Transcription
[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 math, 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 are 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.
[Music] 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 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 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.
[Music] 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, 7, 5, 1, 2, 4, 8, 7. in fact, we'll keep on going like this forever. 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 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 half is equal to 0.5. one quarter is 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 3, 6, 9 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 the 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 math 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 math videos. funny, that.
[Music] [Applause] [Music] 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 nine, you just add its digits and check whether this digit sum is divisible by nine. 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 nine, and so 527 is also not divisible by nine. 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 nine exactly if its digital root is nine. in our example, the digital root is 1 plus 4 is 5. so the digital root is not nine, and we come to the same conclusion as before. 527 is not divisible by nine.
in school, they usually don't teach this simple digital root extension of the standard divisibility test for nine. 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 nine, then this digital root is simply the remainder of that number on division by nine. 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 math, 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 is seven. 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 six, the two on the right becomes an eight, and five plus eight is 13, which is greater than nine, and so not one of the possible remainders on division by nine. but that's easily fixed. what's the remainder of 13 when you divide by nine? well, four, of course. this means that the remainder of 527 plus 44 on division by nine is four. 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 nine, and nine only, we have this extra niceness that remainders essentially correspond to the digital rules. 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 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 two and digital routing on the right, right? let's double-check this. okay, starting with one 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 nine 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 meanest 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. one quarter is equal to 0.25. one eighth 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 two is the same as first multiplying by five and then dividing by ten, right? five divided by ten, that's one half. and of course, dividing by ten only moves the decimal point. this means, digit-wise, we end up with the powers of five and a couple of zeros. neater. and now the rest, i'll leave as a little challenge for you. why are the digital roots of the powers of five looping the same way as the powers of two, 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?
[Music] 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 nine at the top by zero. remember that little difference, right? and what about the other diagrams that i showed you earlier? well, those are visualizations of multiplication by two, modulo 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 cardioid 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 in 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 mathematicians 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, nine is special because we're writing numbers in base 10. wait, what? yes, divisibility test for nine 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 nine 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 2 times 1000 plus 5 times 100 plus 6 times 10 plus 7. 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 nine. and so the remainder of our number on division by nine 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 nine, and why nine 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 nine 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-fingered 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 four, 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 diagram, 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 the fact that 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 their 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 math 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 math.
okay, here's 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 integers. 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. 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?
[Music] 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 this sorry state of affairs and fill in the gaps in our knowledge in this respect. i know of proofs that the curve that materializes in 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 multiplier 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, four, 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 multiplier 240, there are 240 minus 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 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 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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