Welcome to another Mathologer video. Have you heard of vortex maths or the 3-6-9 Tesla code?
I have to admit that in half a century of obsessing about mathematics I had never come
across these terms, until quite recently. There, that’s what comes up when you search
for the combination “3-6-9 Tesla” on YouTube. A pile of seriously viral videos. 7 million views!
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 Nicola 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 centred around the number 3? For example, Tesla would walk
three times around the 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 3,
6, and 9 hold the key to the universe. According to the champions of vortex math,
the vortex is that key. That sounds a bit, hmm, nuts :) But, as I said, I’d never heard of vortex
mathematics and so I was curious to find out more. And 7 million people can’t be wrong, right?
Also, although vortex math was new to me, I had 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 9 points on a circle labelled 1-9. Innocent enough :) Then there is this infinity shaped loop
connecting 6 of the points. The remaining three points... 3, 6 and 9 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 is part of a famous sequence of diagrams, one diagram for each positive integer.
This sequence of diagrams starts out like this. That’s the diagram for 1. For 2, 3, 4 and so on.
9, that’s the vortex and then things continue like this Does this all look a little familiar?
No? Well, then let’s keep on going. What is going on here? Well, I am sure a lot of Mathologer
regulars will recognise these diagrams and the pretty curve that is starting to materialise.
We already encountered these diagrams in the Mathologer video on the times tables,
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.
And it’s the curve that takes centre stage in the Mandelbrot set.
Actually, there is more, much more. Every positive integer does not only give rise
one of these cardioid 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
colour the segments in the diagrams according to their length. 8293
But that’s enough pretty pictures. We are on a mission, remember? Our goal is
to make mathematical sense of the vortex, which includes making sense of its mysterious relatives.
Interested? Well, why wouldn’t you be :) So let’s get going!
Okay, all the Tesla code videos I mentioned earlier introduce the vortex in essentially
the same way. Here we go. Start with the powers of 2. So, begin with the number 1,
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+2+8=11.
11 is not a one-digit number yet and so we keep on going. 1+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 7 5 1 2 4 8 7 In fact, we will keep on going like
this forever 124875 124875 and so on, over and over. Pretty cute and pretty surprising, right?
Now instead of doubling let’s look at the sequence we get by halving, again starting with 1.
1, 1/2=0.5, 1/4=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: Tesla’s 3, 6 and 9.
Whoa. At this point of the discussion, the absence of Tesla’s 3, 6 and 9 from the cycle
is usually interpreted as a tell-tale sign that we are 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 am 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 is 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 are told to look at what happens when we keep doubling and halving
starting with 3 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, ... and there 3, 6, 3, 6, ... The alternating 3 6 then corresponds to a line between
those numbers in the diagram. Definitely the key to the universe, right? And what about the 9?
At this point in time the makers of all those videos get really excited.
We get all 9s! Amazing :) To round things off, it’s then also pointed out that a lot of other
famous numbers in maths have digital root 9. For example, 360, 180, 90, 45 degrees, the number 666,
etc. all those numbers have digital root 9. Key to the universe. How can it be otherwise? :) Okay,
okay, I can see plenty of you screaming at your computers, trying to get a ‘but’ in. And yes, of
course you are 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 math videos. Funny that.
Right, where in school mathematics do you add the 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+2+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+4 = 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. Nice, huh? So,
the remainder of 527 on division by 9 is our 5 up there. Great! 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 the digital roots counterparts of those properties
in the special case of 9. Nothing scary, really just primary school level math.
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 this works. Okay? Let’s just stick with division by 9 and let’s pick two
random integers... 527 and 38 Now 527 is 9 times 58 + 5 And therefore when you divide by 527 by 9
you get a remainder of 5 Similarly, 38 = 9*4+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 quicker way. Have a look. Again, 527=9*58 + 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 5 and 2. So what
this shows is that the remainder of the sum on the left is simply the sum of the
original remainders 5 and 2. 5+2 is seven and so the remainder is seven. Super simple, right?
Well, mostly, but sometimes we have to deal with a little hiccup. For example, if we replace 38 by
44 going up by 6 the 2 on the right becomes an 8. And 5+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 + 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 on division by 9 is 4.
Finally, what about digital roots? Same thing :) 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 summarise, at the level of remainders this sum and product property holds
for division by any number, division by 2, 3 4, 5, 666, division by any number. However, in the
special case of 9 and 9 only, we have this extra niceness that remainders essentially correspond
to the digital roots. All clear? Okay, 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 2 starting with 1. But now it’s clear from our discussion that to generate
that sequence of digital roots highlighted in green we can also just keep multiplying by 2 and
digital rooting on the right. Let’s double-check this. Okay, starting with the 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+6
that’s 7. 2 times 7 is 14 and 1+4 is 5, 2 times 5 is 10 and 1+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: multiply
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 forever. And, of course,
the same is true if we start with any number and keep doubling and digital rooting. Eventually,
things are bound to repeat and from there on will loop forever. Starting with 3 we get a very
small loop 363636. And starting with 9? Well, doubling keeps producing numbers divisible by
9 which all have digital root 9. So just the minniest of mini loops in the case of 9.
Okay that’s great. Now what about those halving sequences? They are 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 sequences of digital roots as before, running in reverse.
Right, to get those decimal fractions we keep dividing by 2 to get 1/2 = 0.5, 1/4= 0.25,
1/8= 0.125, and so on. Now I am sure you’ve all seen these numbers a million times.
