couldn't find the
contradiction so proof by contradiction
also failed in total mathematicians
spent more than 2,000 years trying to
prove the fifth postulate but everyone
who tried
[Music]
failed then around 1820 Janos buoli a
17-year-old student started spending his
days and nights working on the mystery
his father became worried and he wrote
to his son you must not attempt this
approach to parallels I know this way to
the very end I have traversed this
bottomless night which extinguished all
all light and joy in my life I entreat
you leave the science of parallels alone
learn from my
example but the young buol ey didn't
listen to his father he could not leave
the science of parallels alone after
years of work he realized that maybe the
fifth postulate can't be proven from the
other four it could be completely
independent see according to uclid you
could have only one parent parallel line
through a point but bul I imagined a
world where there could be more than one
parallel line through that point but how
well who said you needed to have a flat
surface on a surface that is curved like
this you can draw more than one line
that is parallel to the original line
but wait a second those lines don't look
straight well what makes straight lines
special is that they're the shortest
paths between two points on this surface
those shortest paths just look bent
because the surface is curved here's a
more familiar example airplanes always
try to fly the shortest path between two
cities they're basically flying in a
straight line but that line doesn't look
straight on a map because the surface is
curved these shortest paths on curved
surfaces are called
geodesics so all these lines are
straight they just don't look it because
the world B ey had imagine turned out to
be curved we now know this as hyperbolic
geometry
you know when I used to think of the
hyperbolic plane I would just imagine it
as one giant saddle but that's not
really what it is the hyperbolic plane
is much more like this piece of
crocheting so it starts out pretty flat
and even in the middle but as you move
outwards more and more fabric is created
and that would push parallel lines apart
and the further and further out you go
the amount of fabric grows exponentially
and that ends up causing this crumpling
effect so if you really want to think
about the hyperbolic plane I think
you've got to think about saddles on
saddles on Saddles like it's a infinite
crumpling
mess but that little piece of crochet
isn't the full hyperbolic plane to show
that we need to make a map one that fits
the entire plane into a disc to show how
this works we're going to fill the
entire plane with these triangles
starting in the middle just as with the
crochet things look pretty normal but as
you go farther out from the center you
get all this extra space so you can fit
more and more triangles so they appear
smaller but they are actually the same
size now since the hyperbolic plane is
infinite you can keep adding triangles
forever and they all need to fit on the
disc so as you get closer to the edge
the triangles will appear smaller and
smaller and smaller infinitely smaller
never ending and you can never quite
reach the edge
this is known as the panker disc model
here straight lines are arcs of circles
that intersect the disc at 90° and just
like on our original shape a straight
line down the middle appears straight
while straight lines next to it appear
to curve
away what's remarkable is that Bali ey
didn't have a model of hyperbolic
geometry yet he was just drawing ukian
triangles with the assumption that UK's
fifth postulate didn't hold and while
Bal ey found that the beh behavior in
hyperbolic geometry is very different
than ukids mathematically it seemed just
as
consistent in 1823 the 20-year-old Janos
wrote to his dad I have discovered such
wonderful things that I was amazed out
of nothing I have created a strange new
universe but bullly ey had been doing
more than just tackling ancient math
mysteries in his 20s he joined the army
where he continued developing two of his
other passions playing the violin and
dueling he had mastered both but with a
sword in particular he was unmatched
perhaps because of his many talents buol
ey grew arrogant and found it difficult
to accept Authority from his superiors
that made him hard to get along with
this reached a peak when during one of
his deployments 13 Cavalry officers from
his Garrison challenged him to a
duel Bola accepted their Challenge on
the condition that after every two duels
he could play for a little while on his
violin Voli fought each of them in
succession winning all 13 Duss and
leaving behind all his adversaries on
the
Square while bully ey loved dueling his
first love was still mathematics In 1832
9 years after he discovered his strange
new universe he published his findings
as a 24-page appendix to his father's
textbook
extremely proud and excited about his
son's work farcus boli sent it to
perhaps the greatest mathematician of
all time Carl Friedrich gaus after
careful examination gaus replied a few
months later to praise it would amount
to praising myself for the entire
content of the work coincides almost
exactly with my own meditations which
have occupied my mind for the past 30 or
years years earlier gaus had wandered a
