Times Tables, Mandelbrot and the Heart of Mathematics

NEW (Christmas 2019). Two ways to support Mathologer
Mathologer Patreon: / mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)
The good old times tables lead a very exciting secret life involving the infamous Mandelbrot set, the ubiquitous cardioid and a myriad of hidden beautiful patterns. Time for the Mathologer to go on a serious fact-finding mission.
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). (Added 8 November 2015 check out the proof at www.qedcat.com/nephroid_proof.pdf )
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.
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/Time...
Enjoy!
P.S.: The music we are playing at the end is called Shoulder Closure by Gunnar Olsen. It's part of the free KZbin music library. A really nice piece , isn't it?

[@DanielGonzalez-ff2df]

I’ve never had a KZbin video assign me homework

[@clasticish4565]

thats funny man

[@emadgergis6710]

Go watch Mathantics.

[@ts4gv]

btw, if youre stumped on that problem, consider negative numbers. specifically, any number 1-9 subtracted by 10.

[@specialpetzpetz5020]

A comment of note

[@jarredmattingly5369]

I'm dead🤣🤣😱👻

[@21CenturyBreakdownX]

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

[@Kenshinhugo]

well you should read into 3d fractals :-)

[@werone7010]

If you take psychedelics it becomes clear the universe is fractal

[@karthik4495]

This comment explains more about you than math lol @steve

[@specialpetzpetz5020]

so true... the older i get the more i want to know (now only! F), but time unfortunately is running out....

[@twobob]

No, "math" just "is", people are trippy

[@hyypersonic]

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

[@lightspiritblix1423]

Jormungandr Honestly, me too.

[@discreet_boson]

Same

[@emingevorgian]

same

[@luigikart2225]

this math is way more engaging and fun than school math. absolutely same

[@admi_nw]

Same ;(

[@amenx1143]

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

[@ernestjohnson3261]

I'm gonna notice them now all the time

[@sizzleshnizzle9484]

When you look anywhere in life there is a geometry to it. Nature uses the force in which we use a tool called mathematics to measure and understand. Fractals, fibonacci spirals, golden ratio the cardioid when the is light and a circular reflective object. Everywhere you look you can use mathematics to understand and that is the language of the universe. For me that proves to me that there is a creator of this existence whether is is any of the traditional religious diety or something we have not considered there is definitely a creative and destructive force at play within our universe.. we are here to figure out what this place is and how to utilise the lessons to make our coexisting but seperate worlds work in synergy.

[@lockpickrogue-yc5wc]

🍑

[@Sejf_]

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

[@raffimolero64]

editing to match the finger?

[@jenniferhagler5718]

You ever watched the weather channel?

[@Raiom.]

Redstoneboi he is obviously not doing that because the circles are not moving.

[@TheBaldr]

He has a monitor out of view that shows him the composite of himself and the graphics, just like most meteorologist when they give the weather.

[@sarahjaynewilkes3604]

haha didn't think of that

[@stevenvanhulle7242]

"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

[@michaellinner7772]

At first I thought that you were waxing poetic but read to the end and realized you were just plagiarizing, lol!

[@stevenvanhulle7242]

@@michaellinner7772 I'd rather that you'd say "quoting" instead of "plagiarizing". Plagiarizing is stealing others' work and pretend it's yours. I DID say it's by Bertrand Russell. Have a nice day.

[@michaellinner7772]

@@stevenvanhulle7242 sorry I didn't mean to give offense. It was a poor choice of words.

[@paulbarton4395]

This Russell quote is on the Alan Turing memorial I believe

[@seekergraphix2487]

Then you say then that mathematics is the greatest art form, made by the greatest artist.

[@LivesavingSuccess]

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.

[@light-master]

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.

[@secondarycontainment4727]

Pretty sure they either projected video there or simply edited out a circle drawn on a whiteboard (in post) ...take note of his arm sorta disapearing when he points into the circle area.

[@xXJ4FARGAMERXx]

Have you also considered the fact that the computer can also convert the 99% white pixels? And he can just point to the part that's 99% white because he has eyes. And the computer can still convert it

[@kylebtlr]

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.

[@andyschnell8842]

Makes ya wonder why we are not sold the beauty n design of other wise boring numbers???

[@andyschnell8842]

,,,at school in the conditioning years of our evolution,,,mmm???

[@Deliquescentinsight]

Very true, i like your thinking

[@maythesciencebewithyou]

As if this would make people like Math. We did this in Math class and also in art class, but guess what, people still hated Math. They may like to watch the patterns, if at all, as most people would even be bored by them. Even if they love Iphones and find out that you need to know bunch of Math to develop one, they'll still hate Math.

