[Music]
welcome to another mathologer video
have you heard of vortex maths or the
369 tesla code
i have to admit that in half a century
of obsessing about mathematics
i'd never come across these terms until
quite recently
there that's what comes up when you
search for the combination 369 tesla on
youtube a pile of seriously viral videos
7 million views or better
a lot of these videos feature a curious
diagram in their thumbnails
this diagram is usually referred to as
the vortex and is one of the main topics
covered in these videos
you've all heard of nikola tesla the
genius inventor of the tesla coil and
other early electrical devices right the
tesla car that tesla
but did you know that tesla also had a
host of idiosyncrasies centered around
the number three for example tesla would
walk three times around a block before
entering a building he would only stay
in hotel rooms with a room number
divisible by three and so on
in general he was convinced that the
numbers three six and nine hold the key
to the universe
according to the champions of vortex
mats
the vortex is that key
that sounds a bit
nuts
but as i said i'd never heard of vortex
mathematics and so i was curious to find
out more and seven million people can't
be wrong right
also although vortex math was new to me
i'd already stumbled across the vortex
diagram before in a different context
anyway our mission today is to have a
closer look at the vortex
there nine points on the circle labeled
one to nine innocent enough
then there is this infinity shaped loop
connecting six of the points
the remaining three points three six and
nine form an equilateral triangle and
depending on which video you watch some
extra lines get added like this
or like this
or like this
the last version of the diagram is the
one that i have been familiar with for
many years it's part of a famous
sequence of diagrams one diagram for
each positive integer this sequence of
diagrams starts out like this
there's a diagram for one for two for
three for four
for five and so on
nine that's the vortex and then things
continue like this
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okay
does this all look a little familiar
no well and let's keep on going
okay
what's going on here well i'm sure a lot
of mythological regulators will
recognize these diagrams and the pretty
curve that is starting to materialize
we already encountered these diagrams in
the methodology video on the times table
the mandelbrot set and the heart of
mathematics the curve is called the
cardioid the mathematical heart curve it
pops up in mathematics and nature all
over the place for example it's the
curve you get when you roll a circle
around another circle of the same
diameter like this
it's also the curve that you often see
in cups on sunny days
okay
and it's the curve that takes center
stage in the mandelbrot set the
mandelbrot set and there it is
actually there's more much more every
positive integer does not only give rise
to one of these cardiod infused diagrams
but to a whole family of line diagrams
and a lot of these extra diagrams are
also incredibly complex and beautiful
here are just a few examples
pretty spectacular isn't it it gets even
more impressive when you color the
segments and the diagrams according to
their length
whoa not better
but that's enough pretty pictures we're
on a mission remember our goal is to
make sense mathematical sense of the
vortex which includes making sense of
its mysterious relatives interested
well why wouldn't it be right so let's
get going
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okay
all the tesla code videos i mentioned
earlier introduced the vortex in
essentially the same way here we go
start with the powers of two so begin
with the number one and then every
number is double the number before it
next for each number in this table we
calculate its so-called digital root
for this we keep adding the digits of a
number until we end up with a single
digit for example in the case of 128 the
sum of the digits is 1 plus 2 plus 8 is
11. now 11 is not a one digit number yet
and so we keep on going 1 plus 1 is 2.
and so the digital root of 128 is 2.
easy right
now let's do this for all the numbers in
our sequence there
let's string up those digital roots on
the circle first 1 2
4
8
seven
five one two
four eight seven in fact we'll keep on
going like this forever one two four
eight seven five one two four eight
seven five and so on over and over
pretty cute and also pretty surprising
right
now instead of doubling let's look at
the sequence we get by halving again
starting with one
one one-half is equal to 0.5 when force
equal to 0.25 and so on let's calculate
the digital roots of these numbers
looks familiar right let's check what
happens on the circle 1
5 7 8 4
2
the same digital roots as for doubling
just in reverse order
again repeating forever and ever after
also pretty cute right but notice that
there are three numbers that never get
visited
3 6 and 9 tesla's 3 6 and 9.
