what up ding dongers my name is michael
and today we will be listing off the
eight breeds of cats that might be able
to learn to play the banjo number five
might surprise you this is michael's
toys
thank you for tuning in to the dong
channel it has been a wild ride this
year
i've had a lot of fun
i think last year in total i made like
four videos but
this year because of dong i've made like
more than 20. we're talking relativity
morse code homonyms density balls
i built every strictly convex
deltahedron i also listed off prime
numbers for three hours that was a treat
for me and i really hope it was for you
now should those videos all have gone on
vsauce one maybe i don't feel like they
were big enough but they were certainly
fun and without the dog channel i
wouldn't have made them so thanks for
tuning in here exciting news is that
minefield is out on vsauce1 right now
and for this season i was able to
convince youtube to make two
of this season's episodes free for
everyone to view the first one about the
cognitive trade-off hypothesis which i
hope you've already seen and the fourth
episode which is about the stanford
prison experiment
pretty exciting stuff but we're here
today to talk about the mandelbrot set
it is a famous set of complex numbers
named after mathematician benoit
mandelbrot i got together with our
designer john laser and we designed this
poster it is a beautiful thing it
imagines the mandelbrot set as an island
of finite land and infinite coast it has
all kinds of information about the set
and it imagines all the different
features
of the set as actual geographic features
point is what is the mandelbrot set well
today we're going to talk about that
first of all you may have noticed that i
said complex numbers what is a complex
number
well
it's
a little bit complex just kidding it's
not really uh we all know and love the
number line it's a way to visualize all
of the real numbers on a line a
one-dimensional line like this where
zero can be placed somewhere like that
and then numbers like uh you know one
might be here two might be here over
here we might have negative one oh i can
imagine three being there and four there
this contains all real numbers not just
the integers but all rational numbers
for instance right here well that's
going to be a half well it's actually
going to be one and a half 1.5 the ratio
of every circle circumference to its
diameter in euclidean space is about
right i don't know there that's pi
3.1415 approximately
uh the square root of one it's a real
number it's on here it's just one one
times one is one but what isn't here
well is a lot of stuff for example where
is the square root of negative 1 where
on this real number line is that number
well here's the problem any real number
times itself is positive negative 1
times negative 1 is just 1.
negative 2 times negative 2 is 4. so
the square root of negative 1
may not exist but mathematicians have
found it very useful very powerful to
say that the square root of negative one
is a number it's a number called i
i for
imaginary and probably someone said yeah
but i is a letter not a number and they
were like
now you can look as hard as you want at
the real number line you will never find
i
on that line but we can put i on the
line by drawing an axis that is
perpendicular to the real number line
this way
instead of saying well is i to the right
or left of zero we can just say how
about i is above zero there's i which
means that the same distance in the
other direction from zero we have
negative i
and then right here we've got 2i
3i and so on what we now have is not
just a real number line but a complex
number plane a complex number is a
combination of a real number and a
complex component that involves i
actually every real number is a complex
number here's an example of a complex
number just to get us started
if i pick any point on this plane it
will be a complex number this one right
here this is
two the real number
two
plus
i it's one up from zero this is two plus
i
a number like
this one right here that point that
point is
negative 1 that's the real component and
the imaginary component is 3i negative 1
plus 3i is this number right there every
number here is a complex number and it
is these numbers that either belong or
don't belong in the mandelbrot set by
picking all the numbers that belong and
plot plotting them on a complex plane
what we wind up with is a beautiful
shape known as
the mandelbrot set visualized like this
if you look on our poster this right
here is the imaginary axis and this is
the real axis how do you know
if a number a complex number belongs to
the mandelbrot set or not whether it is
a piece of land on this island or if
it's like out here in the ocean not part
of the set i could spend a bunch of time
talking about julius sets and we can
start more generally but i want to go
right to mandelbrot so let's talk about
functions a function is just a mapping
from numbers to some other numbers and
the function that we're going to be
using to determine whether a number
belongs to the mandelbrot set or not
looks like this it's a function of a
variable
z
such that
its value at z equals z itself squared
plus
c
where c is any complex number
c
is the number that we have located on
this plane and we want to know whether
it belongs in the set or not
what you have to do though is you have
to start this with z
equal to zero
and you need to iterate this function
and if the value that we keep getting
out just grows bigger and bigger without
bound the number is not in the set if
however the number sort of stays
somewhat small it doesn't grow to
infinity then the number belongs in the
set let's do some examples we begin by
applying this function to z where z
equals zero and let's say that our
complex number
is going to be uh the number one how
about that we'll just try this remember
one is a real component but there's this
kind of invisible imaginary component
because we can imagine that one is just
equal to one plus zero times i anyway uh
what is f of z
well we have to begin with z equal to
zero that's the rule for creating the
mandelbrot set
z squared zero squared all right and
then we add
1 which is c in this case we want to
know if 1 is in the set so
0 squared plus 1.