[@howardbaxter2514]

@@maythesciencebewithyou some people will always hate math. There is no changing that. It's like how some people, such as myself, don't find biological sciences that interesting. No matter what you do, some people just won't find it interesting or compelling. But, by showing the beauty of math, and how it goes beyond the cold rigid structure most people see it as, you will grab more people's attentions. Some people think math is not useful in the world, and that it is strictly confined to pen and paper, but when you show them simulations and graphs of basic mathematical functions, and how these functions can create extremely complex systems that can correlate to what we may see in the real world, then more people will realize how math is the language that defines our universe.

[@neomonk5668]

My childhood Spirograph was a math engine of art.

[@user-bq9yy3bt9k]

Damn, i thought only soviet russia children had spirograph as a toy

[@BeatrizRuco]

Юрий Назаров i’m brazilian and also had one. I didnt know URSS invented it :0 awesome !

[@garyvallone5393]

My girls are playing with a spirograph right now. Which is what inspired me to look into it further. I'm a plumber and no mathematician but I was always intrigued by this toy and knew it was phenomenal math

[@DSimonJones]

I spent hours doing spirpgraph as a child I loved it. I never thought about this math connection. It's awesome thank you so much for sharing.

[@migtvill]

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

[@sibbyeskie]

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

[@alquinn8576]

beep, beep, beep. back up those big bootied bitches

[@halasimov1362]

Mandelbutt

[@NStripleseven]

...

[@mathphysicsnerd]

And now for something completely different;

[@PIC18F]

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

[@CalvinHikes]

It did feel that way.

[@miletech]

u wot m8

[@haniffrazri6023]

Its somehow make me remembering sigil.. like the sigil they said solomon king hide..

[@aidengary5748]

Hypmotized

[@Deliquescentinsight]

Yes it can have that effect, as if there is an underlying pattern which if we could only grasp it, would reveal all mysteries

[@JarodBenowitz]

Mathematics is the landscape of all possible structure.

[@Insight-music]

Jarod Benowitz do humans use numbers or do numbers use humans?

[@climbtheladder9440]

@@Insight-music Both? Biologically, we sprang out of these structures through evolution, but it also seems like we (our consciousness) are an intregal part of the drive that keeps all of this flowing. Like the universe is a constant river of math that we are in, a part of, and also the current manipulators and eyes of this flow...at least on this planet.

[@dhanmonee]

Sounds like nerd bs

[@clos4021]

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

[@shumeister1059]

Thank you for showing math in such a beautiful way.

[@SehnsuchtYT]

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

[@grantaum9677]

Reality follows this pattern on every possible scale 🕉

[@mrsirman2177]

Haha exactly

[@bruzote]

Humans strive to fit observations to patterns. For humans, recognizing patterns can improve survival. Since it is that important, you evolved to FEEL that understanding relationships (including seeing patterns) is good. That's why. Sure, there might be applications for this math, but it doesn't matter. When you see a pattern emerge, you are programmed to feel pleasure at some level.

[@NickC_222]

Geometry is the language of god.

[@mrsirman2177]

@@NickC_222 Damn, that is an interesting quote, I'll remember it forever

[@michaelfox9852]

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

[@eboubaker3722]

Comon dude, be serious

[@TommyGunsXL]

mASSematic

[@bruzote]

The universe is doing nothing of the sorts. You are personally seeing butts when humanity in total sees and feels so much more.

[@VideoGameStormers]

well i and 130 something others have found his comment worthy of a like while your comment doesn't have much appraisal at all.

[@SaintNath]

thanks for this comment :D - nice lecture though it made me think a lot

[@tomsullivanchannel]

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.

[@nothinginteresting1662]

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

[@Mr440c]

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

[@marcoottina654]

So I'm not the only one figuring it out !

[@SantNabi]

That was The Whole point

[@foobargorch]

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

[@firefly618]

I suppose it would have to be a trumpet bell-shaped coffee cup, with the sides following an x² curve. I don't know if it exists.

[@mailliw94]

multiple light sources, bro.

[@rohitgulati2500]

I think it would be inverted conical shape..

[@robertlorenz2492]

@@rohitgulati2500 so a beer stien

[@troglovi]

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.