similar path in 1824 he wrote a private
letter to one of his friends in which he
describes discovering a curious geometry
one with paradoxical and to the
uninitiated Absurd
theorems for example gaus writes the
three angles of a triangle become as
small as one wishes if only the sides
are taken large enough yet the area of
the triangle can never exceed a definite
limit in other words you can have a
triangle that's infinitely long but the
area is
finite you can see why by using the
planker disc Model A small triangle
looks pretty ordinary but as you make it
bigger the angles start to become
smaller and smaller eventually all those
angles go to zero because all these
lines intersect the dis at
90° now these lines are infinitely long
but because of the geometry the area is
finite in the same private letter gaus
wrote all my efforts to discover a
contradiction and inconsistency in this
nonukan geometry have been without
success just like buol ey gaus had found
that this geometry seemed thoroughly
consistent he named it non- ukian
geometry a name that stuck it describes
geometries where ukids first four
postulates hold but the fifth doesn't
but gaus decided not to publish his
findings for fear of
ridicule this aversion to a different
kind of geometry should at least be a
little surprising because there is one
other geometry that we should be very
familiar with spherical geometry since
we all live on a
sphere on a sphere straight lines are
parts of great circles these are the
circles with the largest possible
circumference on Earth the equator and
Circles of longitude are examples of
great
circles and we can use this to see how
straight lines behave these lines seem
to go in the same direction but as you
keep extending them you find that they
intersect once and then again on the
other side of the earth and this will
always happen for any two great circles
because they must each have the largest
possible circumference so on a sphere
there are no parallel
lines gaus had long been fascinated by
spherical geometry he was also a geodist
and frequently took measurements of the
Earth in the 18 20s he was tasked with
surveying the kingdom of Hanover to help
make a map as part of this survey he
climbed the mountains near Gooding with
the help of people situated on other
landmarks they were able to carefully
measure the angles of several triangles
which would then be used to determine
the position of one place relative to
another as a reference for the survey
and to help determine the roundness of
the earth they also precisely measured
the angles of a large triangle formed by
three
mountains but for for all of gauss's
romantic Notions of taking measurements
on top of mountains he was not the
kindest
correspondent when boli received his
Hero's response he was devastated
believing that gaus was trying to
undermine him and steal his ideas he was
so embittered by gauss's response that
he never published
again in 1848 Bola had to endure another
hardship when he found out that Russian
mathematician Nikolai luchesi had
independently discovered non Ian
geometry several years before boli had
published his 24 Page Long
appendix when boli died in 1860 he left
behind 20,000 pages of unpublished
mathematical manuscripts he would not
know that gaus did independently
discover non- ukian geometry nor would
he know that upon receiving the appendix
gaus had written to a friend I regard
this young geometer Bali ey as a genius
of the first order
[Music]
while bull ey was embittered non- ucan
geometries continued to develop until
1854 spherical geometry wasn't actually
considered a non- ukian
geometry that's because on a sphere
lines can't be extended indefinitely
this is what earlier mathematicians had
stumbled upon and therefore dismissed
this geometry since ID's second
postulate wouldn't hold but in 1854
reman changed the second postulate from
an infinite extension to something that
is quote unbounded so that the second
postulate still holds on a
sphere with this change spherical
geometry became another valid non-an
geometry by using the generalized four
postulates and taking the fifth as there
being no parallel lines you could now
derive spherical or elliptic
geometry would you consider the fifth
postulate a mistake would it have been
better if he just never wrote that down
if he never wrote that down he would
have stunted his geometry because he
wouldn't be able to prove a lot of the
things that he claimed to have it's
beautiful that he wrote this it's
beautiful that people spent 2,000 years
trying to refute him only to discover
that in fact he was right in writing
this in the first place so while uclid
was right to write down the fifth
postulate he did make a different
mistake so here's the problem with with
what ukl was doing definition one a
point is is that which has no part what
does it mean to have a part what is a
part what does it mean not to have a
part a line is a breadless length what
does it mean to have breath lying evenly
within itself within the points in
itself what the hell is he talking about
we read this two minutes ago and we were
nodding along like yeah completely makes
sense what he's saying it's all nonsense
don't give me a definition that's going
to have an infinite recursion if you
give me a definition in terms of other
