Times Tables, Mandelbrot and the Heart of Mathematics 

I’ve never had a RU-vid video assign me homework

I'm more impressed in how precise he is at pointing something that he's not seeing.

In grade school: Math is pointless! In College: Math is useful! Now: Math is flippin trippy.

I was studying for a test and all of a sudden I was like "oh my god is there a cardioid inside my coffee cup" it's fascinating to see math in real life! there's something really appealing about this weird shape, for some reason I find it very aesthetically pleasing

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty-a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." -- Bertrand Russell

Mathematics is the landscape of all possible structure.

This man has instilled a great deal of happiness back into my life.

I’m seriously procrastinating from doing my math homework to watch a video about math 🤦‍♂️

I became hypmotized and for a moment I thought I'd solved the mysteries of the universe.

For me, this could be the most beautiful video ever made. I keep coming back to it.

Once i took lsd and i saw patterns like these moving in my head. I didnt See them more like felt. It was like Time, conciousness, vision, hearing and feeling were Working together to create geomatrical patterns you cant usually see With your eyes. Ive never heard of this stuff before but it seems like it plays a Major role in our World.

I'm a simple man. I see creation expressed geometrically, I click.

This should be taught in every high school geometry class. We're so fixated on base 10, this is the way to address that in math ed. Everyone hated times tables, this is the gateway to showing how math is cool and past learning wasn't worthless.

The more complex the figure becomes the more it looks like it starts to have another dimension. They start to look spherical!

Considering he's really looking at a white wall, he does a very good job of looking at where the animation will be later on in post.

I'm an artist You have NO Idea of how important this was I have been looking for years on HOW the math of making fractals worked and Mathologer just gave me the answers I was looking for in an easy to understand explanation! THANK YOU!

For those of you who’d like to play around a bit with the stunning times table diagrams that we discuss in this video, download the .cdf file www.qedcat.com/cardioid.cdf and open it with the free cdf player which you can download from Wolfram Research (the people behind Wolfram Alpha and Mathematica). If you have access to Mathematica you can also open my .cdf file in Mathematica and play with the code. For those of you who are looking for a bit of a challenge, ponder this: 1) Starting with the fact that the nephroid arises from parallel rays being reflected inside a cylindrical coffee cup, try to convince yourself that the 3 times table really does produce the nephroid (some really neat geometry at work here, very similar to the argument for the cardioid that I talk about at the end of the video). 2) Why do the diagrams for all the times tables have a horizontal mirror symmetry? 3) Try to explain the pretty patterns corresponding to the 51 and 99 times tables modulo 200 that I display in the video (around the 9:30 mark). 4) (For those of you with a very strong math background) Try to figure out why the cardioid shows up in the Mandelbrot set. The discovery of the stunning patterns that I discuss in this video is due to the mathematician Simon Plouffe. Check out this article tinyurl.com/o2hbtsa and his website plouffe.fr for other stunning visualisations using modular arithmetic. Enjoy! Quite a few animations have been contributed by various people and linked to in the comments: Here is one of the nicest ones by Mathias Lengler: mathiaslengler.github.io/TimesTableWebGL/

My childhood Spirograph was a math engine of art.

so when grandma was crocheting her doilies to put under lamps, she was teaching us about math, thx grandma!

I like the way he kept the scope of his presentation so precise and to the point and anyone with no maths background can enjoy- excellent work

The explanation with the lightsource also explains why the pattern is symmetric, as the source of light in all the directions has an exact opposit on the other side of the horizontal axis. Very interesting video :)

Math is beautiful and can explain the natural world.

Math, geometry, poetry... that blows my mind

Quite beautiful! One of my primary school projects was to construct a string version of a similar pattern. As I approach retirement, patterns emerge everywhere. Never in my wildest dreams did I think the wheel would turn so completely. :D

the ending was really satisfying

I feel like this is really significant but I don't know why

The flow within the patterns is a beautiful dance and I love the way that you can see it in the math and that you are teaching what's hidden from most of us, without a hint. Thank you.

You must be the only mathematician with swag.. 🙃That was really cool. One of the few times I actually enjoyed some explaining math. Childhood trauma slowly healing..

Oh no, I was really looking forward to the coffee cup example for n = 4

the patterns that dance in the middle looks like gazing into a crystal ball, beautiful

As a kid I used have a game called Inspiro. It contained several plastic cog-wheels and other curved shapes with holes, in which you put a pen and rotated around to create beatiful shapes. I definitelly remember some of those you presented in this video.

Thank you for showing math in such a beautiful way.

Those animations were awesome....you must have worked hard on them.nice video and good luck for your future

Humanity: Let's decode the universe. Universe: Look at all the cool things you can do with butts!

This is awesome. I'm more impressed the more I watch. Thank you for sharing your knowledge in such a simple and beautiful way. Maths is beautiful!

Oh my goodness, as an artist this is a cool new way I can utilize precise patterns Math always had cool ways to do art

Now I got it why they say, "Maths is beautiful, Maths is not a dry subject, Maths is very interesting". Thank you for this video.

How did I fall into this rabbit hole? This stuff is an absolute trip.