Yes, BUT did you ever notice the powers of 5 in these numbers? Wait what? Yes, powers of 5.
Just get rid of the decimal points and the 0s and you get 5, 25, 125, and so on, the powers of 5.
Where do those powers of 5 come from? Well, actually, that’s also not hard to explain.
You see dividing by 2, is the same as first multiplying by 5 and then dividing by 10. Right,
5 divided by 10 is 1/2? 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 0s. Neat, hmm. A Now the rest I leave
as 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 equals 10. And what’s the digital root of 10?
Okay, getting there. So, as far as mathematics is concerned the vortex is simply a visualisation
of what happens when we multiply the remainders on division by 9 by the number 2.
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 vortex in terms of remainders and not digital
roots, we should replace the 9 at the top by a 0. You remember that little difference, right?
And what about the other diagrams that I showed you earlier? Well, those are visualisations of
multiplication by 2 modulo other numbers. Here is the diagram for 5 again. Right,
just a quick check. There 4 times 2 equals 8 and when you divide 8 by 5 you get a remainder of 3.
Taking that 3, timesing it by 2 gives 6. And when we divide 6 by 5 we get a remainder of 1.
Works :) And then, as I already showed you before, as we raise the modulus,
the number by which we divide, the first real magic occurs,
with the cardioid materialising out of nowhere. But all this is really just multiplication by 2.
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 modulo 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’d
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.
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 close inspection,
there is 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 9 occupying a special role in our discussion so far. Right,
only for a modulus of 9 can we use the digital root algorithm to construct these diagrams. This
does not work for any other diagram. What exactly is it that makes 9 special in this respect?
Well, vortex mathematicians will probably tell you that 9 being special is just part of Tesla’s
3-6-9 being the key to the universe. Of course, 9 has to be special, right? Actually, on closer
inspection it turns out that in the first instance 9 is special because ... we are writing numbers in
base 10. Wait what? Yes, the 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 make the digital root work, right? Well,
2567 is just 2 x 1000 + 5 x 100 + 6 x 10 +7 And 1000 is 999+1, 100 is 99+1 and 10 is 9+1 Okay,
now expand and collect all the repeated 9 numbers together. 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 9 times something
plus its digit sum. But then, when you are interested in the remainder of the number on
division by 9, we can just forget about the whole yellow bit since it is 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.
Tada, proof complete. And that’s why the digital sum does the trick for 9 and why 9 is special.
Okay, but now what happens if you are an alien with b fingers and you write numbers in base b
and not in base 10 like we 10 fingered Earthlings? Well, then everything I said in my little proof
remains true except that 9 changes to b-1 and b-1 becomes the special number. In turn,
the times 2 diagram for b-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
8-fingered Tesla we have this vortex. For example, for the 8-fingered Tesla we have this vortex.
And you can check that the digital root in base 8 gives exactly these connections. For example,
starting with the 4. we calculate. 2 times 4 is 8. 8 in base 8 is one zero. And 1+0 is 1.
Another example, starting with 5 2 times 5 is ten. in base 8 ten is one two and 1+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 diagram the powers of 2 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 2
is a so-called primitive element modulo this prime. If you are familiar with these terms,
you’ll also recognise that these loops illustrate the fact that for a prime numbers, the finite
field Zp has a cyclic multiplicative group, a fact 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 is clear that the vortex is really not as special and amazing as it is made out to be
in all those Tesla videos and that 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 am 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 are fascinated by that tiny, tiny part and
are interested in a real understanding of the universe well then simply familiarise yourself
with more and deeper maths :) Here is a nice challenge for you suggested by Tristan. Take
one of these diagrams. Let’s just stick with the vortex. Multiply the modulus by some integer. Say,
let’s multiply the vortex 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. Here is the infinity shaped loop and the horizontal mini loop of the vortex
hiding inside this diagram. Whoa a super vortex :) Super key to the universe. Anyway,
can you explain why our diagrams have this mysterious modulus multiplication property?
What about all these other spectacular modular times tables? What is knows about the crazy
structures inside them? Actually, I have not been able to find much about these diagrams in the
mathematical literature. Maybe some of the pros among you can do something about this sorry state
of affairs and fill in the gaps in our knowledge in this respect. I know of proofs that the curve
that materialises in the times 2 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.
Why do these petals appear and why is it always 1 fewer petals than the multiplier? Well, the
details are messy but it’s possible to gain some intuition for the one fewer than the multiplier
bit. Have a look at this animation. This is just the base case where we multiply by 2 that produces
the cardioid. What I am doing here is raising the modulus while at the same time tracing the
powers of 2 that fit in the circle. Notice that the cusp occurs where the last connection goes
straight across. Makes sense, right? Here is what happens when we set the multiplier to 3. x Okay,
so the first cusp occurs at a 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 - x the distance between the two points is half the circle, half the modulus. Now we just
follow our nose and solve for x. 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. And, so, the width of a
flower petal is 2 times x. And that means that there will be a total of 4-1 = 3 petals around
the circle. Tada. The same calculation shows that in general we’ll have multiplier - 1
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. Now, even in this monster diagram
with multiplier 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 lots and lots of little petals. But can you see? Even at the
border there is a lot more stuff going one. 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 really 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 of that makes some progress towards answering this question
is an unpublished writeup by Simon Plouffe that I’ve linked to in the comments. You may know Simon
Plouffe 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
Bailey-Borwein-Plouffe formula for calculating individual digits of pi. Anyway, challenge for
the super keen and capable mathematicians among you. Check out Simon Plouffe’s writeup
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 :)