Pi and Mandelbrot (extra footage)

MAIN VIDEO: • Pi and the Mandelbrot ...
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[@rolien91]

You should really make a Numberphile3 channel in which you go into the deep mathematics that few people understand.

[@michaelbauers8800]

Someone could try to explain the proof for the Taniyama-Shimura conjecture ( which simultaneously proves Fermat's last theorem) such that lay people could follow it. Seems unlikely given the length of the proof :)

[@Human-gc2yt]

You mean this isn't the deep stuff?!

[@theadamabrams]

@@Human-gc2yt Well, this video has some deep *ideas* but not a deep *explanation* (proof) at all. There are lots of curves that look kind of like tan(x) or arctan(x) but are actually something else. In what way does the function at the end of this video actually related to tangent? It's not just tan(n) since the early terms are different and the later terms always increase as n increases.

[@joeyw.7131]

Andrea Palessandro we call that Mathologer

[@StormCrowAlpha]

@@Human-gc2yt not really, no.

[@BobStein]

Notice you added TWO zeros to epsilon each time, so when epsilon is 10^-2*N, then f(pi*10^N) is about where it diverges. This is a very nice visual clue as to why pi is in here (that the f resembles the tangent function, which diverges at pi/2). But I wonder how much more complicated the rigorous math is.

[@samueldevulder]

+Bob Stein Yeah. I'm also interrested to see the exact maths behind the scene, and how he tangent function really pop out of the equations for the number of steps. Any pointer to a scientific pdf about this is welcome.

[@IchBinKeinBaum]

1:10 Hehe, you said pi-ness.

[@giladzxc17]

lol i was about to comment that too!

[@bytefu]

+IchBinKeinBaum I wish I could say that in Russian. There is twice as much fun with neologisms if you know two languages :D

[@Sathrandur]

+TheGerogero You made this Australian laugh - and captured Julia Gillard's voice perfectly😂

[@j0nthegreat]

+IchBinKeinBaum don't be naughty

[@dontfucktheduck]

Do you mean "пишество" as similar sounding to "пиршество"?

[@NickiRusin]

I was expecting trigonometry to pop up at some point. This is very cool!

[@MarkusJaeger-itguy]

this is just cool

[@numberphile2]

+Markus Jaeger I know!

[@TheEvolNemesis]

+Markus Jaeger Yeah, I saw it when the little right triangle 'steps' came into the picture as the picture of what the function was doing... The pythagorean theorem is essentially the equation for a circle of radius c formed by all the possible right triangles with hypotenuse length c... and you can see how tangents are involved in the function too, because you can draw hypotenuse lines with these right angle steps which have angles tangent to the curve and whose right angle sides are directly proportional to the sin and cosine of those angles... the graph of f looking like a tangent graph just kind of nails it... very cool stuff.

[@lemue4972]

+Jesse Finnerty those triangles are actually a very commonly used method of visualizing sequences

[@TheEvolNemesis]

Leo Simmel Yeah, I can see why, the way this problem was explained helped me understand it a lot, as well as why pi crops up so much in solutions to other series and integrals of certain functions.

[@peterchan229]

Markus Jaeger '

[@KingKehra]

I've always loved the Mandelbrot set and stuff like this makes me love it even more.

[@5daydreams]

I like how Holly and Hannah currently are rivaling in my standards for best freaking voice/speech ever. And the Mandelbrot set rivals with P vs NP in my standards for most ridiculously hard-to-grasp concepts in all of math, lol.

[@lonestarr1490]

"And the Mandelbrot set rivals with P vs NP in my standards for most ridiculously hard-to-grasp concepts in all of math" Oh boi. You better stay well clear of category theory and algebraic topology. Or geometric scattering and Selberg theory. Or anisotropic Banach spaces. Or Calabi--Yau manifolds.

[@denelson83]

Your Royal π-ness.

[@scrungozeclown836]

Your royal Piness?

[@Angor6495]

still not happy with that explanation. i want to see the "little bit more complicated math"

[@shoutz5872]

the explenation just feels.. incomplete

[@ChrisConnett]

3b1b's sliding block puzzle has something similar where the digits of π pop out of a count of something seemingly unrelated. kzbin.info/www/bejne/fnbJeXmpm6yLoNU My best guess as to why tangent shows up is that it relates somehow to the integral of 1/(1+x²), because that x² + 1 in the denominator looks suspiciously like z² + c. But I am quite unsure of this.