things then you have to tell me what
those things are if you tell me what
that is you have to tell me what the
thing before it is are the
definitions a bad idea you shouldn't
have definitions you should have
undefined terms I'm not going to tell
you what a point is I'm not going to
tell you what a line is I'm not going to
tell you what a plane is all I'm going
to tell you is what the postulates are
that they're assumed to satisfy it's the
relationships between the objects that's
important not the definitions of the
objects themselves and once you free
your mind to that possibility all of a
sudden you realize that there's a
perfectly good geometric World in which
by line you mean great circle and by
plane you mean sphere and by point you
mean a point on on a sphere and then
four of those axioms are satisfied just
not the fifth and similarly you there's
another model something called the dis
model for hyperbolic space which the dis
is the
plane what I mean by straight lines is
arcs of circles that are orthogonal to
the dis and then points are points
inside the dis and the dis is the
plane see see you can think of geometry
as a game the first four postulates are
like the minimum rules required to play
that game and then the fifth postulate
selects the world that you'll play in if
you pick that there are no parallel
lines you're playing in spical Geometry
if you choose one parallel line you're
playing in flat geometry and if you go
for more than one parallel line then
you're playing in hyperbolic
geometry but reman decided to take it
one step further instead of selecting
just one world to play in why not
combine them all into
one during his inaugural speech in 1854
he laid out the groundwork for a
geometry where the curvature could
differ from place to place one part
might be flat another part might be
slightly curved and yet another part
might have a very strong curvature and
this geometry wouldn't be limited to
two-dimensional planes either it could
be extended to three or more
Dimensions another breakthrough came in
1868 when Eugenio Beltrami unequivocally
proved that hyperbolic and spherical
geometry were just as consistent as ID's
flat geometry that is if there were any
inconsistencies in hyperbolic or
spherical geometry then they must also
be present in ID's flat
geometry the prospects for these new
geometries were looking great and it
turns out this was just the
beginning in 1905 Einstein proposed the
special theory of relativity which is
based on just two postulates one the
laws of physics are the same in all
inertial frames of reference and two the
speed of light in a vacuum is the same
for all inertial observers so as a
result space and time must be
relative but that created a problem for
Newtonian gravity because according to
Newton the force of gravity is inversely
proportional to the distance between the
two objects squared but in Einstein's
special relativity that that distance is
no longer well defined in whose
reference frame are we measuring and so
Einstein had to find a way to reconcile
relativity and gravity 2 years later in
1907 Einstein had the happiest thought
of his life he imagined a man falling
off the roof of a house and what made
Einstein so joyful is that he realized
that while the man is falling he would
feel absolutely weightless and if he let
go of an object it would just remain in
uniform motion relative to him
it would be just like being in space not
near any masses floating around in a
spaceship at constant velocity and that
is an inertial observer now here's the
big breakthrough Einstein realized that
they're not just similar they are
identical because there is no experiment
you could do to determine whether you're
in freef fall in a uniform gravitational
field or whether you're in deep space
not near any massive
objects and so the Free Falling Man 2
must be an inertial observer meaning he
is not accelerating and he's not
experiencing any force of gravity but if
gravity is not a force then how do you
explain things like the space station
orbiting Earth shouldn't it just fly off
in a straight line well astronauts in
the space station also feel weightless
and that's the key it feels just as if
they're traveling at constant velocity
in a straight line it feels like that
because that's precisely what they're
doing they're tra traving in a straight
line how then could that straight line
appear curved to a distant Observer the
answer is because the SpaceTime that
straight line is on is
curved see massive objects curve
SpaceTime and objects moving through
curved SpaceTime will follow the
shortest path through that curved
geometry the geodesic so while
astronauts in the space station are
following a straight line it appears
curved to a distant Observer because the
Earth curves the space time around
it so the behavior of straight lines in
curved geometries is core to
understanding the universe we live in
and in the more than 100 years since it
was published the general theory of
relativity has been remarkably
successful in 2014 astronomers briefly
observed a supernova a violent and
extremely bright death of a star in fact
they saw the exact same Supernova in
four different places how well in
between the Supernova and Earth there
was a massive Galaxy which curved
SpaceTime so light from the Supernova
