Tesla’s 3-6-9 and Vortex Math: Is this really the key to the universe?
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- Опубліковано 17 кві 2024
- Today, a long overdue foray into the realm of VORTEX MATHEMATICS :)
00:00 Intro
04:16 The vortex
08:10 The maths of remainders and digital roots
13:25 Demystifying the vortex
16:30 A matter of base. The 8 fingered Tesla.
19:21 Explanation why the digital root is the remainder on division by 9
24:01 Tristan's challenge
24:44 The magic of modular multiplication maths
25:19 Intuition for multiplier - 1 petals
28:23 Thank You!
Coding competition:
My wish list for the modular times table diagram app:
-Being able to color line segments according to length.
-Indication of the "direction" of multiplication. 1x2 = 2 and so there should really be a little arrow from 1 to 2 not just a simple connection :)
-different loops in different colors.
...
Here is the prize, a copy of my and Marty's new book.
bookstore.ams.org/mbk-141/
That early Mathologer video featuring the modular times tables
Times Tables, Mandelbrot and the Heart of Mathematics
• Times Tables, Mandelbr...
A really nice article about various ways to generate the cardioid by Dave Richeson
divisbyzero.com/2018/04/02/i-...
Nice debunking/demystifying article about vortex math by "Professor Puzzler"
www.theproblemsite.com/vortex/
For a growing pile of implementation of modular times table diagrams see my comment pinned to the top of the comment section of this video.
Simon Plouffe's website
plouffe.fr/Simon%20Plouffe.htm
Articles by him relevant to this video can be found in this directory
plouffe.fr/Inverseofprimes/
See in particular the files
The shape of b^n mod p.pdf
La forme de bn mod p.pdf
What I am talking about in this video is really just the tip of a bizarre mathematical iceberg that most mathematically minded people are completely unaware of. Have a look at this presentation by Marko Rodin on vortex math (beware serious nutty and at the same time truely beautifully presented numerology ahead :) A LOT more than is usually reported on in popular UA-cam videos.
sciencetosagemagazine.com/vbm...
In turn this iceberg is just another tip of an even bigger iceberg of mainly wishful thinking. Have a look: sciencetosagemagazine.com/cat...
Today's music: Aftershocks by Ardie Son
Enjoy!
Burkard - Навчання та стиль
(updated 2 April 2022) Thank you to all of you who contributed a modular times table app. All the apps I am aware of are listed below. The winner of the draw is Mathis Aaserud. Congratulation!
Here are a few implementation contributed by viewers so far. Look at these first:
Adam Abrams: theadamabrams.com/modularmultiplication
Ed Collen: vortex-rho.vercel.app/
Andrew “Ash Mystic” Herman: codepen.io/hippiefuturist/full/NrvqgZ (check out the preset animations on this one. Also check out his fractal tree generator codepen.io/hippiefuturist/full/KRromj )
Man Hin Li: mandelbrot.vercel.app
Liam Applebe: tiusic.com/vortex.html
Owen Bechtel: owenbechtel.com/games/times-tables/
William Ward: scratch.mit.edu/projects/647469837/
Артём Маевский: tinyurl.com/yc8danxx
Baxi: baxi-codes.github.io/mathologer-vortex/
Marc Donis: madc0w.github.io/cardioid/
Rafael Castro Couto: codepen.io/rafaelcastrocouto/pen/KKyoKWm
Laurent Bucher: anceps.net/modularTimesTables.html
Hannes Wendt: htts://math.wendt.sbs/vertex
Hugo Cardoza: Code in p5js editor.p5js.org/hugomosh/sketches/1Sg1NxqI7
john Schoeman: www.doodles.camp/#/doodles/modular-times-table
Banjamin Elo: bnelo12.github.io/vortex-math/
Joe Lucette: jluqu.github.io/modmult.html
Federico Marotta: federico-marotta.shinyapps.io/tesla_vortex
T3CHN01200: victorsohier.github.io/
Tom DeRensis: github.com/tderensis/ModularTimesTableJavascript
Ehsan Kia: ehsankia.com/cjs/vortex
Jayson Vivet: www.geogebra.org/m/cufneprj
Tyler Wolfe-Adam: mathologer-vortex-app.herokuapp.com/
Andrea Coletta: mathologer-modular-time-table.lm.r.appspot.com/app
Mathis Aaserud: sirkular.ispaceyourtube.com/
Justin Kirk: intern-jck.github.io/vortex-math/
Jarred Branch: no online version
Álvaro Silva: mathlogervortexalvaro.web.app/
Rafael Castro Couto: codepen.io/rafaelcastrocouto/pen/KKyoKWm
planck_cst: www.jerpint.io/blog/mathologer-challenge/
Anton Shcherbinin: ch.ant-on.net/modulo/moire?p=1009&m=303
Cristian Merighi: js.pacem.it/2d/vortex
Krischna-Gabriel Schulz: no online version
András Kirisics: kiri-mathologer-vortex.web.app/
relikd: relikd.github.io/Vortex-Math/
Eclectic Gamer: ua-cam.com/video/n_YLB0ncbpI/v-deo.html (Video on using Blender and Geometry nodes to make these diagrams)
Some existing implementations of the modular times table diagrams:
Aymeric Ramiere: www.aymericramiere.com/others_modular.html
Steve Phelps: www.geogebra.org/m/z8wrdret#material/dqKkQEv7
I did this a while ago: www.qedcat.com/cardioid.cdf
Marcus Metzler: github.com/drmocm/Modulo-graphics
Start of a wish list for the modular times table diagram coding competition:
-Being able to color line segments according to length.
-Being able to highlight different loops in different colors.
-Indication of the "direction" of multiplication. 1x2 = 2 and so there should really be a little arrow from 1 to 2 not just a simple connection :)
...
@Mathologer I need your help, I've learnt all the basic integration techniques (By parts, Partial Fractions, Trig Sub, U-sub, Chain Rule) and I cannot find any more integration techniques to learn, can you please give me some more, all the websites say that you will learn more but I cannot find any more.
Zero doesn't exist in the natural world - it's a man-made construction. There is never 'nothing' in the universe - never. There is no 'zero' in reality. The digital root of 10 doesn't exist in the natural world, only in the world of man. That is to say that whatever mathematics governs the natural world it does so without the use of a 'zero'.
Can we devise a base9 system that does not use a zero?
