Today I am going to tell you about the parallel postulate in geometry. Well let
me start by telling you what it is, right? So this originally appears in Euclid's Elements - it's, I
think, published 300 BC. It's what we inherited as geometry, right, and it was in fact for a very
long time, especially in the 19th century, taken to be the example of what it means to know something,
right. We like it because it's deductive - you know, you have a set of rules, and then from
those rules you derive some truths. So what is the parallel postulate? And this is something
that we actually learn in high school geometry. If we are given a line in our plane now, here we
have a point P, a point that is not in the line. Then what we can do is we can we can draw many lines
that go through that point P but what Euclid told us, and that was the parallel postulate, is that
there's a unique line that does not intersect L and goes through the point P. - (Brady: One and only one.)
One and only one. That is that is the key here, one and only one. There exists one and it is unique.
And that's what we took to be true. - (That seems to me like the most obvious thing in the world.)
(Because if I draw any other line it's got that slight angle which means it's going)
(to be- one side of it is going to be tipping towards L.) - Right right right. And in fact, you know, like-
so this is a little bit of a lie because if you open Euclid's Element you're not going to find
this statement the way it is written. So in fact the way he wrote it is more similar to what you
were saying; what he said is, you know, if you have a line and then you have the point P you draw your
line through P then - let me draw just any other line through P to help. What he said is if
the sum of these two angles is less than 180 degrees then the lines, once you extend them to
infinity, they're going to intersect. - (Right, classic Euclid, parallel postulate- seems true to me.) - Yes exactly
exactly and it- well in fact it seems true to you and to most of us because it is actually true. But
it sort of depends on the model of geometry that we're considering, right? So here we're working on
a plane and to do geometry we need among many other things a way of measuring distance. And, you
know, like if we're here in the plane we measure distances with a ruler. Or if you want, you know,
like there's a formula if you have coordinates - um it's basically the Pythagorean theorem. So we have-
we know what we're doing; however that is not the only space in which we can do geometry.
Not even 2-dimensional space, right, so the plane we have a vertical direction and a horizontal
direction, those are our two dimensions right? Let me show you another space where we can do geometry.
Here's what we call a sphere, so as mathematicians call it S2. If you're a mathematician then you
draw R3, 3-dimensional space, and you say that this is the set of points that are at
distance 1 from the origin. If you're not a mathematician you're just looking at the shell
of a football ball. - (Is this a 2-dimensional space or?) - This is a 2-dimensional space because
again we basically only have two directions in which to move right. So if you want we can
do it in terms of meridians and latitude, right, the way we do it to describe location, right. Like
if you look in Google Maps, if you want a precise location, you're going to get two coordinates.
Exactly that. - (Okay so although although we think of balls and things as 3-dimensional objects,)
(because we're only dealing with the shell it's a 2-dimension?) - Exactly yeah and that is very
important, we're not thinking about the filling. We're- you know, we're thinking about a
little ant that is just completely flat. - (So what happens to the parallel postulate here?) - Well so,
you know, here I started by telling you there's a line and there's a point, right? So I have to tell you the same thing here.
I have to define my objects. So a line in here is going to be what we call a Great Circle. What is a
great circle? Take the intersection of this sphere with a plane that that goes through the origin - so
this is a great circle. Or, you know, the way I drew it here this is the equator, that's also going to
be a great circle. - (So every line has to like- through the longest possible line does it? It can't be)
(just a little circle?) - Aha! So that's- so this is key right? So there's a difference between what we call
a line and what we call a line segment, right. And actually so that line segments, that's one of the
first rules that we have in Euclidean geometry or in in the geometry of the plane; which is that if
I give you two points you're going to be able to give me a line. In our plane geometry that's
pretty easy, two points, we get a ruler, here's the line right. Pretty easy, we know how to do it. Here it's a
little trickier but is is not so bad. If you choose a point say over here, and then maybe a point over
here, then how do I find the great circle that goes through them, right? And and I need to find
the great circle and then only look at that little segment that connects them. And that's another- that
little segment. Well, we can do it um by choosing- you know, like we can extend these points, join
them to the origin. There's a little space in between them, let me fill that with a plane.
That's the plane that's going to intersect my sphere, it's going to be something like that.
(So you're trying to find the equator basically that- on which both of those lines sit?) - Exactly, exactly,
except you know like it's a tilted equator right? Like it's not- but yes that is basically it. I could
have- I mean I could have cheated and just chosen two points in in the equator but. But so what I
drew here is a line, right, so the line segment. Well, we have two options: this very clearly short one;
but then there's another one that is longer and it goes around. Sometimes what we say is that a
line segment is the segment that- whose length is less than half of the entire circumference. That
fixes some issues. - (So what does this mean for the parallel postulate?) - Excellent question. So let me do-
let me draw my sphere here again. I'm going to draw a line that is easier for me, again this meridian.
