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this is a riemann surface it's going to
help us think in forward dimensions we
made it by cutting our planes at the
discontinuities and our paths and taping
them together in a way that made our
paths from one plane to the other
continuous Riemann's big idea here is
that the domain the input values of our
multi function should not be a flat
two-dimensional plane our domain should
instead be this a curved surface living
in higher dimensional space a riemann
surface what's incredible here is what
the geometry of our riemann surface is
going to allow us to do
using our Riemann surface as the input
space to our multi function we can
literally fix all of the problems we've
encountered thus far our function will
be one-to-one continuous and our Riemann
surface will even help us elegantly
explain the weird loop behavior result
back in part 11 let's see how Riemann
envisioned these surfaces as sheets
covering the input plane w our Riemann
surface is constructed from two copies
of the complex plane and the idea here
is that each input value on w lies
directly below its corresponding points
on each layer of our riemann surface if
we follow a line straight up from the
value to ion our W plane we find two
points that correspond to W equals two I
our Riemann surface fixes our one-to-one
problem just as our two complex plane
solution to each of our two solutions on
Z corresponds to it's very own copy of
the W plane these are called branches
so our riemann surface makes our mapping
one-to-one just as our two complex plane
approached it last time but what about
continuity as we saw last time a big
problem with our two complex plane
solution is that it introduced
discontinuities we constructed our
surface in such a way that our colored
path was continuous but we encountered
this weird self intersection so let's
dig a little deeper remember the main
idea using our riemann surface as the
input space or the domain to our complex
multi function should give us clarity so
instead of drawing paths or shapes on
our w plane you should really be drawing
on our riemann surface drawing on
three-dimensional surfaces can be a
little challenging so we'll make use of
a tool that wasn't invented until over a
century after Riemann's death a computer
just is with our paper version we'll
start with our W plane line flat on the
ground and place our riemann surface
directly above before we start drawing
paths all over our surface let's make
sure we know what we're looking at we're
trying to understand the complex
function W equals Z squared or taken in
the other direction Z equals plus or
minus the square root of W the mapping
between W and Z is the same in both
directions the visualization challenge
here is that our mapping is
four-dimensional both Z and W have real
and imaginary parts back in part 10 we
called these XY u and V what we're
seeing when we visualize our riemann
surface is a two-dimensional surface in
three-dimensional space in this case
since we positioned our surface directly
above our W plane two of our three
dimensions correspond exactly to the
real and imaginary parts of W we called
these U and V part of Riemann's idea is
that our third dimension should
represent the Z value of our function
but as we know Z is a complex number it
has both a real and imaginary part so
there's no way to show both of these on
a single
what's often done when visualizing these
surfaces and what we're going to do here
is simply pick the real or imaginary
part of Z and use this value as the
third dimension the height of our
surface here we're using X the real part
of Z doing this has a nice visual result
each point on our Riemann surface lives
in 3d space at a location corresponding
exactly to its UV and X values so each
point on our surface represents a single
solution to our equation and three of
the four values needed to describe the
solution are represented by the points
location in 3d space this is a nice
result but we must remember that this is
not the whole picture there's another
variable the imaginary part of Z we call
this Y that is not included in our
visualization this becomes important
when trying to figure out if we've
actually fixed our continuity problem if
we follow a path along a single branch
of our riemann surface we run into a bit
of a problem when we hit this self
intersection after all which way should
we go up or down
to answer this question let's try to
figure out why our surface self
intersects in the first place this
intersection happens along the negative
real axis of our W plane let's consider
a point on this axis W equals minus 1
plugging in negative 1 for W yields two
solutions Z equals plus I and Z equals
minus I these two solutions are clearly
different but they have the same real
part zero since we're only visualizing
the real part of Z we have no way of
seeing that these are in fact different
points this is the danger of visualizing
high dimensional mathematical concepts
what we're really looking at here is
just a projection a shadow of our full
four-dimensional surface so our line of
self-intersection actually isn't this is
exactly like the two-dimensional shadows
of three-dimensional objects giving the
appearance that the objects intersect
there are inherent limitations to the
types of structures we can visualize in
the 3d space we live in
however there are some very clever ways
to get a feel for what's happening in
our missing fourth dimension one
approach is to expand our visualization
to include another dimension of human
perception such as color will color each
point on our surface with a color that
corresponds to the value of the fourth
dimension of our function in this case
the imaginary part of Z Y to do this we
need to decide which colors to map to
which numbers this is called a color map
our colors now give us a nice idea of