I'm no expert in this - just a curious layman - but I did do some thinking on this. Could we not begin counting at 11? The first iteration and the first 'number'? The first "one" so to speak? One One or 11. Then we'd have the first "two" so 12 would be next - 13, 14, 15, 16, 17, 18 and then ,19 - the first "nine". Then we would go to 21 - the second "one" and so on totally eliminating the use of a zero. I understand that multiplication and division become very difficult _without_ the use of a zero so I assume that more complicated mathematical functions also become very difficult. I also assume that this is the reason we invented "zero" in the first place - to make is _easier_ to calculate - is this right? Is there any other purpose for the zero? I don't know. I am simply curious as to whether or not we _can_ do mathematics without the zero.
And I think that if we could do this that doing this vortex math in such a system would yield different results. As I said, I'm no expert and could be talking out of my ear but I do think that the mathematics we use and the mathematics of the natural world are not the same and that perhaps using the maths that governs the natural world might yield interesting results.
@@alflud what number of lions live on mars?
* Is that a question about the natural world?
* What is the answer to that question?
* Is this an example of 'zero'?
My answers:
* I would say so.
* Zero, given what i think i know about mars.
* I would say so.
What are yours?
@@alflud
So base 9 but formatted differently.
11->10 (base 10 “9”)
12->11
13->12
…
19->18
21->20
22->21
Pretty sure what you have invented is just base 9 but with all 0s replaced with 1s, 1s replaced with 2s, 2s replaced with 3s, all the way up to 8s which are replaced with 9s.
I could create a mathematical system where every digit is indicated with a color and that would be functionally the same as one which used digits.
@@alflud
Also, how would you express 1-1? That was most likely the reason 0 was invented.
0 means “nothing”, and that’s why it is something.
I once created a small computer program that used exponentiated polar conversions to achieve an infinite number of possible cardioid shapes (like the ones you found), and suddenly it started showing flower petals and other "real" things we see in day-to-day life, all from equations in my code. The universe is a wild place and exploring it with your own hands has to be one of the most satisfying experiences!
@RoddiyKnox
Yep
Our education system is more of an indoctrination system to train social obedience & compliance within the bounds of capitalism
Early in high school we had our essays sent to a university for evaluation. The commentary I received was that I was writing above my level & using too advanced language.
Sad.
:)
Would love to see a video if you could upload. That would be cool to see
Does it make you wonder that we are in a computer simulation ?
When I was a kid and learning polar coordinates, I did the same thing. I wrote a program to graph things like cartoids using polar coordinate formulae.
In school whenever I noticed patterns such as these divisibility tests, my teachers discouraged me from pursuing them because they themselves were not sure if they'd always hold and were concerned they'd lead me astray. Another example that I recall is my noticing that each power of two is equal to one more than the sum of the lesser powers of two. That's well-established and taken for granted in computer science, yet was unknown to my teachers and regarded with skepticism. I remember also my mom pleading with my teachers to stop counting my work wrong for my daring to use techniques I developed myself from having explored the mathematical foundations of the rote mechanisms they taught. I understand that the pressures on elementary school math teachers drive them to stick with safe techniques, but for them to feel threatened by a student privately moving beyond that is frankly an indictment of the whole system of education.
What a damn shame... Here in France, a lot of school teachers are just "failed researchers" - uni students who wanted to get into academia but just weren't good enough. So we end up with quite a few unmotivated teachers who just rotely follow the program, without much passion at all. I wish the schoolteacher career were seen with more prestige, so that more qualified people would sign up.
You are so right! I myself am a math teacher teaching in advanced (higher level) classes. I share the same experience like you.
I see so many students being discouraged because they had incompetent teachers in middle school killing all the fun one can have with math. I always try to encourage my students to "explore" a topic, not just feeding them subjects.
I myself had a bad experience when I was a high school student. My math teacher once kicked me out of the class, because I presented him an alternative solution to a problem. He simply couldnt stand this, didnt let me prove my solution. Turned out I was right and he never apologized later. How pathetic! But this never discouraged me. It had the opposite effect. Otherwise I never would have studied math at university later :)
I was lucky to have the mother I did (herself a teacher, though not of math) and many fine books at home (including math books). Also getting into computer programming at a very young age gave me an appreciation for math and its applications not shared by my fellow students or even my teachers. Thus I was able to tough it out.
You noticed these patterns, but did you prove that they hold? Without some explanation for why it works you can't say for sure that it will work in a given problem.
"each power of two is equal to one more than the sum of the lesser powers of two" adding 1 would make it an odd number. Makes no sense. Did you mean "two more"?
[edit] Now I see that you start with 2^0, and it makes sense.
I am just a little younger than you, but I hated math in school because every teacher was so dry and boring, I love numbers now, where were all the people like you back then that could have spurred my curiosity much earlier in life. I love when you show your true passions and giggle about it.
When I was in school my math teachers would get together in the halls to play golf. I don't know where your math teachers came from, but mine must have busted outta tha looney bin at some point. :)
My maths skills have always been lacking but I find this absolutely fascinating. Just learning about digital roots, the shortcuts for dividing by nine and finding the remainders is blowing my tiny mind. I’ll probably have to watch it a few times.
Yep, definitely have to watch it a few times.
math is super ez. you take a numba and you take another numba and than you do something with the numbaz to get a new numba. you can also come up with non existing numbaz to do theoretical operations with those to get theoretical numbaz. that's how I calculate how my piss moves thru our dimension into the next. it's quite ez, you just need some numbaz. now go out there and become the next Einstein. you can do it Mr. OmegeProxy!
Looks like I unknowingly introduced this to myself and my wife and daughters with a little game we used to play while travelling. We would add up the numbers on license plates and see who came up with the "digital root" the quickest, even though we didn't know that was the term to use. We saw very quickly that any combination of numbers that add up to 9 could be eliminated so 572 would be 5 without going through the process of adding. Later, as 3 or 4 number plates lost its challenge, we included letters. The letters "I" and "R" could automatically be eliminated since they corresponded to the number 9 and 18 respectively. This expanded the challenge because you had to figure out the numbers corresponding to the letters. As you played the game this became more intuitive when you could eliminate combinations of letters that added up to 9 for elimination. Example GSP562 would be 1. One of my daughters got so good at it that within seconds she could get the digital root of signs with just letters such as names of towns or short sentences.
fascinating sounds like you are all natural at decoding
Ask yourself are you right handed left handed what are your daughters you have two daughters Plus Sons they will give you up for the sons.
wow. an incredible game. we kearn and see patterns so quickly.