[@SequinBrain]

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

[@jacksainthill8974]

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

[@Mathologer]

That's great :)

[@deanhedges2849]

It would be interesting if the konnaticol tones could be brought into string art of the Fibonacci sequence ... kzbin.info/www/bejne/o4CwfYWDm66MnJI

[@Davidamp]

Just got this recommended

[@Mathologer]

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/

[@PaulOMalleyDublin20]

As was said on the video. For any N the reflection and thus petal like structure is similar to N-1 light sources. The internal patterns look like interference of those waves. While the petals are at the edge the reflections in the center are perhaps additional symmetries caused by reflections of ever more complex surrounding shapes. I've no special knowledge and so as I'm running out of words to express my idea my hope is it's understandable. I've watched it a few times, there's much to think about in this one.

[@Mathologer]

+Mathologer Also, check out these cool animations by Johan Karlsson codepen.io/DonKarlssonSan/full/meQOvp/

[@PaulOMalleyDublin20]

Lovely stuff.

[@psionic0]

+Mathologer With regards to 2 and 3: 2 - The equation for the target point P is: P = (N x T) mod M Where N is the starting point, T is the multiplier factor and M is the number of points (the modulo) The horizontally symmetrical starting point of N is (M - N), so the equation of target point P' from horizontally simmetrical starting point is: P' = ((M - N) x T) mod M which can be written as: P' = (M x T) mod M + (-N x T) mod M = (-N x T) mod M since (M x T) mod M is always 0. (-N) is N when we're labelling points in counterclock-wise direction, hence P' is P in the same labelling system, so, like the starting points, P and P' are also horizontally symmetrical. 3 - Provided that (4N x 51) mod 200 = 4N, for 51 pattern we got four kind of lines: I) 1 + 4N points go to 51 + 4N points, which is a quarter of circle ahead II) 2 + 4N points go to 102 + 4N points, which are diametrically opposed (they trace the diameter lines in the pattern) III) 3 + 4N points go to 153 + 4N points, which is a quarter of circle before and can be written as -1 + 4N so is symmetrical case of I), and then we know that if 1 + 4N goes to 51 + 4N, also 51 + 4N goes to 1 + 4N so it's there are no extra lines. IV) 4 + 4N (or just 4N) points go to 4 + 4N points, which is the point itself So, other than the diameter lines, the pattern traces a rotating line from 1 + 4N to 51 + 4N, which resemble to a cicle Finally for 99 case, we see that (N from 0 to 99): (2N x 99) mod 200 = (2N x (99 + 1) - 2N) mod 200 = (200N) mod 200 + (-2N) mod 200 = -2N We then got two kind of lines: I) 1 + 2N (odd points) go to 99 - 2N. It's the horizontally opposed point, and all trace horizontal lines: 1 -> 99, 3 -> 97 and so on. I) 2N (even points) go to -2N. It's the vertically opposed point, and all trace vertical lines: 2 -> 198, 4 -> 196 and so on. I also tried to figure out point 4 about mandelbrot set, but cannot get to a conclusion so far. I started from cardioid parametric function but I suspect the explanation is easier that what it looks at first.

[@Mathologer]

+psionic0 Excellent work :) Did you already check out the answer to 1 that I added to the description of the video ? In terms of 4 I actually don't know of a really easy explanation myself. I also suspect there is one and it's on my long list of things to ponder.

[@fab4fan173]

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!

[@Armagedon.]

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

[@tjmmcd1]

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

[@stephendavis7112]

Careful of mistaking models for realities. You only know the isomorphisms.

[@maxwellsequation4887]

You don't know

[@joostieman10]

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 :)

[@junaid-vc3js]

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

[@antonyjohnson4489]

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

[@NovaCyn]

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

[@Mathologer]

+Phijkchu_Cute_Phijkchu Full marks for that explanation :)

[@brainoutyakabrainout]

That was very helpful to me, thank you. A guy named @Clay_Odem and I (@brainouty) in twitter, are discussing this in the context of a de novo review of Unified Field Theory and whether energy is or is not wholly separate from mass, and if is, whether energy creates mass, and if does, how dow QM fit in. He's an EUT proponent. Thread with this video attached, is here: brainout.net/frankforum/viewtopic.php?f=20&t=549&p=3642#p3641 Thank you again.

[@poopcatapult2623]

Brain Outy energy and mass are very different things. If you somehow had the idea that mass is energy, you were probably confused by the famous E=mc^2 formula. Energy creates gravity and only in a certain sense mass, as in quantum field fluctuations which increase with increasing energy density.

[@ronkirk5099]

Math is beautiful and can explain the natural world.

[@ProDigit80]

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.