That's my line, so this is my line L, and then I'm going to get my point P. Let me choose this
point P right. So lines that go through P: there's this one, so here we have a problem, right we have
this intersection and this intersection. So maybe we need another one right? Okay so maybe maybe
we draw the line segment, or the whole line, that goes through P and then another point over here.
Okay, we draw it. - (No good.) - No good. So turns out that what happens in the 2-sphere is that there
are no parallel lines. So before we said that in the plane there existed one and it was unique,
so here we don't even have that first part of existence - we're done. - (Okay. But I mean that's)
(not- that's not Euclid's fault. He wasn't talking about-) - No, no no no no no no no absolutely not. It's
actually kind of interesting because there was a lot of research being done in in the geometry
of the sphere, or trigonometry maybe to be more precise, because people needed this for navigation
and for astronomy - or astrology for some people too. You know, like everything that had to do with
the skies was a sphere. So there was a lot of knowledge but somehow nobody thought that we
could think about these two spaces as similar, as both sharing a set of rules, right. It wasn't until
maybe later that we- in the 19th century that there was an upheaval in geometry when we realized that
in fact if you consider the first four postulates of Euclid, there are older models where you can do
geometry that holds- you know, that has those four rules except for that fifth one. - (So that- so his)
(first four rules hold up in other spaces? But this is the one that doesn't transfer across?) - Yeah. It-
may- you know maybe to be precise uh it might be a modern restatement of his axioms or postulates.
Right, like we we had to get rigorous. The one model, or the one geometry, that people started to
think about to show that the parallel postulate was just not true in general was not spherical
geometry but hyperbolic geometry. This is going to be a disk, let me draw the center. And again I have
to tell you what it means to have a line. So a line, it's either going to be a straight line that goes
through the center or if I move a little bit away from the center it's going to be the segment of a
circle that is orthogonal to the boundary circle of the disk. So are there parallel lines, right?
Well in this case we are going to have parallel lines but we're going to have too many. So that's
sort of like the other problem right? Like we said: there exists, and is unique. We already saw a place
where they do not exist, now we have one where they do exist but they're not unique. So what happens
here is um- so let me say again that this is my line L, why not, and let's say this is my point P.
Well that's one line that goes through P, doesn't intersect L, great, but I can draw more. I can sort of
move my circles. I could just do many many more because in fact they're going to be infinitely
many. So the length here; again, you know, going back to this distinction between a line segment and a
line - this circle is a line. It sort of looks like it's like finite, right? Like here a line is like
the one that extends forever. The way we measure distances here is such that if you have two points
that are very close to the border, these two points are very very very far away from each other. Even
though you know like we're looking at it, and if we were using the distance measurement that we had in
in the plane, they look very close to one another; but the way we measure things in this new geometry
makes them be very far away. - (This postulate has failed us again) - Yes exactly, except you know it
failed us um in the sense that it- it doesn't hold right, so that immediately means that it- it is not
a consequence of the first four rules which was actually a question that people had for a very
long time. Could it be derived from the first four postulates? And now we see that's just not the case.
(Clearly this fifth one was a bit of an add-on, it's not part of the family.) - It's not a part of the
family no, no not exactly. And in fact- - (Can the- can the other four be derived from each other?) - No, no no
no no those are sort of- you know, like the the four of them are the minimal set of rules that you need
um to play the game of geometry if you want. Yeah but, you know, like in fact in in Euclid's
Elements, if you analyze it you know like maybe in a philosophical or historian of science way, you
realize that Euclid waits a while before using the parallel postulate, right? He uses the first four
back and forth, back and forth, but the the fifth one he waits a little bit. - (That was a clue.)
That's what some people think, yeah, so maybe he was uneasy about it a little bit.
(Does the- does this make you feel more affectionately towards the fifth one or less affectionate towards)
(it? Do you think of it as an ugly aberration or does it make you feel it's more charming?)
I think it's- well I don't know if charming but you know like it's a lot more important,
right, because it just allows us to think of more than one way of doing geometry in two
dimensions. That's- that's just fascinating.
...the angle is less than two right angles.
(It's not catchy is it?) No, there's a lot of words, it's wordy. And
let's let's see what it means. Two straight lines for instance, we're going to put a third one in...