Wow!
What a brilliant game? I’ll have to keep this in mind when we pile in the van for our next trek.
I was taught the concept in elementary school, under the name “casting out nines”. Sadly, it was presented as a trick or technique, without real explanation, which I had to discover for myself. So much is lost when mathematics is taught as a bag of techniques without the underlying beautiful patterns!
@JoelNeely, I fully agree to your comment. I was also taught this at elementary school, for a later confusion as follows: Since these divisibility rules are Base-10 dependant, I had thought for many years that the divisibility of a number with another was Base dependant, and that perhaps on another base those same numbers were "conmensurable". A gross mistake that hindred developing intuition on numbers theory.
I loved the explanation where Prof. Burkard decomposes a base 10 number in: a (9+1) + b (99+1) + c (999+1) ... seen it that way is so straightforward !
my grandfather taught me "casting out nines" about 50 years ago. he used it to verify this hand calculations (pre-calculator) (+ - * /). it is a way to find single digit errors, however it has a weakness, it cannot detect the error of having a zero instead of a nine (and vice versa) . Also, it is only for numbers expressed in base ten.
@@marklarsen9894 Yes, the single-digit issue is shared by many check-digit schemes used to protect "numbers" from transcription errors. (I used the scare quotes because these "numbers"-such as account "numbers"-are really just identifiers made up of digits, not intended for use in numeric calculation.) Such schemes were especially important before computer networks were so pervasive, and data were captured and coped by hand. There are other kinds of errors-such as transposing adjacent digits-to which a simple digit-sum check digit is blind. That's why some check-digit schemes also applied weights to the individual digit positions.
I was also taught "casting out nines" in high school back in the 90s. My math teacher was great, but didn't have significant post-high school math education, and didn't know any of the deeper meaning behind anything. So the only application she knew for "casting out nines" was what it said in our textbook (error checking), and everybody hated it because we were lazy high school kids who were prone to saying stuff like "in the real world I'll just use a calculator".
While it's easy to be amused or annoyed at some of these silly viral math things that miss the forest for the trees, I feel like there's something very important we need to learn from them about how to engage with people about math and education in general.
I'm a math teacher, and, believe me, we would love to teach the real explanations alongside the techniques. Unfortunately, we simply do not have the time, and we have to prioritize. The techniques are more useful for getting students to pass the tests, and the tests determine our ratings as teachers. Additionally, we are training for the workforce, so the technique and ability to get the correct answer is pragmatically more important than understanding the correct answer. Knowing your bridge will stand up is more important than knowing why it stands up. Proper dieting is essential to good nutrition even if you don't know the chemistry or biology involved.
At my school, I've got 45 minutes a day for 187 days to teach students everything in Algebra. That 187 days does not account for student holidays (at least 14 days), bad weather days (at least 3), and the many interruptions caused by events such as Pep Rallies, ACT-testing, SAT-testing, MAP-testing, emergency drills, professional learning conferences, etc. I would estimate that I only have about 113 hours (two full 8-hour-day weeks) with my students over the course of the entire year. Can you imagine learning EVERYTHING there is to know about Algebra if you were given only given 8 hours a day for 14 days? It's simply not feasible. Especially if you're sharing the instructor with 150 peers.
Tesla didn't throw the 3-6-9 principle out there because there was anything super special about the numbers themselves, but because they correspond with certain realities about electromagnetism, waves, oscillation, vibration, spin, and curvature as found in nature. Or in other words, it's not about the bare math, but how well 3-6-9 applies in the context of physics. It's the basis of 240 VAC @ 60 cps as the most efficient formula for producing electrical power.
Exactly, he says he wants to debunk it but then doesn't talk at all about the relationship between the 3 numbers what so ever
Tesla also explicitly said the electron wasn’t real and thought people who did were stupid. He didn’t particularly have the best physics takes
@@Jearbearjenkins Your hindsight is just sharp as a tack. Wow. So impressive. I mean...if only you could have been there with your brilliant mind to illuminate him on electrons. Glad we didn't go forward with alternating current! Could you imagine the state of the world?! Wow-we-wowzers!
@@markomus1 Tesla also believed in the ether (already discredited back then) and said something brilliantly stupid about Einstein's curved space in 1932.
@@martingisser273 Your hindsight is just sharp as a tack. Wow. So impressive. I mean...if only you could have been there with your brilliant mind to illuminate him on the ether (which was NOT discredited universally back then OBVIOUSLY) or informed him about Einstein's curved space. Glad we didn't go forward with alternating current! Could you imagine the state of the world?! Wow-we-wowzers!
I wonder what sort of scientific things YOU believe RIGHT NOW that someday will be part of someone's, "Martin Gisser also believed in blah blah blah so yeah there's that," narrative.
Thank you for your clear, intuitive, and expansive explanations, illustrations, and wry sense of humor!
9:23
I'm a high-school maths teacher. I have taught that dumbed-down divisibility-by-9 test without going into digital-roots and the remainder property. I shall never do it again. Props to your deliciously made videos.
I the 1950's I was taught to check math problems by something the teacher called "casting out nines." I didnt know why it worked but was intrigued by it. 60 years later I stubble across the answer.
@@BezalElle tell me too!
Don't have the answer but one of my finance managers used to use the technique to check the added up data from his spreadsheets.
@@watwthmot I had forgotten about it until I read your post, but in accounting some similar process was used.
Divisible by 9 errors in accounting are many times a transposing error ..eg writing 187 instead of 178 ..the difference is 9( 187 -178)..back in the days of adding up columns of numbers w a calculator .
I was in grade school in the 60's in NY and was taught the same thing and they also called it "casting out nines". Later in life when we were using printing calculators and you had to "double tape" every bank deposit, if your two results were different and the difference was divisible by nine, you knew you had a transposition error.
Wow, I could listen to you all day! If I had had you for a math teacher in high school for a semester or two, I would have majored in mathematics in university! I hope you are teaching young people somewhere. Thank you!!
What i found amazing about the doubling sequence is that the embryon (and all the cells ) is using this exact sequence of doubling as well as the processor architecture.
You, sir, are brilliant. I’ve never seen something so complex, presented in such a simple way, that was so incredibly easy to follow. Please don’t ever stop making these videos. They, and you, are terrific. Thank you.