[@ProDigit80]

Like a disorderly pattern, would just fall apart, like in a DNA sequence, if the sequence isn't properly aligned, the creature won't be able to grow, or exist. The further along the chain of patterns you go, the more strings are used to connect the points. These strings need to enforce one another, for a stable picture. Just like DNA, if we see a complex image far down the scale, but the rays aren't enforced by other rays, in nature, the DNA would fall apart. Which is why chaos, doesn't breed in nature. Only order does. It's that order that could help us solve certain things of life and universe, if we get a better grasp on them... While the Mandelbrot is just a mathematical equation, it can illustrate in simple ways how the universe and life works; If someone could just gaze long enough upon it, and find the correlation...

[@gasun1274]

you're describing chaos, which is deterministic. life isn't all that special from a statistical perspective, check out sabine hossenfelder's video on free will to know more.

[@saintmay1952]

@@ProDigit80 you're assuming a lot.

[@sabastiankilgore781]

Oof

[@brucedickson6019]

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

[@democratic_chocolate2067]

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

[@gammazzz3894]

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

[@Rene_Voortwist]

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..

[@yahccs1]

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!

[@Astronut128]

Its strange how KZbin 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.

[@nightvisiongoggles]

KZbin hears your thoughts

[@haroldbn6816]

Spooky

[@duh4572]

You sir, never fail to amaze. Outstanding!

[@aglees2b]

For sure. That was most excellent

[@Kamau2012]

Simply the best spirography video on KZbin. It explains how and why the formations happen.

[@mckillalaberry]

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

[@delandreth]

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!

[@Mathologer]

Glad this worked for you and thank you very much for saying so :)

[@dewdewism]

kzbin.info/www/bejne/boLUfKN8druerck

[@mconway0003]

this changed me

[@bunbunnbunnybun]

Theres a but i my tea

[@josephgrossenbacher7642]

"multiplication - exponent - connection" : see complex analysis v functiontheory (also with several complex variables) & riemannian-surfaces --> an instance of the ambiguity of some complex valued holomorphic functions.

[@robinwells8879]

Memories of my Spirograph set in my childhood.

[@maroofsultan]

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

[@studiohq]

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!

[@hamidhshaikh5707]

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.

[@AnjaHuebel1]

Yes, it's a little-known secret that mathematicians just like playing around with numbers, functions, sets and things like children with lego! The amazing thing is that so many really useful things emerge from their play! Or maybe that's because maths is truly at the heart of this universe.

[@angsterweltschmerz779]

Maths is astounding beyond alls!

[@NocturnalJin]

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

[@McEffinHat]

They decided "arseoid" was too sophomoric and settled on "cardioid" instead.

[@btat16]

Jeff Benson Jesus mate I spilled my cereal on the table

[@OolTube02]

"Rectoid."

[@Deseptor]

hahahahahahahaha, theres a butt in my tea. hahahahahahaahahah (i LOVE carioids and fractals and stuff)

[@overloader7900]

@@Deseptor It is badass tea

[@EquaTechnologies]

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

[@Galileo2pi]

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.

[@SonariNeiracchen]

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 ^_^

[@ibacon1999]

the ending was really satisfying

[@mathematicsolympiad775]

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

[@ellenfalk1040]

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

[@billytheweasel]

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

[@aarongreenfield9038]

When you see that shape in your coffee, it is screaming drink me.

[@wewladstbh]

Couple of years studing number theory and group theory ought to do it

[@user-yt2tu7qd5e]

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

[@diegosourux]

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

[@mattiefee]

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

[@kosewiz]

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!!

[@NASkeywest]

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

[@Kiboz2000]

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

[@venkatabalajite4988]

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

[@theowild2524]

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

[@blackpoolram]

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

[@gdubb6905]

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

[@xavierplatiau4635]

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.

[@robfurnari1]

Really impressive! Fantastic!

[@deplorableamerican9451]

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. 😊

[@JudyTepley]

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.

[@bobmcarthur5468]

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

[@thadsuits3835]

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.

[@Mathologer]

Great idea :)

[@nerdalert226]

Wow...

[@Hunbatz95]

I thought of doing that too, do you have an animation of that?

[@vegut]

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

[@XxfishpastexX]

Jorge Verona why would you waste an ayahuasca trip on KZbin

[@vegut]

Alejandro Reyes I didn’t, do you do that son?

[@fartkerson]

​@@XxfishpastexX I think he meant before he had seen the video, this appeared in a trip, but he didn't have the words to describe it before finding this video.

[@XxfishpastexX]

martk fartkerson ohhhh!! Yeah definitely makes more sense. I’ve seen cool geometric structures on my trips as well. I hate the internet when I’m tripping though, which is why I made my comment.

[@vegut]

@@XxfishpastexX yes is best to do it far from the city, all electronics have an effect on it. I don't do it often either, too much information that needs to be understood, I let coincidences happen, like finding this video

[@ivanovicsharapova2402]

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

[@Onkarr]

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

[@joshthomson1998]

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.