what's happening in our missing fourth
dimension Y if we look at our line of
self-intersection now it's much more
clear that since our colors are
different our four dimensional function
actually doesn't intersect itself the
apparent self intersection is just an
artifact of our visualization technique
so as we follow paths on our riemann
surface the right thing to do with these
self intersection lines is to ignore
them if we now follow our path around
our surface we see that it's perfectly
continuous even at our weird self
intersection line excellent so that's
great we have continuity but there's
still one missing piece of the puzzle
what about the weird behavior we saw
back in part 11 where some paths ended
up in new locations and others didn't
let's recreate these suspicious paths
first just on our W and Z planes now
let's do the same exact thing but this
time using our riemann surface to help
us see what's going on to keep our
surface from getting too crowded we'll
choose one of our two paths to visualize
first all right and ready we'll draw the
same exact paths on W and see how they
show up on our Riemann surface as they
are mapped to our z plane
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so why does our green path start out in
one location on our Z plane only to end
up in another simply because our green
patch leads us to the other layer of our
service from the perspective of our W
plane it appears that we've returned
exactly to our starting point but
actually we have it the W plane is just
a projection a shadow in reality our
path has led us to a completely
different branch of our function with
different Z values this of course is
only half the picture each point on our
W plane has two solutions on Z this is
why our single path became two paths
back in part 11 on our riemann surface
our second solution is a flipped version
of our first the important thing here of
course is that for either set of paths
our riemann surface allows us to clearly
see that some paths on w lead to the
other branch and some don't more
specifically paths that go around the
central point on our riemann surface end
up on new branches this point is called
a branch point branch points occur
wherever the two branches of our
function have the same exact value and
can tell us a great deal about how our
complex function behaves for us this is
the point W equals zero
so our riemann surface not only fixes
the trebles are ran into earlier but
beautifully explains the strange path
behavior we saw and this is just the
beginning
Riemann surfaces are a huge part of
modern mathematics and there is way more
to say than we have time for here all
right
or finally ready to answer the question
what is this
our our entire discussion has centered
around a single function f of Z equals Z
squared plus 1 and so far we've looked
at one way to visualize the riemann
surface for our function by plotting
three of our four variables in 3d space
the two-dimensional plot we started our
discussion with the one most of a.cian
math class only shows two of our four
variables the real parts of Z and W the
surface we created back in part one is
the result of including one more
variable the imaginary part of Z as the
vertical dimension of our visualization
when we first saw this surface in part
one many of you asked a very good
question if according to the fundamental
theorem of algebra our function is
supposed to have exactly two roots why
does our surface appear to equal zero at
way more than two locations this
apparent contradiction has everything to
do with the shortcomings of living in
three-dimensional space that we've been
discussing when we visualize
four-dimensional functions in
three-dimensional space we must remember
that what we're really seeing is a
projection a shadow of the functions
full four-dimensional form
let's have a closer look at our surface
in our opening shot half of our surface
was hidden behind our paper and the
colors we used were chosen somewhat
arbitrarily to roughly correspond to the
surface height now that we know a bit
more about functions of complex
variables let's change the color of our
surface to correspond to the fourth
variable we were forced to leave out of
our three-dimensional visualization V
now that we have some idea of what all
four variables are doing let's look for
the two routes predicted by Gauss
remember routes are where the output of
our function equals zero for this to be
the case both the real and lateral parts
of our output variable W U and V must be
0
seeing where u equals 0 isn't too bad
this is where our surface intersects the
Z plane now where does V the imaginary
part of our output variable W equals 0
if we look at our color map this should
be where our surface is green it's
difficult to see exactly which shade of
green corresponds to 0 so let's add a
blue line to our surface where V equals
exactly zero now if we look closely we
see that both the real and imaginary
parts of W equals zero at exactly two
points
positive and negative ion our complex Z
plane exactly as our algebra predicted
we have finally found our missing roots
so our friend Gauss was right all along
our function does have exactly two roots
of course finding these roots took some
effort we had to journey deep into
mathematics and ask ourselves what a
number really is this led us to the
strange but necessary conclusion that
the numbers we should really be using in
algebra are the two dimensional complex
numbers this result dragged us down
deeper into the four-dimensional world
of complex functions and Riemann
surfaces when we finally emerged we saw
that the algebra many of us learn in
school is only a shadow of an elegant
powerful and higher dimensional
mathematics that has everything to do
with the numbers that have been given
the terrible name imaginary thanks for
watching
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you