(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: ukposts.info/have/v-deo/po-KgHhnpZmT0X0.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 :) ...

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!

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.

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.

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.

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!

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.

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.

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.

Division by 7 always produces 142857. Spare key to the Universe, if you lost the master copy

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)

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!!

Thank you for your clear, intuitive, and expansive explanations, illustrations, and wry sense of humor!

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.

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.

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.

"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!

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.

That's mind blowing 🤯 I really didn't want the video to end. Thank you so much 👍

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 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 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.

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.

Super cool as always. This looks pretty accessible. I hope to find time to look into Seymour's explanation. Thanks!

I wasn't even taught the divisibility test, let alone the remainder function! This is really cool!

I really appreciate the fact that you spend time watching other UKposts 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!

You blew my mind just now! Wow! Thank-you!

4 minutes in and i must say your presention skills are impressive. Keep up

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!

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.

Man this is mind-blowing. Is this the key to the universe? It certainly seems so.

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?

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!

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

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.

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!

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.

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!

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.

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.

I think this is the kindest, most conscientious debunking of mathematical mysticism I've ever seen. I love this channel so much lol

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!

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.

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!).

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 👏

Thanks for such a clear explanation and understanding driving me too to it.

Truly fascinating, thank you ✨

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'm convinced Tesla thought this was a neat mathematics trick, and someone picked it up as "mystical" because Tesla is smart.

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.

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.

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!

Wonderfully narrated and illustrated. Your channel is quite addictive! Thank you for your efforts, more please !

Great delivery of semi-complex information. Thanks

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.

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?

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.

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.

Literally the first video in a long time I actually had to listen to at normal speed my man should be an auctioneer lmao

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.

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. 😆

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.

"Does this all look familiar?" Yeah. The Mandelbrot set!😊❤🤓

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.

I LOVE THIS. I have been terrified of maths but I now sit in wonder. ❤️🙏

The division is found in layout of musical scale on the same Enneagram, a Pythagoras scale, that is, where each next note is a result of division of previous string length by 2.

Happy to see you again . Thank you.

Man, I was never taught the divisibility test. Christ almighty that would have saved so much time.

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

Amazing video thanks for explaining this and finally making sense of it

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'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

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.

in opening a kaleidoscope ; the irony that keloid variables are not heart rate measures indicating ambient light doesn't adjust from angularity adjacent in linear grid

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!

Thank you for maintaining such a commendable pedagogical level in your videos.

I didn't learn the "remainder rule" in school, but I discovered it on my own and it blew my little mind

Cartesian circle is also during the process of analyzing radio signals.

Amazing you blew my mind... Great vid.

Had an incredibly rough week, this was precisely the pick me up I needed!

When I was a kid, I learned the "casting out 9s" trick in some puzzle book (possibly by Jerome S. Meyer) my older brother had. I didn't learn why it worked until I read Martin Gardner's column about divisibility tests.

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 !!!

Crazy to know the significance of this number existed thousands a year before him, inspiration, answers are already here, we just need to discover it, like he did!

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.

I've been watching your videos for years and only today realized that I have your book! Q.E.D.: Beauty in Mathematical Proof. My mother gave it to me for Christmas maybe 10 years ago and I enjoyed my first read through immensely. Now I peruse it from time to time, it really is a gem! If you like the Mathologer videos I thoroughly recommend the book!

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"

I invented the digital root stuff myself but mixed it up with some cryptography because i was working with mod 26. a*b=ab mod 26 was so beautiful to see. That was when i was in my primary school

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.

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.

just a friendly thank you from an out of field listener, curious of the properties that make up our life and universe,

On behalf on everyone like me who tried hard to grasp what you're exposing us to, thank you.

absolutely loved your video! I think the vortex is still very special even though it works in all kinds of base systems. Surely the fact the vortex remains strong in all counting systems adds to its beauty and power. It's maybe just only about 9 in the base 10 system

Those circles remind me of the Spirograph game I played with for hours on end as a youngster!

Similarly, if you numerize (consecutive integers beginning with 1) the alphabet, for example, A to Z corresponds from 1 to 26 And from 26 to 1 so that every letter of a word is represented by two numbers (A = 1 and 26, R = 18 and 9, T = 20 and 7); take the digital root of each of sum of the word {2: digital root 1, digital root 2}; add the digital roots; then the final digital root of the sum will always be 9. 9 is the final digital root of any word.

I've always done everything in Threes!!!

The vortex is just the formula for calculating counter frequencies. It's really interesting what vibrations can do...

It's those diagrams and cool connections that make me love number theory. Please do more number theory videos.

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.