I agree! Unfortunately, or not, now I'm going to theorize and write proofs that freak people out, and that never ends well...but it's so fun!!!
he is just copying and pasting every other video out there.
He’s just copying every other single video that talks about 369z
@@cecilyschneider3631 epic, show me them lol
@@surferxblood isn't every other video the ones saying they're the secret to the universe?
Division by 7 always produces 142857. Spare key to the Universe, if you lost the master copy
I just tried it, and it got stuck in the lock. The digits aren't in the right order, so I guess it makes sense. Gonna have to go flip a coin to make a new universe and try again...
Yeah, they’re in the same order, just starting in a different location in the sequence.
If you lose your key to the universe, call the LockPickingLawyer. He'll open it for you then he'll lock it back up and open it again just to show that it wasn't a fluke.
I noticed the matching digits as well (in a different order). I'm nowhere near smart enough to figure out if there's a connection, but it's a cool coincidence.
Wait, are you assuming that base ten is a fundamental part of the universe…?
Hi, I’ve been looking all over the ‘net, libraries etc. and written text is so slim on the ground! I recall being amazed, when as 9 yr kid, finding the symmetry of the 9X table. Then as time wore on, through school and on, I stumbled across Teslas’ Vortex diagram. Watching this video has opened my eyes to more patterns!!
There are so many insights in this lucid lecture that I have to repeat it and reflect on several excerpts.
As an American born-and-raised who was in the public system as both student and teacher… our math education is disgustingly deficient in number theory.
High school graduates (even some going into stem fields) do not even know the Euclidean algorithm. They have almost no experience working with modular arithmetic.
Too many decades of parents complaining about this “useless” math subject has led to them and their children being mystified by the simplest of number theory diagrams.
Thank you mathologer for making so much explanatory content paced for victims of the US public school number theory book banning.
(Inb4 some other American tells a story about their one teacher that taught them number theory)
Yeah, every time there's an attempt to teach more theory in American math classes, a lot of parents get angry because they don't know how to help their kids with their homework. It happened with "New Math" in the '60s, and it happened with Common Core in the 2010s.
@@WarmongerGandhi Another problem with "New Math" was that it focused very strongly on the axiomatic method even in primary schools, where that isn't really appropriate. It tended to put theoretical foundations before practical examples, which is the opposite of how people normally learn math (and how it was historically developed). However, Common Core actually corrects these mistakes in a lot of way, focusing much more on comprehension and on solving problems in multiple ways. That still makes parents livid though, because now they complain "my student knows how to get the right answer, why does he have to do it a particular way? Isn't getting the right answer good enough?"
As a tutor, I see these complaints all the time, and it is very frustrating. Because no, getting the right answer is definitely _not_ the point. Nobody cares if you can, say, long-divide two decimals. Your calculator will always do it faster and better. People only care if you _understand_ how the algorithm works, which most kids don't, and just following a list of instructions doesn't show you understand.
You were a math teacher in the US and think the US has a national education system? Did you think you were a federal employee or what?
@@Dziaji you know roughly 10% of any US public school's funding comes from the federal government, right? We don't have a national education system, but the federal government still has a lot of control.
Where did he even say the system was nationalized? The comment wasn't edited and I don't see what you're even talking about.
I'm a high school senior taking discrete mathematics at UC Berkeley. I just learned the Euclidean Algorithm.
"A conspicuously simple and universal pattern is more likely a feature of the observer's perspective than the universe being observed." ... seems a more profound lesson than anything one could wring from an obsession over Tesla circles. Thanks!
Great quote! Who said it?
"an obsession over Tesla circles"
So we are going use quotes as math proofs from now on?
Observer and object are both parts of the same totality.
now prove it.
That's mind blowing 🤯 I really didn't want the video to end. Thank you so much 👍
Super cool as always. This looks pretty accessible. I hope to find time to look into Seymour's explanation. Thanks!
I have always been intimidated by math. But this video has been eye opening. For the first time in my life I am interested in math. It was engaging and made me want to know more.
I suck at match and got lost halfway thru vudeo
Yes, math can be intimidating but also beautiful when observed with an open mind.
For me too
In the great words of Billy Mays.. 😆
Same with me Katherine. Astonishing vid.
When I was in 8th grade back a 100 years ago😁, I remember a kid telling our math teacher about the mysterious 3-6-9 numbers! Our math teacher explained to us every single number could be magical if we deeply look for it. As a matter of fact he give us a group project assigning all different numbers to different groups to come up with the uniqueness of a particular number. By the end of the week we found out that every single number 0 to 9 can be unique and magical! So everytime I see these Tesla 369 videos, they remind me of my old math teacher!
This is linked to the following paradox. "n is the smallest positive integer which is not unique." But the very fact that it is the SMALLEST such number makes it unique!
Could be your teachers is from future to to teach you about secret code of universe, but she/he doesn't want to teach you directly to beyond our understanding because it will hurt.
@@kingki1953 there is no secret code for the universe! We are here for no reasons, just a result of a freak accident called the big bang!
@@mooneymooney251 our unknowledgeable is the secret of universe
You blew my mind just now! Wow! Thank-you!
I don't know much about math, but that was very fascinating and quite intriguing. Thank you for the time and effort you invest in your productions they are very much appreciated and enjoyed -Thank you!💌🇨🇦
I remember learning this in 1st grade. She said it was a short cut and I assumed it was being taught to everyone. I remember taking longer than other students to learn long division because I couldn’t find a reason I should use it since I could do the problems in my head using these techniques. I didn’t t know it was rarely taught until you said it.
seriously, you can do this in your head? I'm soooo phuked in the head with math! I'm jealous of everyone here in this lobby!
I was accused of cheating throughout grade school and really math classes in general bc I did everything in my head and writing my steps down made zero sense and would often mess me up. No work apparently means cheating instead of being logical and easy
he he!
@@Just10_Dime Agreed! Hell , it took me longer to "show my work" than it did to just do the problem... smh
Glad you said that. In the 1950s I couldn't learn long-division too, as I could do it in my head.
I wasn't even taught the divisibility test, let alone the remainder function! This is really cool!
You must've been educated in u.s.! We gotta great math program right?! But no really teachers are great once you get into those higher level highschool classes. It's where the American education system actually works. Problem is you got phys. Ed majors teaching math classes at critical levels like algebra and their just not suited for the job.
This reminded me a very old way to prove something called *9 prove* were we subtract 9.... details were gone
That explains it. School doesn't do cool, so they nixed that lesson real quick. God forbid it be teachers and the education systems job to make learning interesting.