which was spreading out in all
directions had several different paths
to reach the Earth and four of those
reached Earth at approximately the same
time the Galaxy had functioned as a
massive gravitational lens the
astronomers realized that other galaxies
in the cluster might also lens the light
from that Supernova but with different
path lengths and gravitational
potentials so the light would reach
Earth at different times after careful
modeling they predicted that they should
see a replay of that Supernova just a
year
later and on the 11th of December 2015
just as predicted they saw the same
Supernova once
more in addition to being able to
observe the effects of curved SpaceTime
we can now even measure the ripples of
SpaceTime itself gravitational waves
formed by Cosmic events far far away
like the merger of black
holes and according to a recent survey
by nanograv the fabric of SpaceTime
seems to be buzzing with the remnants of
grand Cosmic events in the hundred years
since general relativity was published
countless findings have supported its
predictions and at its very core are the
curved geometries of boli and reman but
so far all the effects we've looked at
are local distortions of
SpaceTime what is the shape of the
entire
Universe using the differences between
the geometries we can find that out too
in flat geometry we expect all the
angles of a triangle to add up to 180°
without fail but in spherical geometry
the angles don't add up to 180° but to
more similarly in hyperbolic geometry
the angles add up to less than
180°. so to determine the shape of the
universe you just need to measure the
angles of a triangle and measuring a
triangle is precisely what gaus was
doing 200 years ago in fact this led
some to speculate that he was actually
trying to measure the curvature of space
itself the angle he found 180° within
observational
error but that shouldn't be very
surprising take this balloon for example
which approximates a sphere if I draw a
small triangle on it well the surface
I'm drawing on is basically flat so the
angles inside the triangle will add up
to essentially
180° only if I make the triangle large
enough will the effects of curvature
come into play and then the angles in
the Triangle will add up to more than
180° and this was the problem with
gauss's experiment even if he was trying
to measure the curvature of space itself
for which there is no solid evidence the
triangle he measured would have been far
too small relative to the size of the
universe so in order to overcome the
scale issue that gaus encountered we
need to scale the triangles formed
between mountains up to the largest
triangles we can and since looking
further and further away is the same as
looking further back in time we need to
look back as far as possible to the very
first light we can see the cosmic
microwave background or CMB a picture
from when the universe was just 380,000
years old while the CMB is almost
completely uniform there are some spots
that are slightly hotter or colder now
we know how far away the CMB is so if we
can figure out how large such a spot is
then we can draw a cosmic
triangle it's thought that the first
density and temperature variations
originated from Quantum fluctuations in
the very early Universe which were then
blown up as the universe expanded but
due to this rapid expansion not all
regions were in causal contact with each
other so using the information we have
how the early Universe evolved
astronomers can predict how often spots
of different sizes should appear in the
CMB this is what this power Spectrum
shows essentially a histogram of how
often each spot size should occur if the
universe is flat so now we have
something to compare our measurement to
if the universe is flat the angle we
measure on the sky should be the same as
we'd expect but if the universe is
curved like a sphere the angles of the
triangle should add up to more than 180°
so the angle we'd measure would be
larger than predicted and this peak
would shift to the left similarly if the
universe has hyperbolic geometry then
spots should appear smaller than
predicted and this peak would shift to
the right so what did we measure this is
the data from the plank Mission which is
almost exactly what you'd expect if the
universe were flat this Mission also
gives us the current best estimate for
the curvature of the universe which is
007 plus-
0019 so that's basically zero within the
margin of error so we're fairly certain
that the Universe we live in is
flat but living in a flat universe seems
to be remarkably serendipitous right now
the average mass energy density comes
down to the equivalent of about 6
hydrogen atoms per cubic
meter if on average that was just one
more hydrogen atom the universe would
have been more spherical curved if there
were just one less the curvature would
be hyperbolic geometry and so far we're
not entirely sure why the universe has
the mass energy density it
has what we do know is that general
relativity is one of our best physical
theories of reality and at the very
heart of it are those paradoxical and
seemingly absurd geometries ones we
found because mathematicians spent over
2,000 years thinking about a single
sentence from the world's most famous
math
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