whoa
at this point of the discussion the
absence of tesla's 3609 from the cycle
is usually interpreted as a tell-tale
sign that we're dealing with some sort
of divine message some secret code the
powerful key to the understanding of the
universe that tesla was raving about
well maybe i have to admit that i'm not
quite able to follow the line of
reasoning here however judging by the
zillions of likes these videos attract
and the enthusiastic comments pretty
much everybody else watching appears to
be in awe and fully on board with what's
going on here
must be just me i guess
of course there is more evidence that
the vortex is the key to the universe
much more now we're told to look at what
happens when we keep doubling and
halving starting with three and
evaluating the digital roots of the
resulting numbers there
whoa the plot thickens both for doubling
and halving the digital roots alternate
between 3 and 6 there
3 6 3 6 3 6 and so on and there 3 6 and
3 6 and 3 6 again the alternating 3 6
then corresponds to a line between those
numbers in the diagram
definitely key to universe right and
what about the nine
at this point in time the makers of all
those videos get really excited
we get all nines
amazing
to round things off it's then also
pointed out that a lot of the other
famous numbers in maths have digital
root nine for example 360 180 90 45
degrees the number 666 this guy here
all those numbers have digital root nine
g to the universe how can it be
otherwise
okay okay i can see plenty of you
screaming at your computers
trying to get a button and yes of course
you're correct everything we've seen so
far has a fairly simple mathematical
explanation
part of which just about everybody has
been exposed to in school an explanation
that by the way never gets mentioned in
vortex matt's videos funny that
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right where in school mathematics do you
add digits of a number remember
divisibility tests if you want to find
out whether or not a number is divisible
by 9 you just add its digits and check
whether this digit sum is divisible by
9. that's how it's usually taught in
school as a refresher let's do an
example 527 okay 5 plus 2 plus 7 is 14
and 14 is not divisible by 9 and so 527
is also not divisible by 9. well of
course if that works then you can repeat
adding the digits until only the digital
root remains and then very simply
a positive integer is divisible by 9
exactly if its digital root is 9.
in our example the digital root is 1
plus 4 is 5.
so the digital root is not 9 and we come
to the same conclusion as before 527 is
not divisible by 9. in school they
usually don't teach this simple digital
root extension of the standard
divisibility test for 9. of course they
should teach this in school but they
don't
in fact what they should also teach is
that if the digital root of a number is
not equal to 9 then this digital root is
simply the remainder of that number on
division by 9. nicer so the remainder of
527 on division by nine is our five up
there
great
now just out of interest were any of you
out in viewer land taught this remainder
extension in school
anyway very much worth knowing and
teaching don't you think
i'll show you the simple proof for all
this in a little while but for the
moment let's keep on going
to explain what's really going on with
all that vortex math i need to remind
you of two more super important
properties of remainders on division by
any number
as well as a digital roots counterpart
of those properties in the special case
of nine
nothing scary really just primary school
level maths promise
what are those properties well the first
property is that the remainder of the
sum of two numbers
equals the remainder of the sum of the
remainders of the two numbers and the
second property is the exact
corresponding statement for products of
numbers
that's all a bit of a mouthful but a
quick example will make it clear what i
mean and at the same time show you why
it works okay let's just stick with
division by nine and let's pick two
random integers 527 and 38. now
527 is 9 times 58 plus 5. you can check
this and therefore when you divide 527
by 9 you get a remainder of
5.
similarly 38 is 9 times 4 plus 2 and so
the remainder is 2.
okay
what if you are also interested in the
remainder of the sum of 527 and 38
well of course you can just add the two
numbers divide by 9 and this way find
the remainder yes you can do that
but there is a much much quicker way
have a look again
527 is 9 times 58 plus 5
and 38 is equal to this okay adding the
stuff on the right we get this
so what this shows is that the remainder
of the sum on the left is simply the sum
of the original remainders five and two
five plus two is seven and so the
remainder 7.
super simple right well mostly but
sometimes we have to deal with a little
hiccup for example if you replace 38 by
44 going up by 6 the 2 on the right
becomes an 8
and 5 plus 8 is 13 which is greater than
9 and so not one of the possible
remainders on division by 9 but that's
easily fixed what's the remainder of 13
when you divide by 9 well 4 of course
this means that the remainder of 527
plus 44 on division by 9 is
4. clear
again this is the sum shortcut if you
know the remainders of two numbers
the remainder of their sum is simply the
sum of the remainders or the remainder
of that remainder sum
that's also good to know right what's
even more important for us is that the
same also works for products so 527
times 44 has the same remainder as 5
times 8.