It's amazing when we connect our knowledge we can observe the logic behind everything around us

I am totally showing this in my computer Science class and all of my math classes!! This is so cool.

I really like how his hand goes through the diagram. Gives it some nice perspective. Most people don't see it as professional, but I personally believe the combination of CGI and real life images created in this manner are quite satisfying to watch. Well made video, great editing, and the explanation was rounded very well! Also, great voice ^_^

Memories of my Spirograph set in my childhood.

Absolutely beautiful and absolutely fascinating, just simply mind-blowing. Very well explained and illustrated too!

Finally. The mathematic formula to find permutations of multiple-cheeked butts.

This might be the most beautiful and incredible thing I have ever seen. I've been interested in fractals since I was a young child, though I had no idea what I was seeing until I was in my late teens/adult years (I'm now 26). Also being able to see a a new way of looking at my times tables is so fascinating. After teaching a student about logarithms and really starting to understand some more of the patterns found there (I'm sure there's loadsmore that I'm missing, my maths education stops at vector calculus and double integrals), the multiplication->exponent connection is extremely interesting and has left me pondering a lot as to why that connection is there and where else it shows up. I just found your page tonight and really have enjoyed the clear speech and the incredible visuals. Having watched many Numberphile and ViHart videos, this whole way of viewing maths really clicks with me. Thank you so much for making this, Mathologer!

Fascinating. The very foundations of creation are based upon math, geometry, light frequency and the frequency of vibration.

Simply the best spirography video on RU-vid. It explains how and why the formations happen.

Beautiful - it just shows how times tables are a key to so much more mathematics... I've never seen them visualized like this before. Amazing!

I'm not going to lie, this video blew my mind.

The symmetry is clear if you are familiar with modular arithmetics, or the modulo operation. The modulo operation takes a number and a base often written in programming as a%b (a is the number, b is the base) and returns the number that can be found by adding or subtracting b from a any whole number of times. This is actually how we get from 26 to 6 in the first (base 10) circle since 26%10 = 6. Now the symmetry comes when we look at the relation between 1 and 9, 2 and 8, ... n and 10-n: namely that they are just n%10 and -n%10. So when we go from 9 to 18 we could actually say we go from -1 to -2, which is the same as going from 1 to 2 but on the other "side" of the "mirror" Notice that 10 can easily be subsittuted with any other base b and therefor this symmetry holds for all the circles. Yea I know that probably wasn't extremely clear. But its 2AM whatdya expect :P

This man gave me hope to my life. I reborn with new career.

“Folgers coffee, the best part of waking up. Is fractals in your cup.”

Its strange how RU-vid recommended this video to me today, since just a few hours ago I tilted my water glass and was intrigued by the shape the light was making on the table...the Cardioid.

You sir, never fail to amaze. Outstanding!

When I was 10 or so I used to do those kind of drawings using a compas and a ruler. I kept them all this time.

Few times in my life had my mind blown today I found something new to learn 🙂

Heart shape? When I was a kid we used to say "There's a butt in my tea!"

Thats great work and explained exactly what i was looking to have answered. 2 thumbs from this guy

0:48 some strange serendipitous stuff happening! This is like a godsend for a piece of art I was working on 👏👍💪

I really like the kind of educational and informal style of the video with the camera man in the back talking

Watching all the internal patterns in the last animation was like watching what the inside of my eyelids looks like. Wild

Thanks so much for this video! I was able to apply my programming skills and create a simulation of this in Java.

I can always expect Mathologer to make my life fascinating by bringing these amazing mathematical discoveries to my notice.

I like to see a re-visit of this, over the years he has allowed himself to give nore details to his audience. A new version could be epic.

I had my Y6 Primary classes investigating these fascinating patterns way back in the 1970's. Wish we had the interactive whiteboards back then!

:0 I saw this on ayahuasca! I couldn't find a language to it until now, cheers!

It's so weird when you discover these things for yourself and then look it up and other people have found it already. I was messing with geometer's sketchpad as a freshmen in high school by using two concentric circles and a line between the two and animating both points around the circle while tracing the line. Kudos to all those who discover these things for themselves even if they've already been found by someone else. Discovery is so much more rewarding than having someone else tell you everything.

Absolutely beautiful, showing how much the symmetry and beauty of the Universe is hidden in seemingly boring or laborious mathematical pursuits

I'm not good with mathematics. I simply cannot think clearly with numbers, However; I do wish to I be good at Mathematics some day. I envy you, Thank you for the videos. 😊

Como la simpleza de las tablas se relaciona con la sutileza y la hermosura de los patrones formados que bien se encuentran en el mundo que nos rodea, un privilegio poder verlo. Gracias por su trabajo y por compartir conocimiento!!

This is a fantastic, excellent, pretty lovely example of the power of the natural numbers, thank to share his sort of ideas, I do love it, thanks.

This channel has considerably improved my existence.

Wow, Mathematics can be THAT beautiful!

