Follow-Up: Finite Difference Method

[@polygonc4538]

I'm really glad that you're making videos quite regularly again! You have been inspiring me since even before I went to study Mathematics!

[@adheesh2secondsago630]

Yes this is amazing!

[@roygalaasen]

It has been missed!

[@clout123]

Two things I love here. I'd never heard of Finite Difference before, so that was amazing to learn about. But also, I love this idea of the scripted video and then the informal follow-ups. Always enjoy your videos, but this last few have been superb. Thank you!

[@theadamabrams]

15:33 Okay, I have a guess as to the next video's topic. In case I'm right and in case someone doesn't want to see this "spoiler", I'll put it after some dots: . . . . . . . . . . . . . . . . . . . Shamir Secret Sharing. It uses polynomials fitted to points, so it goes nicely with the previous video. And James did say "it's a secret", which might just be natural language but might be a clever hint.

[@VincentZalzal]

I really like the format of having a first video, then an unscripted feedback video that includes comments and responses. Very fun to watch!

[@valentintheil3587]

Cheers for the relaxed follow up! The new background looks great. I, for one, have never seen anything like this before.

[@princechisanga7717]

Thank you. I'm requesting for you to be making your videos more frequently Mr James I really enjoy them. You inspire me a lot.

[@j_blue6784]

We used FDM to solve PDE on the computer, can be used to simulate discrete membranes on a triagonal mesh for example. Good video 😁

[@SmileyMPV]

Gildbreath's conjecture is fascinating! Thanks for sharing that.

[@singingbanana]

I've added a link in the description for more information.

[@SmileyMPV]

@@singingbanana Haha, I already found that link referenced on wikipedia :P I'm surprised by how hard it seems to be to make a more general statement on which sequences have this property. I would have expected some randomized model of the set of primes would lead to a non-zero probability of this property.

[@PhilBagels]

@@singingbanana And if Gilbreath's Conjecture is true, it also demonstrates how there is no pattern to the primes, at least discernable by the Difference Method. Since every difference other than the first 1, will be an even number (which is provable and obvious, since all primes other than 2 are odd numbers, and the difference between two odd numbers is always even, and the difference between two even numbers is always even, and 1 is odd. Which means you never get to a constant row until you're down to a single number - which would always be 1, according to Gilbreath. And since you have to go all the way down to a single number, it doesn't give you any useful information about the next prime after the ones you started with.

[@kedarsharma487]

Proving Gilbreath's conjecture would require us to get to know some special pattern among primes. Thank you for such amazing work

[@daviddoran1682]

That would be amazing.

[@BlackSwanized]

I very much enjoy this format of scripted video with an unscripted follow-up, great job!

[@nekogod]

Production value of these videos is definitely up love em.

[@danielzitnik4247]

Hi James. I'm really happy to see you making more maths videos. You explain things very well with lots of enthusiasm and you teach me something new every time. Today a friend asked me a probability question and it stumped me. It doesn't seem hard, but I can't grasp it. I think lots of probability questions are easier than they appear, but we get lost in the language of it all. I'd love to see you make some videos on probabilities (if they interest you).

[@pegy6384]

I like your regular scripted videos, and now I'm *really* enjoying these unscripted feedback videos. It's a bit like having a reply video feature again. The new decoration is nice--is it just decorative, or is it also functional (sound dampening)? I hope the Minard and Turing posters have new homes, they're lovely as well. You have also found another formula here--you found a way to thank people for the nice comments without having to do any patting of your own back. Win-win on that.

[@singingbanana]

I miss video replies. Yup, the tiles are sound dampening.

[@deliciousrose]

Love these follow-up videos!

[@aliskprado]

Loved this follow-up video!

[@fmontanari]

Thank you so much for what you do, I love your videos!

[@KitagumaIgen]

As a student I heard that the possibility to fit any finite sequence with any number of polynomials was attributed to The Big CFG.

[@I.____.....__...__]

2:43 Exactly. The best you can do is find something that fits for the data you have available. There are plenty of ways multiple different sequences of numbers can start out the same for a while before diverging. If you only have the parts that are in common, there's no way to know which is the next. Remember when Derek did a Veritasium video about black-swan cognitive bias, where he asked people to figure out the pattern of three numbers, but it wasn't what they assumed? (On a side note, it's never okay for teachers to give students trick questions, there's very little tangible value in trying to trick students. 😒) 6:58 I had to do a double-take. 😂 9:43 Yup, that's what got me interested in primes in the first place. I thought of doing this to see if I could find a pattern, I figured there'd have to be _some_ sort of pattern, they just _can't_ be random. That's what got me interested in primes so many years ago after not caring for so long.