perfect what does that equal well 0
squared is 0 plus 1 is 1. wonderful now
we iterate this function meaning our
result becomes our new value of z so now
z is equal to one so what is this
function
at one well if uh if we need z squared
that means we're now using one squared
and we're adding c which is one and the
answer we get is one squared one plus
one which is two now this is our value
for z two all right so what is f of two
well that's two squared plus one which
is for five uh-oh all right let's plug
this one back in now z is equal to five
f of five equals five squared plus one
five squared is 25 plus 1 is 26. my gosh
you guys this is just going to keep
getting bigger and bigger and bigger it
is going to grow without bound and so
one is not part of the mandelbrot set
but now let's try this for the complex
number
negative one which can also be thought
of as negative one plus zero i
now we begin with f of z where z equals
zero
zero squared all right cool and then we
add c which is negative one so we're
adding negative one what does this give
us well zero squared is zero and then
we're adding a negative one which means
we're subtracting one so we get a
negative one okay so now our new value
of z is negative one it originally was
zero now it's negative one so what is f
of negative one well it is negative one
squared since we need to do z squared
and then we need to of course add c
which is in this case still negative one
negative one squared is one plus
negative one one minus one equals zero
uh oh
now i think you can already see what's
about to happen we now need to plug in
zero as our next value of z but we
already did that we already did that we
know that zero squared plus a negative
one equals negative one we're just going
to oscillate back and forth between
negative one and zero negative one and
zero forever this will not grow without
bound therefore how you like that
infinity symbol it probably annoyed you
that i drew it like this i just did that
for kicks because on michael's toys
you never know what's gonna happen
okay so we can see that because at
negative one this function does not
grow without bound as we iterate
negative one is part of the mandelbrot
set and sure enough if we look on our
little graph here
not graph our map
negative one which is actually the very
center of the main disk here that's
certainly part of the mandelbrot set now
let me point out a couple of the
features i love here
right here uh there's a point at uh the
real component is negative three
quarters and the imaginary component is
zero this is a single point it is the
only point that belongs to the set that
is along this line that intersects the
real axis right at negative three
quarters
that i call a point bridge it is an
infinitely thin bridge perfect for brief
romantic walks because it's just a
single point okay anyway we've got other
things like the seahorse coast beautiful
seahorse shapes adorn this side of the
main cardioid this is a cardioid this
wonderful heart shape you can find
cardioids all over
for instance on some of the baby brats
the baby brats are very similar looking
to the entire mandelbrot set
the main disc okay we talked about that
but on on the seahorse coast side we
also have this main disc
spiral coastline it's a beautiful place
i really wish it was real a really cool
thing that this poster points out is
that while the perimeter of mandelbrot
island may be infinite but its area is
not we're not exactly sure what its area
is we do know that a circle can
completely enclose the set that is
centered at the origin with a radius of
two
but
right now our best estimate for its area
is let me see here about
1.506484 square units
so there you go maybe someday we will
find a way to find the exact area it may
not be
rational uh in fact it might not be i
don't know boy there's so much more to
learn such an inspiring poster really
really love it thanks to john for
helping me design this i love this
poster if you want it well you got to
subscribe to the curiosity box this
poster comes in the latest box box 10.
we only have a few hundred left if
you're a subscriber you got it or it's
coming if you haven't subscribed we have
a few left we always make a few thousand
more than we have subscribers but the
only way to be guaranteed that you will
get our latest inventions and
discoveries and favorite math and
science objects you need to be
a subscriber i am
loving working on this box so far we
have raised nearly 150 000
for alzheimer's research it's awesome
lots more facts on this poster a lot of
cool things in this most recent box
including this shirt which uh i wanted
to wear in the episode but uh jake took
mine this is his which is a small
and i'm not gonna try to fit into this
just kidding this is me wearing the
small
how you like that
kraken kraken mare lunar park
there's a moon of saturn that has liquid
hydrocarbon lakes that literally have
waves they're only like a fraction of a
centimeter large but i mean you could
surf that right so we designed this
shirt that's like
a vintage national parks poster
imagining
that uh kraken mare the the one of the
big lakes on the um on titan saturn's
moon was a was a park which could
actually happen in the future sometime
i think this really tight shirt makes
gravity stronger or it just makes me
cooler it's like look at me like i don't
care
sit the way i want also a puzzle now the
inventor of this puzzle
called it uh i think foursquare i
believe uh i got a copy
i loved it and uh the point of course is
to get these uh pieces out but after i
sort of um you know solved it and
learned its wiles i said whoa
we need to print
ien
on the bottom
ien you see that
why is that printed on the bottom of
this puzzle it won't make any sense
until you solve the puzzle and even then
it might not
i'm excited to see what you guys think
about that puzzle there's obviously many
more things in the box including oh boy
i can't show you this actually this is a
prototype of a product coming out in the
next box box 11. can i
i don't want to give away what it is
actually i can do whatever i want
but i choose not to
that's nice
this is the only clue i'll give you
do you hear that
[Music]
all right hopefully that titillated you
i don't know
um
i don't want to show it off because i
think the final product is going to be a
little bit different
um but the point is that the mandelbrot
set is beautiful we actually had a giant
version of this poster made to put up in
our office and it's a beaut you can
celebrate mathematics and new
discoveries and put this up in your room
or dorm room or
um keep it in your car for reference
during when you're in traffic i don't
know your life
point is i do love you i'm glad you are
here watching today
and as always
thanks for watching
[Music]
you