[@Ja55yK]

pdfs.semanticscholar.org/dbed/13dae724fed20356b81be91c63fc13b1e1b8.pdf It still seems so unintuitive to me

[@ricobarth]

Just re-write the equation to x² + (1/2)² = y - epsilon. Then you can see how it's related to the equation for a circle.

[@harrisonwhorf3173]

Thinking of the Mandelbrot set as a function of the number of iterations starting at c=1/4 allows you to approximate it to the tangent function as c gets nudged larger and larger. The width of the tangent function is pi which is how that number pops up.

[@sizzla123]

=) Ingredients 1 cup whole almonds, toasted, cooled and chopped 1/2 teaspoon salt 1/2 teaspoon baking soda 2 teaspoons baking powder 3 1/2 cups flour 1 1/2 teaspoons vanilla extract 1/2 cup vegetable oil 1 cup sugar 3 eggs 2 teaspoons grated orange zest

[@omp199]

+AUSPICIOUS KONDOR I see what you did there. Now, where's the vegan recipe?

[@sizzla123]

omp199 TA DA!!! =) Almond Chocolate Chip Mandelbrot 2 tablespoons flaxseeds 6 tablespoons water 2 cups all-purpose gluten-free flour blend (I used Bob’s Red Mill All-Purpose Blend) 1 cup almond flour 1 teaspoon ground cinnamon 1 teaspoon xanthan gum 2 teaspoons baking powder ½ teaspoon salt 2/3 cup coconut sugar or vegan dry sugar of your preference 2/3 cup slivered almonds 1 cup fair trade, non-dairy chocolate chips ½ cup plain, unsweetened non-dairy milk (I used almond) ½ cup neutral oil (I used safflower) 1 tablespoon pure vanilla extract 1 teaspoon almond extract

[@omp199]

Yay! Thank you, AUSPICIOUS KONDOR! You are a hero / heroine / non-gender-specific valiant person! :)

[@sizzla123]

omp199 LOL I AM just a HERO. =) Enjoy!

[@keich.3938]

AUSPICIOUS KONDOR You are either: 1. German, 2. not German but can understand the language 3. or you saw someone in the comment section saying "mandelbrot" means "almond bread" in German.

[@tahmidt]

1:11 so..haha piness :P

[@jorisbressan6412]

hapPiness ;)

[@kcthewanderer]

I was hoping this follow-up was coming. Once again, Brady delivers!

[@terapode]

Brady, these numberphile videos are getting better and better. It´s really entertaining.

[@Partysize2]

Your explanations are very clear and understandable. I'll be looking for more of your math topics.

[@danielnobrega1009]

Mindblowing!

[@Infinitesap]

Thanks for all your great videos. It's awesome to follow. Holly is so ....

[@zakirreshi6737]

This is nothing once a legend found pi hiding in colliding blocks.

[@alexanderlevakin9001]

Another legend says that one of those blocks is exactly the small block who collide between y=x*x + 1/4 and y=x, looking for its way to infinity. That's why we found Pi there. Who lets the bloks out?

[@goemon4]

Daaaamn This is still really awesome

[@conorbmcgovern]

Seeing this for the first time - absolutely amazing!

[@AKorigami]

I do not understand any of this, but loved it!

[@Quadflash]

Elegant, beautiful insights into the ways that math describes the universe. Thanks!

[@tjejojyj]

I'm glad you posted this one too; I thought the main video was a bit light on for the maths. Great video. Really clear and concise explanation. Thanks.

[@narrotibi]

This is so amazing.

[@tarekalkattan]

Thank you , That was amazing.

[@CengTolga]

This is awesome!

[@seven0929]

1:11 Best reaction ever!

[@youreinacoma3009]

Pi-nes HA HAAA: double entendre

[@samueldevulder]

Holly & Mandelbrot Set... Love both of them.

[@johnvonhorn2942]

Let's talk about this over dinner. After a great meal, "would you like to finish with a slice of pi?" With moves like that, why am I still single?