Same :(
My 7th grade teacher taught us about the divisibility test and I've used it ever since for division by 9 and 3.
I have looked at Math and Engineering in a longhand way until I realized that keeping things simple in the beginning will find a successful result in a physical form. Understanding is a never ending experience of relationships that is endless.
4 minutes in and i must say your presention skills are impressive.
Keep up
A friend and I had a driving game where we raced to see who gets the digital root from random cars' driving plates (back then the standard local driving plate contained 6 digits). Eventually I realized that 9's were inconsequential and could be ignored, immediately afterwards both of us sped up our game by "distributing" values, forming 9's and disregarding them.
An example of what is our sped up mental process: 166384 = 1+8, 6+3, 6+4 = 1
I also just learned from this video that what we were doing is called Digital Root
I think this is the kindest, most conscientious debunking of mathematical mysticism I've ever seen. I love this channel so much lol
Agree. Wonderful channel. And that's how it should be done. I immediately lose respect for or won't continue listening to someone if they're acting like they're on the offensive, regardless if I think they are right or not or if they align with what I am inclined to believe. There's enough condescension and inflammatory behavior out there, plus you're not going to win any minds or hearts by acting like that, it's just pandering to people who already agree with you. Cheers.
You should get out more. 🤓
So what did Tesla mean when he said that then? Was it just an elaborate troll? Do you think Tesla would say something like that with nothing behind it?
why do you people talk like that
@@peaceenjoyer sorry, talk like what? I'd love to discuss the content in the video, I think I'm just missing some context
If you look at drawings from the father of geometry, Euclid was born around 300 B.C. and he has 369 theory shapes exactly like you describe all over his work. Thank you for the video we enjoyed it.
In relation to 3-6-9.
Ken Wheeler made an experiment with molten bismuth cooling down over a powerful magnet. He predicted that the cooled metal would exhibit "bubbled cavities" at the extremities of the puck in this triangular, 369, formation.
For those interested, I recommend.
Well that's just an equilateral triangle, it's not systematically related to 3-6-9
I really appreciate the fact that you spend time watching other UA-cam videos, in addition to creating your own. This is what makes Mathologer not merely “yet another maths channel”, but something of higher value; your videos don’t just provide yet another explanation of the same thing, but provide further explanation _in context_ of existing explanation attempts. Love it!
Way, way back in the early 1980's, I had to make a choice between continuing Math or Art studies, for my final 2 years of high school. I chose my artistry and have always kept a little candle burning for my love of numbers and equations. I have to tell you, I watched as I am interested in Tesla and was curious of the vortex diagrams you might display. Now I musts thank you as I am SO EXCITED by what you have shared, despite being in my late 50s, I am going to return to study math. So, thank you, thank you for inspiring me!
Math should never be optional in school. It is always the most important discipline and the basis of all else (including art).
I chose math. I always felt art was just welfare for idiots.
@@edwinwebber5776 If that makes you feel like a bigger person go for it.
@cdanielh128
The primate brain tricks itself into releasing a temporary burst of dopamine. The sensation fades and they feel worse than before.
As with any drug, the rush fades as tolerance builds, as they ever increase the dosage.
The anger that Mr. Webber was cultivating, starts to fester more and more until it becomes this very hate-filled and self-destructive loop seen here.
That's when you realize that every hateful and ugly reply they made was that of a sad and very lonely person whom were subconsciously screaming to just be noticed.
He doesn't feel any better or bigger. Never will acting like this either.
I do hope @Edwin Webber gets his life all figured out and treats everyone, including strangers on the web, with respect. A little decorum can go a long way.
@@edwinwebber5776 Thats simply because you don´t understand it. That you don´t understand something doeasn´t make others anything:D Art has LITERALLY saved lives:)
Great delivery of semi-complex information. Thanks
Man this is mind-blowing. Is this the key to the universe? It certainly seems so.
It’s nice to know new people are trying to understand a new topic of math I remember when I came across this when mark Rodin was first starting off when we where still trying to work out coils and the potential of the math! And we’re constantly learning new things!
I am from Austria and we never learned that the number, which remains actually is the remainder (9:45). When I learned about modular arithmetic in math Olympiad, I guessed that fact to be true while doing an example. Not even my highly invested teacher was sure, whether the solution was right. Infuriating, that you do not learn these deeper truths about mathematics at school.
depends on your school. we were thought about that.
we were also given homework to come up with divisibility tests for other numbers like 7 and 11.
@@dsdsspp7130 Dude is your teacher Flammable Maths? I think he talked about giving his students that exact homework. I'm just asking because I don't think that is going to be a common task to give to students.
@@samuelmahler5961 Out of curiosity what's the answer?
@@samuelmahler5961 no, it's from years ago back in high school. it might have been a textbook question, it's not hard to figure out if you know modular arithmetic which was a part of our curriculum.
we certainly did have very passionate maths teachers though.
@@mathlegendno12 just search divisibily test for 11 and 7 in google you'll find it in no time.
if you want to solve it yourself here is a hint:
just calculate the remainders of powers of 10. here's how to do it for 11:
remainder of 10^n to 11 is always 1 (if n is even) or -1 (if n is odd)
example: 432 = 4*100 + 3*10 + 2*1 ===> 4 - 3 + 2 = 3 , so remainder of 432 divided by 11 is 3)
so just like 9 you sum up all the digits except you have to negate every other digit with the rightmost digit being positive.
Truly fascinating, thank you ✨
In the US of America. As an industrial electrition. We use 3 colors for 3 phase power. You add the circute number's digits. When divied by 3. the remainder. Will tell you what color to use. If we had 9 colors. This would work as well. Bravo!
Thank God someone's finally talking about these things! I stumbled upon one of these Tesla 3 6 9 videos ages ago, and I knew that everything they said definitely wasn't magic or anything and probably had a simple explanation in number theory, but I could never put it into words myself. I'm so glad someone finally put together an understandable and informative response to those things.
But I believe that the root cause is simply using 10 as base is quite intuitive. As 10 is an arbitary choice, the whole stuff is nothing general
Thank you! I was excatly the same, I figured it was just some base 10 shennanigans, even found the pattern was almost identical in base 8, but I don't have the education/smarts to prove anything. So they just laughed at me in the comments when I tried to explain to them.
Maybe the real conclusion is that number theory is magic.