5 times 8 is 40 and when we divide 40 by
9 we are left with a remainder of 4. so
the remainder of 527 times 44 and
division by 9 is 4.
okay what about digital roots same thing
right the digital root of the sum or
product of two numbers
is equal to the digital root of the sum
or product of their digital roots
very nifty again to summarize at the
level of remainders the sum and product
property holds for division by any
number division by 2 3 4 5 666 division
by any number whatsoever however in the
special case of 9 and 9 only we have
this extra niceness that remainders
essentially correspond to the digital
rules all clear
ok now let's use these properties to
explain what's going on inside the
vortex
remember
we started by looking at the sequence of
powers of two
in other words we kept multiplying by
two starting with one
but
now it's clear from our discussion just
now that to generate that sequence of
digital roots highlighted in green we
can also just keep multiplying by 2 and
digital routing on the right right
let's double check this okay starting
with 1 on the right
1 times 2 is 2 2 times 2 is 4 2 times 4
is 8. 2 times 8 is 16 and the digital
root of 16 is 1 plus 6 is 7. 2 times 7
is 14 and 1 plus 4 is 5. 2 times 5 is 10
and 1 plus 0 is 1 and so on works and so
looking at it this way
it's actually not such a big surprise
that the numbers on the right will
eventually repeat why well we are always
doing the same thing over and over right
multiplied by 2 followed by finding the
root multiplied by 2 followed by finding
the digital root and since there are
only 9 different possible outcomes of
this operation things are bound to
repeat and then loop one forever
and of course the same is true if you
start with any number and keep doubling
in digital routing
eventually things are bound to repeat
and from there on will loop forever
starting with three we get a very small
loop three six three six three six and
so on and starting with nine well
doubling keeps producing numbers
divisible by nine which all have digital
root nine so just the meaniest of mini
loops in the case of nine
okay that's great
now what about those halving sequences
they're a bit unusual and in fact i'd
never seen anybody calculate the digital
roots of decimal fractions before
watching these tesla videos having said
that with what we know it's also not
hard to explain why we end up with the
same sequence of digital roots as before
running in reverse
right
to get those decimal fractions we keep
dividing by two to get one-half is equal
to 0.5 1 4 is equal to 0.25 1 8 is equal
to 0.125 and so on
now i'm sure that you've all seen these
numbers a million times yes
but
did you ever notice the powers of five
in these numbers wait what yes powers of
five just get rid of the decimal point
and all the zeros and you get 5 25 125
and so on the powers of five where do
those powers of five come from well
actually that's also not hard to explain
you see dividing by a 2 is the same as
first multiplying by 5 and then dividing
by 10
right 5 divided by 10 that's one half
and of course dividing by 10 only moves
the decimal point this means digit-wise
we end up with the powers of 5 and a
couple of zeros neater and now the rest
i'll leave is a little challenge for you
why are the digital roots of the powers
of 5 looping the same way as the powers
of 2 just in reverse
leave your answers in the comments hint
again the key is that 2 times 5 is equal
to 10 and what's the digital root of 10.
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okay getting there
so as far as mathematics is concerned
the vortex is simply a visualization of
what happens when we multiply the
remainders and division by nine by the
number two
in technical lingo what we are doing
here is multiplication by 2 modulo 9.
actually i almost forgot if we want to
think of the vertex in terms of
remainders and not digital roots we
should replace the 9 at the top by 0.