[@austinkelsch1985]

The inverse tangent function graph at the end of the video blew my mind

[@2Cerealbox]

"pi-ness" - oh boy, I don't think you know what you said.

[@webhead66]

Yeah but Holly did ;)

[@tom7]

Numberphile2 forever!

[@MartiniComedian]

This is mind-blowing!!! :O

[@dennismiddleton8483]

Yes, estimation of pi can be achieved through iteration of real value c + very small number E. If E is set to values of (1/10) raised to the nth power, we can create run sets for numbers like (.25 + (.001, .0001, .00001, ....)). Each such run starts with a power of 10 to the -3 (referred to as run 3 below in the data). Completion of each even numbered run reveals an estimation of the digits in pi, while the odd runs reveal pi * SQRT(10). I ran 18 runs and this seems to hold true. I know that the idea of SQRT(10) has been "debunked" as not having any relationship with pi, but this seems to suggest otherwise (at least up to 7 digits). Anyone ever look into this relationship? ========================== Run # 3 c=0.251 Iterations i = 97 Ratio to last = -0.1 ========================== Run # 4 c=0.2501 Iterations i = 312 ========================== Run # 5 c=0.25001 Iterations i = 991 Ratio to last = 3.1 ========================== Run # 6 c=0.250001 Iterations i = 3140 ========================== Run # 7 c=0.2500001 Iterations i = 9933 Ratio to last = 3.16 ========================== Run # 8 c=0.25000001 Iterations i = 31414 ========================== Run # 9 c=0.250000001 Iterations i = 99344 Ratio to last = 3.162 ========================== Run # 10 c=0.2500000001 Iterations i = 314157 ========================== Run # 11 c=0.25000000001 Iterations i = 993457 Ratio to last = 3.1622 ========================== Run # 12 c=0.250000000001 Iterations i = 3141591 ========================== Run # 13 c=0.2500000000001 Iterations i = 9934586 Ratio to last = 3.16227 ========================== Run # 14 c=0.25000000000001 Iterations i = 31415925 ========================== Run # 15 c=0.250000000000001 Iterations i = 99345881 Ratio to last = 3.162277 ========================== Run # 16 c=0.2500000000000001 Iterations i = 314159263 ========================== Run # 17 c=0.25000000000000001 Iterations i = 993458825 Ratio to last = 3.1622776

[@estevebyt]

Probably this video is more interesting than the "Pi and Mandelbrot" one :)

[@michaelbauers8800]

+Esteve Boix I can't speak for the site, but I think they want to keep to the basics in the main videos, and then add the details in the supporting video. That way people can get the main idea quickly. It also gives them more hits, possibly, with more videos ;)

[@ryanedwards5545]

Esteve Boix 1 put context. steps to 2.

[@SchuldinerLA]

This is fascinating, great video! Do you have any link to some article that could explain it in a more detalied/mathematically rigorous way?

[@aryanarora7046]

+SchuldinerLA if u are really interested in the article u can google it yourself

[@citizenphnix]

+Aryan Arora So helpful of you.

[@aryanarora7046]

citizenphnix lol i try

[@SchuldinerLA]

+Bob Stein thank you! I found it while looking on Google (I did it on my own lol) but maybe it requires a level of knowledge I don't own yet. I'll try anyway.

[@SchuldinerLA]

+Vedvart Hmmm seems very interesting, I'll read it properly, thank you very much! :D

[@amaysaxena2]

quick suggestion Brady, maybe you could put a link in the descriptions of each of your videos to papers/articles describing the subject in more mathematical detail for the interested viewer. thanks for the wonderful content!

[@PeterAbt]

is there like an extra extra footage of the interview? :0 :D

[@tanyekai1975]

Brilliant

[@monkeybusiness673]

It's nit "Pi-ness" in the Mandelbrot set! It's "Pi-ety" of course! :-D What a Pi-ous set.

[@thermotronica]

Shes an amazing teacher

[@Pe6ek]

She's.

[@gaabinubatrafinulifilit122]

I am in love

[@bsuperbrain]

I love you.

[@guilemaigre14]

definitely want to know more about it.

[@IsaacLevy]

Could you share a link to a mathematically rigorous explanation of this?