[@recklessroges]

This singingbanana maths club is great! Thank you James.

[@xxMrPHDxx]

The most beloved math professor on Numberphile (for me, at least)

[@lkjhfdszxcvbnm]

At last, but not least

[@Lovuschka]

Infinitesimal differences are very good explained in a book series on which chess endgame studies composer Richard Guy worked (who sadly died recently at 103 years old), among others: "Winning Ways for your Mathematical Plays"

[@singingbanana]

I met Richard Guy two years ago at a sprightly 101. He was amazing for 101. Super nice and total with it.

[@QuantumSpark]

Great Video! I noticed something on the past video on the pentagonal numbers that I wanted to comment on. When I worked the formula for the number of points on a n sided pentagon (just one pentagon) the formula was 5*(n-1) which means the very first pentagon that was just one point had actually zero points according to the formula. Of course that means the formula just breaks down on n=1 and it doesn't mean that 0=1, and it shows that sometimes it's our intuition breaking the formula rather than the other way around. Cheers!

[@taavimetso9178]

It is nice to see your videos again:)

[@theludvigmaxis1]

I would’ve love if you one day did a video on spectral methods. Very similar yet much more complicated in my opinion.

[@reidflemingworldstoughestm1394]

Awesome, love it. I first noticed the differential thing trying to work out a way to predict time requirements in an online building game... starting with a column of build times for the first few levels, then finding each subsequent layer of differences until a constant popped out, then working the algorithm backwards to finish the original sequence. Game like that are a goldmine of reverse engineering math.

[@topdog5252]

Nice to see the words finite difference and fully understand

[@Tvyghhkd2399]

You are the best in saying the poetry of mathematics

[@josephmelnick3446]

Before you even mentioned it, I was admiring your wall artwork with the yellow/gray/white squares... simple, pleasing, and reminiscent of many cellular automata. Simply pretty. If there's significance to that particular pattern, I'd like to know what it is.

[@rogerkearns8094]

Yes, I'm interested too.

[@darreljones8645]

The finite difference method only works if the unknown function happens to be a polynomial. Here's a beautiful example of where it doesn't work: If the function is f(x) = (2^x), then it will start 1, 2, 4, 8, 16, 32, 64, 128, ... If you take the differences, you'll get 1, 2, 4, 8, 16, 32, 64, 128, ..., which is exactly the sequence you started with.

[@vsm1456]

you can always go down to a row where there's only one number, and work out the polynomial anyway. so we can generalise it by saying that this method helps you find a polynomial that fits the sequence, no matter how big the sequence is or what was the formula to generate it (if there was one)

[@tomkerruish2982]

Conway and Guy in The Book of Numbers mention an extension of the method, which I'm pretty sure they labeled a difference fan. Essentially, if you have a sequence like n^k, then the left-hand diagonal behaves as (n-1)^k. Using it as a new top row, you can keep reducing the base by 1 with each iteration, until you arrive at a constant left-hand diagonal, which is simply a multiple of 1^k. (Please consult their book for more information; it's a really, really good book.) Edit: I've since seen that this is mentioned in the video's description.

[@BiggyJimbo]

Thank you for the brilliant videos.

[@ArmstrongSteve-n4s]

😊Thanks for the informative talk!

[@Galakyllz]

I've tried different rules for generating sequences and found some interesting things. For example, if you start with 0 and 1 in the sequence and only permit yourself to use -1 or 1 in your table with the goal of approaching 0 differences throughout the table, it turns into a series of ones. 0, 1, 1, 1, 1 _1, 0, 0, 0 __-1, 0, 0 ____1, 0 __ __-1

[@j.vonhogen9650]

I love the term "Pythag". I think I'm gonna use that instead of "Pythagorean Triple" from now on.

[@andrew17641]

So excited about my shout out!

[@pegy6384]

Dr. Grime--if Gilbreath's Conjecture is proven some day, do you think that would make an argument for 1 being considered a prime number again? It would make that table more elegant for the first row to start with a 1 also.