remember that little difference right
and what about the other diagrams that i
showed you earlier well those are
visualizations of multiplication by two
modular other numbers here's a diagram
for five again right just a quick check
there four times two equals eight and
when you divide eight by five you get a
remainder of three
taking that three times it by two gives
six and when we divide six by five we
get a remainder of one and so on works
and then as i already showed you before
as we erase the modulus
the number by which we divide the first
real magic occurs with the card
materializing out of nowhere there
very nice
but all this is really just
multiplication by two the other diagrams
i showed you you get when you multiply
by other numbers so for example have a
look at this crazy diagram here this is
multiplication by 240 modulo 7417
who would have thought now
multiplication modular different numbers
is super important mathematics with
myriads of applications within and
outside of mathematics finite fields
cryptographic algorithms number theory
in general and so on
in an app for drawing these diagrams we
then have two basic controls one for
setting the number we multiply with and
one for setting the modulus in fact
let's have another coding competition
whoever among you submits an online app
that implements drawing these diagrams
enters into a draw for marty's and my
new book that one there
anyway for the vortex we have multiplier
2 and modulus 9 then changing the
modulus to 50 we get this and if we now
change the multiplier to 3 we get this
actually on closer inspection there's at
least one more aspect of these diagrams
which we could also change
can you guess what i've got in mind here
it's a tricky one and easily overlooked
let me give you a hint it's got to do
with the nine occupying a special role
in our discussion so far right
only for modulus of nine can we use the
digital root algorithm to construct
these diagrams
this does not work for any other diagram
so what exactly is it that makes nine
special in this respect
well vortex mathematician will probably
tell you that nine is special it's just
part of tesla's 369 being the key to the
universe
of course
nine has to be special right
actually and closer inspection it turns
out that in the first instance 9 is
special because
we're writing numbers in base 10.
wait what yes divisibility test for 9
and all the other digital root magic is
a direct consequence of us writing
numbers in base 10. interesting huh you
want proof no problem
let me show you why the remainder of
this number on division by 9 is the same
as the remainder of the sum of its
digits
that's what makes the digital root work
right
well
2567 is just two times one thousand plus
five times one hundred plus six times
ten plus seven
and 1000 is 999 plus 1 100 is 99 plus 1
and 10 is 9 plus 1.
okay now expand and collect all the
repeated nine numbers together
there
now 9 99 999 are all divisible by 9 and
so the whole yellow bit is divisible by
9.
and the green bit is just the sum of the
digits
and obviously the same is true for any
integer any integer is equal to nine
times something
plus its digit sum but then when you're
interested in the remainder of the
number on division by nine we can just
forget about the whole yellow bit since
it's divisible by 9.
and so the remainder of our number
on division by 9 is equal to the
remainder of the sum of the digits ta-da
proof complete and that's where the
digital sum does the trick for 9 and y-9
is special okay but now what happens if
you're an alien with b fingers and you
write numbers in base b and not base 10
like with 10 fingered earthlings
well then everything i said in my little
proof remains true except that 9 changes
to b minus 1 and b minus 1 becomes the
special number
in turn the times 2 diagram for b minus
1 becomes the special vortex diagram for
an alien tesla
it's now this new diagram that can be
constructed using digital roots for
example for the eight finger tesla
we have this vortex
there
and you can check that the digital root
in base eight gives exactly these
connections for example starting with
the 4 we calculate 2 times 4 is 8 8 in
base 8 is 1 0 and 1 plus 0 is 1.