[@zesalesjt7797]

I'm not sure who will see this or if it even matters. However, I didn't get the answer directly from the video, but I realized towards the end that you can measure any line (all straight and curves) with a circle. The circle traveling the line would use pi as it's unit of measure. Fractional pi units would equate to circle sections. However, in the case of the Mandelbrot set, the approximation is a multiple of pi | x *pi in order to get the exact value at the limit x would need to equal infinity.

[@Cernoise]

This is neat! Also, there's something about Holly's accent or voice or mannerisms that reminds me of Marian Call somehow.

[@AwesomeCrackDealer]

I wonder how did they find this out!

[@ryanedwards5545]

Fuvity u look with novelty or inspiration, do work, see pattern then discern meaning as implication crystalization.

[@cbehopkins]

Presumably this then only works for base 10? Would we be as astounded by this if we worked in base 16/12 etc? Or are there other methods for those bases???

[@shankarsomayajula9310]

Pi is 3.141… in decimal 10 numbering system.

[@marvelous1358]

I like the way she laughs

[@mallong7532]

It would be nice to see it iterated in the set as a zoom.🕊

[@LemonadeMouthSomebod]

Every time there is a video featuring Dr Holly Krieger I fall in love again.

[@enlongchiou]

At x = 0.216988703 approach 1/ 3.141592653, at -0.5032 approach -1/2.718 on fractal.

[@ColHogan-bu2xq]

4:00 : _"Pi is measuring time."_ What the heck ?

[@optimalzeeeee]

I once saw a comment saying the Numberphile2 logo ought to be tau and I can't look at the logo without getting frustrated now

[@traviswaldorf]

You're measuring the period of that final tangent function at the end of the video it seems, that period is the connection to pi.

[@ChrisDjangoConcerts]

How does precisely does the graph at 6:20 'looks just like' the tangent function? Can someone please explain where to find this or how to see this?

[@Number3pt14]

look up tangent graph

[@ChrisDjangoConcerts]

+Number3pt14 but the tangent function looks different. It is never at an angle of zero degrees. How is this related to the tangent function?

[@nathanaelcase2783]

+Christiaan Mantz If you prefer, you can say it looks like the arctangent function. Anyway the arctangent function is just a reflection (about the line y=x) of the tangent function, so you could argue that they "look the same."

[@ChrisDjangoConcerts]

+Nathanael Case Thanks I guess that explains it then

[@apburner1]

Absolutely, positively the most beautiful mathematician ever,

[@danieledwin5754]

I would be grateful if you could explain the lick between sequences and functions in more depth with lots of examples (like the link between the sum, product and quotient rule for sequences and how it is used in the proof for the sum product and quotient rule with functions)

[@opinionsmayvary5009]

I concur.

[@euanthomas3423]

314 comments! Do I see pi sneaking in? Holly is a real star.

[@heruilin]

A much older result (Euler) relating squares and pi states that the infinite sum of the reciprocal of the square integers converges to pi squared over six.

[@zzasdfwas]

Can we go the other way around and use very precise values of pi to calculate the Mandelbrot set, or in this case the lambda map, more quickly?

[@noahchristensen3718]

(Spammed for attention) I wonder why the bleeding heart, a flower formed by natural processes, flourishes into an approximation of the mandelbrot set. It has the cardioid, and the first bit of the tail. I don't know whether the stem has nodes that correspond to other zeroes.

[@maxgoof8605]

Okay, if you think of it as tau/2, and relate to to trig, it makes some sense. But I still have to ask: If you did it in base 8, where the cusp is 0.2 octal, would the number of steps that 0.2000...001 give an approximation of tau/2 (or pi, if you insist) in octal rather than decimal?

[@sac12389]

If you wanted to replace

[@unvergebeneid]

I wonder if there are like millions of similar fancy calculations one could make that just lead to "some number" but this one happens to cancel out in such a way that you get π and therefore it's remarkable or if this actually reveals something about the underlying nature of who-knows-what.

[@Spractral]

Pi pops up all over the place

[@leftaroundabout]

I think you can actually see “pi and circles” in there more directly, by comparing the euclidean path length around the circle tangent to that parabola with the taxcab distance over the same path. I just wonder about the √2 factor you'd suspect from the zig-zag course...

