Hey Brady, we are going to talk about
Euclid's fifth axiom: the parallel
postulate. Really what this axiom is
about is the story of a revolution about
two thousand years in the making.
So the axioms, there are 5, and the axioms are
about geometry so geometrical figures. Ok
let's start with, let's start with number one.
Let the following be postulated, ok one:
to draw a straight line from any point
to any point- we have points and lines
and we can connect- between two points we
can connect the points with a line.
(Brady: Alright, yeh that's pretty simple. Alright what next?)
Ok number two: to produce a finite straight line
continuously in a straight line - so this
might seem a little unclear, what's
happening is that we're creating a
plane. Think of a line segment, that's a
finite straight line, and we're going to
continuously move it in a straight line
that gives breath. So we have from a line,
to moving it in a line we get a plane.
Now we've talked about points and lines
and we've created a plane we can move to
circles. To produce a circle with any
center, that's a point, and any distance
or radius.
We've gone from points and lines and
planes to now allowing some other
objects to live in our world, circles. The
first three seem straightforward enough,
number four things start to get slippery.
Number four is that all right angles are
equal to one another. - (Brady: Well that's gotta be the)
(most obvious of all of them! Why do we even)
(need to say that?) - What a right angle is
requires definition and you could
actually start by defining what points
and lines and these basic objects are.
But the notion of equality is more
subtle because, do we mean equal in the
sense of is, or this is exactly the same
as that or a notion of congruence? - (We've got)
(four. Now we're coming to the fifth, this)
(is this is the one that we should be)
(paying attention to right?) - That's right so
here comes the fifth - the fifth
postulate. Let's write it down the way
Euclid wrote it and then we'll
decide for ourselves what it
means. That if a straight line falling on
two straight lines makes their interior
angles on the same side - stay with me -
less than two right angles, then the two
lines if produced indefinitely meet on
this side which are the angles, the
angles less than two right angles.
(Brady: It's not catchy is it?) - No, there's a lot of 'where
is wordy'. And let's let's see what it
means. Two straight lines for instance,
we're going to put a third one in which
lies on top. Now the wordy bit about the
interior angles being less than two
right angles are those interior angles
there. And then the axiom says that
if these two lines are extended then
there will be a point where they meet. - (Brady: Ahh, so)
(if these aren't 90 they're going to)
(crash?) - That's right. An equivalent way of
saying this, this statement is the
following: so if you have a line and a
point not on that line, say here, and
then there exists a one and only one
line through this point parallel to the
given line. Any other line that we
draw will eventually intersect the given
line. So immediately this stands out, this
axiom stands out from the others and
it's just in its wordiness and and the
fact that it's not completely
straightforward; he was kind of aware
of this and he used it reluctantly; and
even in the analysis of deducing other
propositions in the elements he waits to
use this axiom as long as possible.
(Brady: It's his axiom of last resort?) - If you- yes.
And and although when we write it down
we we can think this is- we understand
why you take it as an axiom and we- it
seems very straightforward. And so here's
the amazing story, the
two-thousand-year-old story of this
revolution, this revolutionary idea. The 5th turns
out to have this kind of departure into
other worlds. The first two had to deal
with them lines, points and planes; and
the third had to do with the circles. Number four
is is also- there's some more hidden
things there but the fifth, this notion
of parallelism is is- turns out to be
the key in kind of unlocking new and
mysterious worlds. In 1820-
1830, independently, two
mathematicians: a Hungarian, Bolyai, and a
Russian, Lobachevsky, they independently
discovered what we now call hyperbolic
geometry and the fifth axiom - so we can
write maybe 5 prime - can be stated
like this: given any any line and a
point not on that line, there exists
infinitely many straight lines through
that point parallel to the given line. So
parallel meaning not intersecting.
(Brady: A moment ago we drew something that showed)
(there was only one parallel line; now)
(you're saying they are infinite?) - That's right.
(Where can these be drawn? What does)
(this- what does this look like?) - So this new world
is the hyperbolic plane, which they've
created. Bolyai actually said 'Out of
nothing I have created an entire world'.
So now it turns out that straight
lines in the hyperbolic plane, in this
model of the hyperbolic plane, look like
this; well Euclidian vertical lines that
intersect this boundary at 90 degrees, or
on Euclidean circles centered on the
real line. So centered at the real line
means that you would intersect at 90
degrees. So these are straight lines in
hyperbolic space. - (Brady: So that's a straight)
(line too?) - In the hyperbolic plane, that's
right. Doesn't look like it is but it is.
Okay so let's take this as our given
straight line and let's take this point
as the point not on this straight line.
So we've drawn one straight line
parallel to the given one, parallel in
that they don't intersect, but now we- - (Brady: Hang on,)
(so that line is parallel to that line?)
Yes that's right. We say parallel lines
are lines that don't intersect, so they
don't intersect. With our hyperbolic eyes,
or the hyperbolic metric, these lines are
and indeed straight and they don't intersect
so they're parallel. And now through this
point we can draw another
straight-line that's parallel to this,
and we can draw another. So all these
circles are perpendicular to the
boundary and going through this point.
[Preview] Ah it's an unfamiliar thing to most people,
and it certainly behaves quite
differently than Euclidean space so at
first it's confusing but I think you can
get used to it fairly quickly if you
think about it.