another example starting with 5 2 times
5 is 10 in base 8 10 is 1 2 and 1 plus 2
is 3 and so on okay so at least from a
mathematical point of view the vortex is
really not that special it's really just
one of infinitely many diagrams that
pretty much do the same thing
and definitely as we've already seen
many of the diagrams with large modulus
are a lot more spectacular from a purely
aesthetic point of view also even
mathematically there are lots of
diagrams that are superior to the vortex
in many ways for example have a look at
the diagram for 11. in this diagrams the
powers of two create a loop that is as
large as possible containing all the
numbers except for 11 yes
that is one continuous loop unlike in
the vortex which consists of two loops
that such a maximal loop exists has to
do with the fact that 11 is a prime
number and that two the so-called
primitive element modulo is prime if
you're familiar with these terms you'll
also recognize that these loops
illustrate affected for a prime number p
the finite field zp has a cyclic
multiplicative group effect which is of
huge importance in mathematics
okay so what about the claim that the
vortex is the key to the understanding
of the universe well today's discussion
was really about presenting a sound
explanation of the mathematics that
comes with the vortex an explanation
that demystifies its supposedly super
special properties
i hope that by now it's clear that the
vortex is really not as special and
amazing as it is made out to be by all
those tesla videos and they're
proclaiming it to be the key to the
universe mainly based on these
properties is simply ridiculous
but of course you knew that already
didn't you
in fact i wonder what you now think of
all those tesla videos and their
creators please share your thoughts in
the comments
having said that i'm convinced that
mathematics as a whole
is the master key to understanding the
universe and of course the maths we
talked about today is a part a tiny tiny
part of that key and if you're
fascinated by that tiny tiny part and
are interested in a real understanding
of the universe well then simply
familiarize yourself with more and
deeper maths
okay here's a nice challenge for you
suggested by tristan take one of these
diagrams let's just stick with the
vertex
multiply the modulus by some integers
say let's multiply the vertex modulus
9 by 3 that gives 27
draw the new diagram then the loops of
the first diagram are contained in the
new diagram let me show you
in this example there the infinity
shaped loop and the horizontal mini loop
of the vortex hiding inside this diagram
there it's an infinity shaped loop and
there's the other one
whoa super vortex super key to the
universe anyway
can you explain why our diagrams have
this mysterious modulus multiplication
property
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what about all these other spectacular
model times tables what is known about
the crazy structures inside them
actually i've not been able to find much
about these diagrams in the mathematical
literature
maybe some of the pros among you can do
something about the sorry state of
affairs and fill in the gaps in our
knowledge in this respect
i know of proves that the curve that
materializing the times two diagrams is
really the cardioid this appears to be
due to the famous 19th century italian
mathematician luigi cremona
also when you experiment a little with
small multipliers 2 3 4 5 6 and large
modulus another striking pattern jumps
out at us
there can you see the pattern i'm sure
you can
why do these petals appear and why is it
always one fewer petals than the
multiplier well the details are messy
but it's possible to gain some intuition
for the one fewer than the multiplayer
bit have a look at this animation
this is just the base case where we
multiply by two that produces the
cardioid what i'm doing here is raising
the modulus while at the same time
tracing the powers of two that fit in
the circle
notice that the cusp occurs where the
last connection goes straight across
makes sense right here's what happens
when we set the multiplier to three
okay
so the first cusp occurs at the point x
such that 3 times x is ideally on the
opposite side of the circle
for the next multiplier 4 this picture
would look like this
now
4x minus x that's the distance between
the two points is half the circle so
half the modulus
now we just follow our nose and solve
for x
there we go of course x is really just
the distance from the top
around the circle and so the width of a
flower petal is 2 times x
very nice
and that means that there will be a
total of 4 minus 1 equals 3 petals
around the circle
the same calculation shows that in
general we'll have multiplier minus one
petals
of course there are still quite a few
details missing from this argument to
make it into a complete proof anyway
good enough for this video what do you
think
now even in this monster diagram with
multiply 240 there are 240-1 that's 239
tiny little petals around the outer
circle let's zoom in on part of the
circle there there are lots and lots of
little petals
but can you see even at the border
there's a lot more stuff going on for
example how about this ring of smaller
petals
challenge for the keen among you how
many of those little petals are there
and how many loops does this monster
diagram have
who can find the answers to these
questions
but of course zooming out that's where
the real spectacular stuff is happening
how exactly is all this complicated and
beautiful structure linked to the
multiplier and the modulus the only
place i know that makes some progress
towards answering this question is an
unpublished writer by simone plouffe
that i've linked to in the comments you
may know simone proof for his
involvement in the establishment of the
encyclopedia of integer sequences as the
creator of the inverse symbolic
calculator and the discovery of the
spectacular bailly bowen proof formula
for calculating individual digits of pi
anyway
challenge for the super keen and capable
mathematicians among you check out simon
proves write up and then go where no one
has gone before and explore the secrets
of these diagrams
and that's all for today i hope you
enjoyed our vortex adventure until next
time
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[Applause]
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so
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you