[@paradoxica424]

If I can get access to the paper relating to this, I expect it to relate to either y = arctan(x)/pi or y = tan(pi*x)

[@kubastachu9860]

1. Came back here for the piness 2. Realised that I actually do not recall what gives the mandelbrot set this piness and got curious 3. Ended up repeatedly watching piness moment

[@brachypelmasmith]

What does it mean that "you must come in in a curve" at that one meeting point? So we do not change epsilon in linear way or something else?

[@michaelbauers8800]

For that case apparently you have to vary epsilon in a specific way. For the x=.25 case , you vary epsilon by powers of 10, like .1, .02, .001, etc. to see increasingly more digits of PI ( A simple Python program can be written to show this property - I had posted an example in the main video for base 2, but it could be changed to base 10 easily)

[@granthicks2030]

IANAM, but I think it may be this: When you're approaching the node at 1/4, you're approaching along the real number line, meaning that all your values for epsilon are real (there is no imaginary component). If you draw a line connecting them, it is unavoidably a straight line. The values of epsilon approaching the node between the two components, on the other hand, are plotted on the complex number plane; they approach the real line but never quite reach it. While it is possible to draw a straight line on the complex plane (its formula would involve a linear relationship between the real part and the coefficient of the imaginary part), for some reason such a line doesn't work for approximating pi. For that purpose you need to choose successive values of epsilon that lie along a curve on the complex plane. Or (IANAM) I could be entirely wrong.

[@kannix386]

i have one question: where did the base10 come from? it was pi * 10^n. but why 10?

[@Gehr96]

I still don't get it.

[@JayFe0]

+Gehr96 You're not the only one. It's quite a hard one to grasp. For me, this is one of those ones where I can kind of see parts of it but I'm missing some background that I'm sure would make it fall into place. Edit: On the plus side, I do understand a bit more about the Mandelbrot set that I didn't get before I watched this.

[@omp199]

Gehr96 It wasn't explained.

[@whiterottenrabbit]

+Gehr96 Me neither... Where's Numberphile3 to explain things further?

[@LuisManuelLealDias]

They keep talking about seeing the pie, but they never show the pie! I'm starving!

[@Xnoob545]

1:10 piness -2018 Numberphile2

[@vsm1456]

Pi definitely has to do something with parabola (quadratic function) because parabola is one of three possible curves you get by dissecting a cone with a plane (other two are circle and ellipse).

[@comfortat]

What is the practical application of the Mandelbrot set? What has this discovery helped us do?

[@user-jn3lg9um5e]

Isn't it possible to define a diferent set with almost the same rule (for example every iteration must be equal or less than 4) and get tau in the same way?

[@michaldvorak8377]

I wrote a simple algorithm in C# to compute this myself, i got this: e = 1, N =2 e=0.1, N = 8 e=0.01, N = 30 e=0.001, N = 97 e = 0.0001, N = 312 etc. with e and then other N's are: 991, 3140, 9933, 31414, 99344, 314157, 993457, 3141625, 9935818, 31430913... I see something close to PI each second term.. this is all weird

[@picknikbasket]

Ahh, that answers my question in the first video

[@DocBlob]

When she says that 'if you put the decimal point in the right place' , do you just put it after the 3, or is there some mathematical formula to decide what you divide the number by corresponding to the value of epsilon?

[@helloitsme7553]

so if you have a function f(x) , where x is the value outside the mandelbrot set, the irations before 2 is pi / √x

[@helloitsme7553]

* f(1/4+x)

[@EcceJack]

Okay, *NOW* I get it :)

[@garethdean6382]

Must... bake... fractal... pie...

[@dillon1012]

i code in lua, and it is fun to code the same processes that thy do in the videos

[@ThunderChunky101]

So many mathematicians have red hair...

[@ThunderChunky101]

***** ...?

[@ThunderChunky101]

*****​ You said England. Not Britain. It's Scotland (and Ireland... And Wales apparently)that's got a high percentage of redheads, not England. I live here! 