Yeah, I did not know the "trick" in the video. But as soon as I saw it I was like "It's just a spirograph" or some other display of a function. So there is an underlying mathematical function, and it's a pretty way of displaying it. Interesting if used to show how leaves/petals form a certain shape, but just interesting and useful, not magic.
Sumerians used base 60
I'm fascinated by all this. I have to watch it multiple times. I have no clue what this is used for. I only use basic math in my work. I ♥️ all these videos. Great job explaining 👏
Happy to see you again . Thank you.
9:46 I remember figuring this out after teacher taught us the usual divisibilty rule for 9 and I shared it with the class. The teacher praised me for pointing it out and it made the topic a little simple for the whole class!
Wow aren’t you a smart cookie
Solution to the problem at 16:14
Start with this equation, which is true for every value of k:
5^k * 2^k = 10^k
The digital root of any power of 10 is 1, so
DR(5^k * 2^k) = 1
Using the multiplication rule you explained earlier,
DR(DR(5^k) * DR(2^k)) = 1
In other words, DR(5^k) and DR(2^k) have to be multiplicative inverses of each other.
Taking the “digital root” of an integer is equivalent to modding it by 9. (The only difference is that if DR(n) = 9, then n mod 9 = 0.) In mod-9 arithmetic, every number except for 0, 3, and 6 has a unique multiplicative inverse. Since the digital root of a power of 2 is never 3, 6, or 9, this means that DR(2^k) completely determines DR(5^k).
As k increases, the value of DR(2^k) cycles as follows:
2 4 8 7 5 1 2 4 8 7 5 1 …
Taking the multiplicative inverse of each number above gives the values of DR(5^k).
5 7 8 4 2 1 5 7 8 4 2 1 …
So DR(2^k) and DR(5^k) cycle through the same values, but in reverse.
Looks good :)
You don't need to evaluate all the numbers in the two cycles to check that one is the reverse of the other;
5 is the multiplicative inverse of 2, so 5^k is congruent to 2^(-k). So as k increases, it runs through the same cycle in the opposite order.
Something I was waiting for but wasn't mentioned: the horizontal symmetry, which can be explained by simply numbering the points with negative numbers going anticlockwise.
I was actually waiting for it as well
We were taught casting out nines which was basically mod 9 arithmetic. It made it easy to check the addition of a long list of numbers.
Thanks for such a clear explanation and understanding driving me too to it.
Wonderfully narrated and illustrated. Your channel is quite addictive! Thank you for your efforts, more please !
He said it was A key to the universe not The key. Their is a very stark difference. With the work he did do, his findings with those numbers is astonishing. Limited but nothing short of astonishing.
Bruh
Pythagoras who was a messianic type figure had this all nailed down in the mystery schools and i think Tesla got his knowledge from the same source
@TrashLid Hi Trashlid, its because I read. Try some Manly P Hall, John Black and mystery school research. It was also my opinion. Certainly not a conspiracy theory. Why do 'smart' people like you whine so much instead of humility?
Amazing video thanks for explaining this and finally making sense of it
Amazing you blew my mind... Great vid.
As an ex Maths teacher in UK the reason this doesn't get taught is simply because it is not in the curriculum, and to actually get through the curriculum leaves no time to play with Maths, or indulge the class in whatever the teacher finds interesting or stimulating (supposing he/she has a higher understanding of Maths in the first place!).
All hail lord curriculum
@Chris Lauden I agree.
Yes it is quite sad that the curriculum does not leave that room. I trust that Burkhard is not blaming the teachers for that!
But it is quite sad that the social status of maths (and therefore the interest and learning speed of children!) is in such a deplorable state. It is something I try to battle against and especially Mathologer does a great job in this respect!
Yes , well said. The broad curriculum is the summation of alot of mankinds work of course. In the real world beyond + -* and / only a handful of the next generation actually find yearly let alone daily applications for Pythagoras, solving a quadratic or calculus. Alot of interesting math has niche applications. We can see the benefit of this , like considering the math the Rubiks cube can throw up , but we have to "weigh" this usefulness.
However "casting out the 9s" IS very useful really. Take the summation of say 5 numbers all with varying decimal place information. Did you punch all the numbers into the calculator correctly ? Some calculators the information may have been lost off the screen...So using the maths here can at least apply a quick check ...
Math teachers be two weeks behind one week after class starts
I was not taught any of this math in school (as far as I can remember). This is so exciting! Thank you so much for your efforts and sharing them!
The connection between mathematics and physics or reality is something physicists are particularly fond of. I think one can overdo that, but underdoing it is also not good because many phenomena can be much more accurately described mathematically than with ordinary words. A balance seems to me to be the best path.
Does it make you wonder that we are in a simulation ?
These principles will be present in the Plasmoid Unification Model, I'm almost certain of it. That will be the new way we understand the universe, it is most assuredly something to look into. Great video.
For a long time I've wondered about the multiplication tests using digital sums when working with systems outside of base 10. You're the first to show a proof that my hypothesis was correct. Thank you so much for this.
I believe it works with any base system above base 10 by using [system_base - 1]
First time I was introduced to this type of math was a different process to verify binary divisions manually. This is also the first time I see it being used with base 10...
Man, I was never taught the divisibility test. Christ almighty that would have saved so much time.
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I found this video very interesting and for someone who is absolutely not a fan of math this is pretty fun. I am glad I picked this video about Tesla's 369 rather than any other because you explained everything in a such good way and showed that there's nothing magical about it. I haven't seen too much about it, but I beleve that people just like to make an elephant of a mouse for anything and that's why it's "a secret of the universe" and all that. It's always easier to just believe the bullshit from the internet rather than do your own research and think with your own head. I am happy that videos like this exist, nice and clear explanation and pure math, thank you so much! I will definetly check out more from your channel!
I LOVE THIS. I have been terrified of maths but I now sit in wonder. ❤️🙏
Thank you for maintaining such a commendable pedagogical level in your videos.
this guy is awesome..wonder the accent?
Fantastic multiples of 3 6 9 create vortices. This is how the fabric spacetime interacts with matter and energy from stomach, venturi, eruptions, corollary, double helix and solar system.
These diagrams remind me of the spiral graphs we used to have as a kid. It also resembles the graphics program I wrote in college where we had to code something like a drawing program. I don't even remember what mathematical formula I used but it was related to sin and cos of x and y and I plotted those values based on the radius. Instant spiral graph like drawings on the screen of any size.