[@xenoalvis7866]

I don't think that 1 will ever be considered a prime since that would break the fundamental theorem of arithmetic

[@tomkerruish2982]

Unfortunately, the third row would then start with a 0.

[@Bibibosh]

id like to learn more about conway's game of life. i have watched many videos recent and past, but it still intrugues me to this day.

[@michaelmam1490]

Could this be used to approximate a non-polynomial function with polynomials?

[@manirs1569]

Dr James, Thanks for your efforts to share Maths knowledge and I watch most of your feeds, I came across a perfect Square series from a Sanskrit Poem which was composed in BC Era . This series is built using no 2 as constant and I would like to hear from you on this please.

[@joebrinson5040]

Please keep making videos.

[@ruben307]

video soon? Maybe it could be for the Tau day.

[@rogerkearns8094]

If not tau day, perhaps tau morrow. ;)

[@killymxi]

So, if Gilbreath's conjecture is true then it might give an infinite series equation that generates primes? I think it could be quite a compact way to define primes. The major issue is that we lost information about signs of differences in the table. Perhaps, looking into this discarded bit of information might be insightful. Or at least produce a pretty pattern...

[@alan2here]

The simplest formula that fits the data is the most likely one to be the one you wanted.

[@danielbarreiro8228]

Since you did a bit of a follow-up of a follow-up re: the British Flag, you might consider one about how Babbage's Difference Engine use of finite differences since the problem it was mainly meant to solve were tide tables and astronomical calendars which are not polynomial problems at all. The DiffEng was meant to be used mostly in the same way they used human computers to interpolate intermediate values from actual values calculated by mathematicians using the real formulas or from the best data they did have. How did they limit the error to less than the last digit printed in those tables? How many actual points needed to be really calculated so that the interpolations were reasonably accurate? Thanks for the videos.

[@singingbanana]

Good questions, I would love to learn more about the difference engine.

[@feralmath]

It is not necessary to regard a finite sequence as a candidate formula. Any dth degree polynomial is fully determined by d+1 numbers. If the next number on the lowest row is taken to be different than the constant, then you have just changed to a different polynomial with higher degree, d+2. Also worth noting that any d+1 distinct values may be used and never any more than d+1. e.g. f(0)=3, f(23)= 34 and f(-7) =44 is a fully determined unique quadratic. Add some other value in the mix and the only way you do not now have a cubic is if you happened on the singular value of the original quadratic for whatever index value you have chosen.

[@ultralowtuba007]

Thanks for these vids!! So cool to learn more about this. I was trying this on a sequence and came upon a difference pattern of 1, -1, 1, -1, 1, -1 and wondering what a pattern like that indicates...?

[@harrisonmorrow7172]

@singingbanana this is the first time I have understood the finite difference method... would you be able to do a video on Buchberger's algorithm?

[@haniamritdas4725]

Really brilliant! I also discovered this sequence-difference-algebra by investigating the power rule for derivatives from elementary calculus. I was simply recovering the ground mapped out by Newton-Gregory, but I did not know that for many years and investigated the territory as an uncharted expedition. When I did discover that history, it made sense of my ignorance. Hard to say if my personal "discovery" was subconscious plagiarism, since I did read Newton, or if it diminished my accomplishment? I had to laugh! It's a silly a question. We are all exploring the frontier together, generation after generation, and our various ways of understanding are all uniquely our own, as well as a heritage of our own legacy. Whatever that may be. Who knows? Not me!

[@dking7985]

If the first row of the prime difference table does always start with a one then the second row can only have a two or a zero. It would be interesting to work backward to see what that implies about row three, four and so on. Just for fun I am going to make a spread sheet to look how a two at the top (a twin prime) works it's way down the table.

[@vsm1456]

did you find anything interesting?

[@ingiford175]

@11:18 The 4th row: Should that start with a -1? because if its a 1, then the 3rd row should start with a 1 2

[@singingbanana]

Gilbreath's conjecture looks at the positive differences only (I say it around 10:20)

[@ingiford175]

@@singingbanana Thanks, missed you saying that it was abs value of differences. Finite Calculus and Continued Fractions are my current two playgrounds.

[@Life_42]

Great decoration!