[@ThunderChunky101]

***** Funny you should say that. That's not the case. It's about the same.

[@ThunderChunky101]

*****​ ;) There's not much data on it to be fair. But go to Ireland or especially Scotland to see what i mean.

[@ThunderChunky101]

theRealPlaidRabbit​ I agree wholeheartedly! Haha. I'd say us chemists are more human, and that's saying something! 

[@mcexcrack]

From all these videos of Mandelbrot, the standards for my potential girlfriends have risen so much that I will probably die alone .-.

[@Shockszzbyyous]

I don't know whether to like or dislike this.

[@CosmiaNebula]

Marry number theory, which, according to Gauss, is the queen of mathematics.

[@kubastachu9860]

That's pretty low standard -- marry math, which is a queen of all sciences.

[@LeeOCGaming]

Why

[@roelin360]

I always roll my eyes and cringe at comments like these. This stuff is genuinely the bane of maths/science videos.

[@SunriseFireberry]

What is the relationship between the Mandelbrot Set and transcendental numbers in general? Or is there one at all, except for pi?

[@unvergebeneid]

+TimeAndChance Well, since the set of transcendental number is not countable, there can't be a "neat" way to approximate arbitrary ones, using the Mandelbrot set or not. Also, it's explicitly mentioned in the video that the relation to π is not accidental either.

[@gammaknife167]

LOOK AT THIS GRAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAPH *im staring at you really intently right now*

[@MaNu50k]

whos this beauty

[@UnimatrixOne]

😍 Holly Krieger 😍

[@christianerbgarten5057]

Is that the equation for estimateing coopers droop?

[@fiddlinmacx]

I whipped up a quick python script in SageMath and wow, do I need a faster processor ;-) I'd like to know if Dr. Krieger uses Sage at all. Awesome software. I'm not a programmer or a mathematician and yet I can test these theorems relatively easily and explore my own, all for free (as in beer and as in speech ;-)

[@rikuurufu5534]

"pi-ness"

[@michaelbauers8800]

I know technical people love technical words, but pi-ness is easy to understand and descriptive

[@zfloyd1627]

Does it work with the cusp -7/4?

[@SidneySilvaCarnavaleney]

Un brasileño, descubrió la "PI" el número absoluto como se hace, la fórmula que se utilizó, los medios para llegar a esta conclusión, ya que algunos pensadores de la época informó que la cifra era "inmutable", era un número "irracional" , una cifra que no se aceptan hacerse en fracciones, por ser "irracional", que es infinito, y no podría ser un número racional, es que Sidney Silva logró desentrañar este misterio de este número gigante; que hasta la fecha nunca se había estudiado para llegar a tal conclusión; se demuestra y soltar toda la "teoría", "teorema" y la "tesis" de la época, que declaró con sinceridad completa que es "cambiante", por tanto, acepta los cambios que es "racional", es compatible hará una fracción (2205 / 700), (3.15), que fue investigado y investigado ser fiable al 100% para los cálculos de las matemáticas, es finito, ya que es un número exacto y constante será una fracción; lanzar este desafío para los académicos (as) alumnos (as), Amigos (as) y compañeros conocidos (as) y para todos los que quieren derribar la "tesis" de Sidney Silva, en este gran descubrimiento de la serie de "PI".

[@maddyIncubus]

need to contact 3b1b man.. he would have explained the pi with a circle.. nice video.. as always

[@LudwigvanBeethoven2]

So if it takes infinite steps to escape from 1/4 and the fact that epsilon gives you a really large number i.e approximated pi without decimal point, Can i say the biggest number, i.e the infinity starts by 31415...?

[@1ucky022]

great, imo its the pi, its approximation to it at first, as you see island on mandelbrot converging at tha point of real-imaginary axis cross and when it reaches i=0 it reaches the pi exact value at infinity

[@Zafoshin]

There's a circle here, that's being approximated. For c=1/4, we get at x=1/2: 1/2 = (1/2)^2 + (1/2)^2 So this point is also the point of the circle centered at (0,0) with radius 1. Now why as this point is approached by this method the number of steps equals sth like pi is beyond me. Maybe if you take the recursive definition of the function for c+1/4 for n and limit that to infinity you can somehow get some equation that shares the geometric properties of a circle or sth...

[@MoreisMor]

Crushing too hard to learn anything here