[@seanhuggins5382]

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

[@Mathologer]

That's great :)

[@Keex11]

So I'm not the only one who immediately thought "yeah I'm going to make that" :-) Fun stuff

[@joogps]

i’m so excited to recreate this in processing tomorrow!

[@csgstormer]

Sean I know this is like never going to be seen by you but if it does, please help with this crazy idea I have: Would it be possible to match up the Symmetry and time table with the symmetry of music? Hopefully you can grasp the seed of the idea but basically I want to visually capture the symmetry of a song using times tables. How can I do this?

[@XHappyKillerX]

Chase Gielda Hey, what exactly do you mean by the symmetry of music and how do you want to apply this to the time table?

[@magpej6613]

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

[@Anima.Libera]

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! 💓🌀♾💫🌌🌐

[@eXtremeDR]

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.

[@paulomartins5071]

+eXtremeDR The Congruence Fractal.

[@pizzy1992]

+eXtremeDR many natural structures are frattal-like,for example,the lugs,plants,shells etc....they have to be,in this 3d world,with the minimum volume and the maximum surface ( Monge sponge)

[@jannikheidemann3805]

Romanesco is the most obvious and beautyful plant fractal.

[@SuperStingray]

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

[@lesromanchuk1293]

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

[@jorgeluiszimmerman3487]

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

[@xX1GuNNy1Xx]

EXCELLENT!!! but um, no yellow on lines anymore please, especially when its high density, hard to see :D

[@saccharinesaccharine9175]

HI BABY memory set game water ocean fan baby

[@saccharinesaccharine9175]

HI BABY memory set game water ocean fan baby

[@doudsbass]

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 !

[@ajaykumarmaruvada9113]

Thank you for al u have done to us . U have literally shown what is math and why it is supreme and beautiful. We need gurus like u there . We wish our education system changes more from learning and knowing and not totally proving u r learning by a bloody damn test which is a blank question paper. HAPPY TEACHERS DAY!

[@Narend1987]

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

[@gsg7354]

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

[@arhamshahversatile]

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

[@1111757]

Why is it symmetrical: Assuming N is the number of points (10 in the first case), the point on the far left is N/2 and the point on the far left side is N (or 0). Multiplying by two will bring you from the left to the right side: N/2 * 2 = N If you go X steps clockwise from the very left, you end up at N/2 + X. Multiplying by two will bring you to N + X * 2: (N/2 + X) * 2 = N + X * 2 If you go X steps counterclockwise from the very left, you end up at N/2 - X. Multiplying by two will bring you to N - X * 2: (N/2 - X) * 2 = N - X * 2 Therefore the lines are always symmetrical around the axis between N (or 0) and N/2

[@tinkeringtim7999]

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.

[@alekskrakoew8584]

this video is so important

[@larryg3446]

how so?

[@SKIP__ADS]

Yes it is

[@NonPhysicalPhenomena]

How?

[@mynameforever1]

Mathematical example of synchronicity

[@file_not_found5606]

you can say that again

[@sfundomabaso3200]

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

[@wewladstbh]

Don't forget the Jacobian!!!!

[@SRAVANAM_KEERTHANAM_SMARANAM]

Sir, What software is sued to draw these cuves shown at 4:00 minutes duration of the video kzbin.info/www/bejne/p5nFpn6Yra-moZo

[@EsmaelM-ze8nk]

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

[@jurisembergs9245]

Incredibly comprehensive! Great pleasure!

[@dimospas]

Excellent visualisation! Well done!

[@Mathologer]

+Dimos Glad you like what we are doing and thank you very much for saying so :)

[@s7eve1980]

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

[@Mathologer]

+Stephan Fabry Bin zwar in Deutschland aufgewachsen aber das Wort kannte ich noch nicht :)

[@bunbunnbunnybun]

What

[@NBryanBlack]

I am really interested in the microphone applications of this, and hope you have already made the mentioned video! (I haven't looked yet, as I wanted to thank you for THIS video!) In this video, I already saw implications of microphone patterns, especially when you drew the patterns using the two circles. Microphones develop their pickup patterns using interference patterns that depend on how much sound is allowed to reach the back side of the microphone. The nephroid pattern is pretty similar to the figure-8 (or bi-directional) pattern as well. If you haven't already made the microphone pattern video, I would like to voice my vote in favor of your doing so! (and in case you have, I'll look for it now!) :) Thanks for making this video!

[@ralph3295]

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

[@alexoftheway8169]

This was a facinating vidio,thanks for putting this together so we could see how these shapes work.