I was not taught the divisibility by 9 test in school. But my father did teach that and many other mathematical concepts to me when I was very young. (Mid 1960's) He used a book called The Calculator's Cunning which used number theory to teach people how to perform complex math in their heads. I still have the book.
Edit: Here's the book information for those who asked.
CALCULATOR'S CUNNING
The Art of Quick Reckoning
Karl Menninger
Translated from the Tenth, Revised, German Edition by
E. J. F. PRIMROSE
Forward by Martin Gardner
BASIC BOOKS, INC., PUBLISHERS
New York
First published in the German language by Vandenhoeck & Ruprecht, Gottingen, under the title, Rechenkniffe: lustiges und vorteilhaftes Rechnen
Tenth, revised edition 1961
English translation copyright 1964 by G. Bell and Sons Ltd.
Library of Congress Catalog Card Number: 65-19543
Printed in the United States of America
Share that book with me
good to know. Nearing 60 Y.O. I still want for the learning of such things. I will look into getting a copy of that book. Thank you for sharing that knowledge.
Please could you also share that book with me?
Where can we get the book?
@@TK-sx4fr I would assume that this book has been out of print for decades, and I have no idea how easy or hard it may be to find.
I just want to say, Thank you for existing and making this video. When my friends ask me why i'm shaking my head and giggleing to myself when they show me tesla vortex videos. I can just send them to this video instead of trying to explain these concepts to people who just don't get it.
Hi Burkard, Nice to have someone so ebullient concerning math. I have a question as to the representation here of a vortex as the definition in fluid dynamics requires at least 3 dimensions to define the flow revolving along an axis inclusive of a curl component. Can you extend this definition to an x,y,z coordinate system to model something akin to a waterspout rather than linear representations?
Love your content, I've learned so much from your videos. I always need to be able to imagine a problem before solving it and your diagrams, illustrations, & animations are incredible. You've also got the best shirt collections I've ever seen
Dear Mathologer, I was not taught this pattern in school. In 3rd grade i observed the digital root pattern of multiples of 9 myself and used it for quick solving of any problems involving 9. Later in college I rediscovered this pattern and obsessed over it for a few years... What I found was very interesting and fulfilling as it relates to quotients, products, and a prime sieve. Eventually I moved to a base18 system of counting to account for the parity of digital roots, (like when a number like 31 adds up to 4, this does not account for the oddness of 31, but 13 does.) It happens then that this pattern takes a much more intuitive form when we allow for digital root as well as parity. The little "hiccups" are practically cured.
At this point I've only watched half this video, but I felt like answering your question about schooling and sharing my own journey with digital root maths.
I was, but not by a teacher. My best friend in school showed me. He is a cpa now, and the only reason I passed trig.
God bless numbers / patterns are in many things
If you explained this on some casual Math blog I would read it.
Have you published any literature around base 18 mathematics? I'm intrigued by the concept and have been looking into it myself when time permits.
@@Frankenstein786 I'm only a hobby mathematician, so no I have not published.
Thank you. Very concise and humorous! I was taught this as part of a "bag of tricks" to speed through math tests in elementary school in post-WW2 Britain. Mental arithmetic was highly prized. I hope you do keep up your excellent videos beyond your 100th birthday. I mean, what's special about 100?
phenomenal content. thank you.
Hi, FYI -
Tesla was discussing numbers that model our Universe in Natural Philosophy studies - perspective - modeling the Universe.
Further, mathematics is only valuable as it is ultimately useful, and correctly applied !
3-6-9 is really about Fractal geometry and its Nature - in Nature !
So first we see Fractal Geometry matrix (microcosm), operating inside a Holographic Geometry matrix (macrocosm).
- So finally this the vortex mathematics of a hyperbolic hourglass shape of the torroid or the vortex of the donut hole - in the torroid !!!
Had an incredibly rough week, this was precisely the pick me up I needed!
I recognized the divisibility pattern around 3's and 9's as a child doing the times tables as explained in the video. I later briefly brought up recognizing the pattern in college during Algebra II (comes in handy doing factors) and seemed to surprise everyone in the room. Didn't know other people like "junk" math. 😆
Actually, here in Italy they teach this "digital root" test to check divisibility for 3 (recursive sum of digits equal to 3, 6 or 9), for 6 (EVEN numbers divisible by 3 by the previous test), for 9 (recursive sum of digits equal to 9); even with the "remainder" detail mentioned at 9:30 for the special case of 9. They used to teach that method in primary school, back in the 90s (born in 1988), and I am pretty sure they never stopped to do that.
The special case of these patters appear if the modulo basis is a prime, in which case the repeating set of the elements is a finite discrete field (of N-1 elements, N being the basis), which is useful in many very commonly used encryption and data protection schemes.
this is something I've not been exposed to before. I'm 68 so I'm a product of what I learned before around 1973. however, to use your vernacular, as soon as you started talking about the number nine I was saying to myself, all that really is is b - 1. I was a COBOL programmer for a few decades I'm familiar with base 8 and base 16. some of the other programmers I worked with used to call me a bit fiddler. in fact, I remember learning about using different bases way back in 1963 in summer school at my grade school. it was very confusing at first.
but the bottom line for me even from the beginning of this video was that what it's really showing is just how fascinating the relationship with numbers is, rather than any kind of a key to the universe.
but as you pointed out later, that the universe is based on mathematics. heck, even music is based on mathematics.
This didn't ruin vortex maths for me, just made it much more interesting and expansive
It *is* interesting, without a doubt. Just not... "mystical."
@@KipIngram It's most definitely mystical. There are other properties he is not discussing. And; did you notice? None of the other base models have the flip in polarity like that of 3 and 6. The others are continuous loops. The 369 does crazy things when applied to electromagnetism and such.
Those videos are excellent to see just before bedtime
they are but I also usually end up staying up late and trying stuff out 😆
I'm not a math teacher, but I am a math adjunct teacher: I teach physics.
I do teach the rule "add up the digits to see if it is divisible by 3 (repeat for 9)" rule.
If you could make a video of "things they ought to teach in school" I promise I will teach all of them.
Thanks, Steve
I love this .
From your perspective as a physics teacher, have you ever considered the possibility that the physics paradime being taught could be an unnecessarily complicated way of modeling the universe? Could some of the most prolific engineers in history have been aware of a greatly simplified model which can be used to explain everything, including that which your existing paradime has no answers for?