12:17 kind of makes me think of the order of the universe, and life, in a simplistic way. In the beginning of this segment, there's lots of moments of order (12:25, 12:29,..). The further you go, the less orderly the ray patterns are, the less frequent there's a moment of order (or pattern) that we recognize (eg: 13:07, 13:13.90). I think in the universe, even life, we can compare amoeba, bacteria, plants (the beginning blocks of life), to the beginning of these orderly states. The further we get along the video (@13:07 minutes in the clip, and further) the more complex the structures become, resembling the more complex animals on this world (mice, dog, ...). Until you come to the a 'last' orderly state or recognizable pattern (13:29); which can be compared to the pinnacle of creation, a human. This state is much harder to find, because the window of time, to recognize this pattern is much smaller. While the patterns in the beginning are easier to recognize due to their reduced complexity, and can be recognized even multiple frames away from their final (and perfected) form. You'd think everything is a mess, but then out of nowhere, there's order again. A very complex, yet beautiful order! Yet, the question is, if beyond this, there are more organized patterns one can find, if one would scan only fractions of a frame instead of full frames; or if this last organized pattern is really the final organized pattern, and anything beyond this, is just chaos leading to infinity? Could it be, that if we re-sequence DNA in a way, could end up with a pattern so complex, a creature far more intelligent than us could evolve, like we would when unraveling new, and more intricate patterns in this scale? If the times tables can reveal another (or more than one) organized pattern, after many iterations, it could be mathematically proven that there's a possibility that there's another state of DNA that is superior to humans that could exist or come from the natural evolving and unraveling of these patterns... After all, all creatures are created with the same building blocks, just a different mathematical formula is applied to them, resulting in a different DNA strand, resulting in a creature with entirely different features and intellect.

This is fabulous, thank you. I make Temari thread ball designs and have done some patterns using simple versions like the 'astroid' as design elements. this connection to Times Table helps to sort out much better in my mind the why of several ''string art'' designs I've not tried yet. I had never had it explained in this manner and relationship before.

wonderful ideas addressed in an interesting way. I never saw such a practical way of teaching mathematics before.

this video is so important

Great stuff. I noticed that if instead of a multiplier you use exponents, you can make the resulting "exponential table" illustrate an interesting property of number theory. Using any prime exponent (in place of the multiplier) along with a modulus of the same prime number, no lines at all are drawn. This a graphic illustration of Fermat's Little Theorem dealing with congruences.

That's awesome! You can see all directions and ever-changing angular patterns. Anything you wanna see is there

Mathologer is one of the greatest exposers of the 'secrets' of mathematics, he makes the abstract visible and intuitive

I am majoring in sound production and deal with Cardioid Microphones all the time. It's really interesting to see the math behind it and I think it could be really cool if you guys went over the math behind a microphone.

Really impressive! Fantastic!

That was mesmorizing, spiral is shown in the animation at the end of video, lots of cool stuff there ...thank you for sharing your knowledge

Wow, this is really a superior. A great seed for a lot of programming fun. Thanks

It was sooooooooooooooooooooooooo Interesting. I like this video so much, that I can't express my feelings!

What’s also pretty kool is when shown moving it resembles cell division

One thing that never cease to amaze me is how complexity arises from such simple rules. This must be very meaningful - but I have no idea what?

As a child I had those classic games to reproduce these shapes (that moves me), one with wool, and one with plastic circles for drawing... beautiful! 💓🌀♾💫🌌🌐

Great lecture and nice presentation - thanks for sharing. When' done over multiple dimensions- then the results are beyond amazing - from detailed plants, flowers over insects (even with detailed wings, legs, etc.) to fascinating singularity structures. Meanwhile I'm sure a DNA sequence is exactly the same - a time table for this fundamental principle.

funnily enough, the cardioid microphone is in german "Nierenmikrofon", which directly translates to "Kidney Microphone"

Extremely amazing mathematics creating extraordinary patterns, thank you for creating such video.

watching this reminds me of the patterns we see when analyzing increasing frequencies in Cymetics.

That animation around 8:20 is brilliant ! Thanks for sharing all that, I think it's even worth a slower and longer version of twenty minutes or more 😊 What's amazing to me in this type of progressions is that it seems endless and alive. It's like a breathing : while at the periphery the number of petals is increasing and they're all rotating in a counter clockwise motion, at the center the petals are spiralling in a clockwise motion and their number is decreasing. It goes in cycles. Both sections (the center and the periphery of the disc) are like mutual conjugates. They evolve anti-symmetrically like a moving illustration of the "action and opposite reaction" principle ! If Nassim Haramein saw that, I think he would say that it needs to be thought of as multiple 2D projections of a moving 3D object. He'd go on with holographic beauties. I'm curious what would say Theoria Apophasis too !

3:04 Am I the only one who has ever looked at a cup of tea or coffee and seen this shape?

He makes a good point about why did it take so long for you and I to hear about this. It's amazing

this is really new for me and shocking, mind-blowing.

3:16 I've seen this in calculus 1 where we had to draw beautiful pictures using trig functions (Polar curves) to determine the area

WOW! I'm seeing the foundations to the earth being laid down.

Math is SOOO connected - every single thing connects to itsself - Pi to Julia Sets to Mandelbrot Set to Times Tables to Fibbonaci Sequence to Newton's Fractals - This is why I love math

Thanks for sharing the CDF file and other links to tools to play around