I believe humanities savoir is outside the realm of physics
I was thought both for 3 and 9 the rule is similar. That would mean the system can be setup in base of 1 2 3 which is just a triangle. Doubling between 1 and 2 as the repeated sequence. An alien duck with 3 toes maybe?
Math is the language used to understand everything that exists from subatomic particles to the universe itself! Your love of math is beautiful. Please continue sharing your enthusiasm for math and sharing your ability to break down items into their various pieces and parts, and of course, the fun you have in combining those things then in various ways.
🥰🥰🥰
get well everybody
Math is just an instrument, made by humans and has fallacy because the human mind cannot create anything that is perfect, we have no perfect knowledge and never we will, it's natural limit, therefore, mathematics is not better than history or philosophy when is about understanding life and the universe. Nowadays there is too much bias for math and STEM in general, a bit of brainwashing I would say. I'm happy I studied math and engineering many years ago when the academy wasn't a brainwashing institution yet.
um maths is a numeric philosophy that can only tell you anything about the model, rather than reality itself.
@@thetimeisland850 thats what you think
@@arthurw1604 thats what i philosophise! it is also technically true. Math can only prove its own internal models, never reality.
my understanding from this video is that "vortex math" is just 1 of an infinite way to draw circle patterns. But, the pattern of the petals forming is because it forms all patterns including vortex math. Does that sound right?
I've always done everything in Threes!!!
I’m terrible at math but have always been interested in mathematical concepts. At the start of this video I pondered “maybe 9 is special simply because we base our integers on 10”. I felt so vindicated when you proved that the diagram works with any other base minus 1.
Base 10 is expressible as base-9 plus null (0). That's why everything seems to revolve around 9 in base 10.
It’s always interesting seeing how close genuine mathematics, particularly number theory, gets to numerology sometimes - the choice of base ten is essentially completely arbitrary.
On a barely related topic, I would love to see a video which covers some of the bignum arithmetic algorithms, like karatsuba, toom-cook, and possibly even FFT based multiplication. We rely on these algorithms thousands of times per day for RSA in TLS, so I feel they deserve some love.
"Once you abandon scientific rigor, you're no longer a mathematician--you're a numerologist!"
From the movie Pi.
+1 to a vid on Karatsuba / Toom-Cook / FFT.
@@capturedflame Arbitrary in a universal, cosmic sense dear, we just happen to have ten fingers, on average, as a species, due to evolution.
@@capturedflame Plenty of cultures have had numbering systems based on 6, 12, or 60. Even odd things like 17, 23 or 27 are not unheard of. There are other body parts than fingers one could enumerate. So, yes, base 10 is arbitrary in the sense that there are other equally valid choices that have been used. The fact that it's the dominant one today is little more than a historical accident.
@@capturedflame I mean that we have (on average) ten fingers to be pragmatic about, yes, but not because of God, destiny or some universal force, only happenstance.
19:48 first time ive heard this but ive know this in my head since a kid cant wait to hear more
"Does this all look familiar?"
Yeah. The Mandelbrot set!😊❤🤓
It's those diagrams and cool connections that make me love number theory. Please do more number theory videos.
What perfect timing! I just remembered these claims about Vortex Math and mentioned the explanations of their patterns in the comments of a math meme about how 9 is only special due to the decimal system. :)
We are the victims of our own curiosity when we try to think objectively about mysterious phenomena. The same thing applies to alien abductions stories, sasquatch, etc. Our fascination with the unknown sometimes tricks our brains into ignoring obvious explanations.
It's Sasquatch for me. I really want it to be true.
That's cause and effect in the laws of nature. The language of the universe as we want to understand is always easier and more exciting analysing with the intellect. Really appreciate your passion! The key to the universe its much more profound and it lays in letters and numbers. I know you know what I mean! גימטריה
Thats cool but I wish you number obsessed folk would stop doing 9/11s
just a friendly thank you from an out of field listener, curious of the properties that make up our life and universe,
Ive never understood math i would literally pound my head but you have helped me more in minutes than my teachers did in years
What a hero this guy was. It's to bad we don't have someone like him around these days?
We do but they are either silenced or don't have the resources to get the knowledge out
@@02SplinterCell02 Could you imagine if Tesla had a UA-cam channel today?!
@@malayaleeking he would have died of a sudden heart attack before his channel got to 10 subscribers
Cartesian circle is also during the process of analyzing radio signals.
I am a software application developer. I realized certain patterns regarding the success and stability of a software solution that actually applies to every system. It is so common that everyone is aware of them, but nobody realizes it. Every system that could possibly occur requires 3 pillars of support. It doesn't matter how simple or complex it is. In fact these 3 pillars not only support the system, but also support each other. Perhaps there is a way to expand these 3 pillars into multiple dimensions, which can easily generate highly complex systems that would likely appear chaotic. Interesting.
I’m doing that with acupuncture
I didn't learn the "remainder rule" in school, but I discovered it on my own and it blew my little mind
I'm convinced Tesla thought this was a neat mathematics trick, and someone picked it up as "mystical" because Tesla is smart.
what's perplexing about Tesla is he wasn't particularly well-versed in mathematics. he engineered devices in his imagination for the most part. It was Carl Steinmetz, an oft-forgotten figure in the history of math and science, who attempted to actually work out the math of what Tesla was doing.
Tesla never explained why 3, 6, and 9 were the key to the universe. This digital root stuff probably has nothing to do with it.
Eh, no. Tesla was a hack who for every 'real' invention had a grandiose, self-aggrandizing nonsense that only existed in his imagination. See death rays, wireless energy transmission, quack diets, pretending he invented radio, or saying special relativity is wrong. It's quite possible he invented some dumb number sequence and thought it's somehow magical.
Tesla had a very profound understanding of everything deeper than anyone on earth today. Numbers speak, but not with words. It is very very hard to explain to someone the importance of the particularities of that number 9 in equations like he showed in the vid for example. It is something you have to see or to do to understand.
@@KuK137 💩
inebriated i am, however... this video is the most entertaining ever. yo, math is so awesome. you my friend are awesome
When you convert the Solfeggio frequencies with this digital math, they all reduce to 3, 6, 9. Those frequencies create harmonic resonance that creates the oscillation proven in the Hopf Fibration, with these other "digital numbers" this vortex math explains the fabric of our universe. How time and space collapse upon itself. Like tesla said " the secret to the universe, is energy, frequency, and vibration"