[@josefmatos2938]

Ok - I have seen the decoration behind him in a few videos. What is the math behind it?

[@samrustan]

Good to see you back in the KZbin saddle

[@FatLingon]

I don't think I saw the previous video, but I have been toying around with differences of differences or differences...etc. I looked at x^1, x^2, x^3, x^4, x^5, x^6... etc and checked the differences for these, and then had x increase. And after some prime number factorization I found that levels of differences increased every time one had a number that was the previous leader multiple with a new unique prime. Lets start with 2, then 6 (2x3), then 30 (2x3x5), then 210 (2x3x5x7), then 2310 (2x3x5x7x11), and so on. For each of these numbers, the levels increased. So 6 would have same number of levels as 10 would, as both factors into 2 primes (2x3 and 2x5).

[@Juarqua]

Is there a similar formula for sin and cos functions, simplifying the Fourier transformation? I've come up with this question because of the next Corona wave which has already hit Germany and is going to hit more countries soon.

[@sarahferraro4641]

Look up Taylor series

[@Maninawig]

Love the mathelete letterman jacket

[@krotenschemel8558]

A secret video... hmmm... So, Emma Thorne recently revealed her secret project and turns out, she was cast in a Star Trek fan production. The way James phrased it, that's probably not gonna be his secret video project, however in my mind the USS Enigma is now a thing. With the experimantal Bananawarp Drive, which reaches non-transitive speeds, so it can always outpace another ship, while being slower than yet another. That of course is accomplished by juggling modular tachyon particles, which have been quantum entangled to different pairs in seperate time lines, which allows the warp drive to fluctuate it's otherwise attumed warpfield to altering temporal scenarios.

[@singingbanana]

Oh good. Then we can piggyback a tachyon signal on a delta wave.

[@adheesh2secondsago630]

I am assuming your next video would be something related to prime numbers.

[@onurkurum]

Finite element method next?

[@11Agamemnon235]

Hey! You’re still making videos! You sent me notes on representation theory once. Well I’m done with grad school now!

[@singingbanana]

Haha! Fantastic!

[@JavierSalcedoC]

I'd like to know what would be the next number for the series made with a limited number of prime numbers. How close they are from the target

[@xeuszzz]

Obviously, as a string of odd primes generates only even numbers (yes, zero is even), and hence except for the 1's and the line of odd primes all the numbers between will be even, and consequently any even number among the 1's will result in a string of odd numbers in between until next to a previous prime. This would mean the next "prime" would be even, and therefore any other number among the 1's has to be an odd number.

[@MichaelWarman]

I feel like I should recognise the redecoration; is it from the game of life? The grey section looks like a glider, but I'm not sure about the yellow. Plugging the yellow pattern into a game of life engine produced a pretty pattern that settled quite quickly into a two stage oscillating cluster, but I couldn't find a name for it

[@shruggzdastr8-facedclown]

Q&A @James: Is there any kind of thematic connection to the theorem being followed-up on in this video and the geometric art piece seen in the background?

[@singingbanana]

None at all! I considered making a mathematical pattern, but more importantly, I wanted 50% coverage on my wall so did a checkerboard pattern with no hidden meaning. And I had a few grey tiles left over so add them to make it a little asymmetric.

[@shruggzdastr8-facedclown]

I like it b/c it makes me think of Parker Square -- also reminds me of '60s/'70s Minimalist/Conceptual/Hard-Edged Geometric Art

[@josefmatos2938]

What is the math of the decoration behind you?

[@rogerkearns8094]

I think a similar problem pertains to the existence of a right cuboid having distinct integer length, breadth, height and diagonals including the internal diagonal.

[@singingbanana]

That's right. That's called an Euler brick. The difference with the flag theorem is that the diagonal is broken, rather than corner to corner.

[@rogerkearns8094]

@@singingbanana Oh, I see. Thanks. :)

[@addymant]

I should've asked this on the first video, but in stats we'll use differences for similar reasons, but sometimes we also use differences between every other point, or every third point, etc. (typically used to account for seasonality). I was wondering, is there was any 'classical' math analog for that kind of thing?

[@singingbanana]

Do you mean, is there a formula when the steps are not 1? There is, have a look in the description.

[@addymant]

@@singingbanana Exactly! Thank you!

[@SuperYoonHo]

great vid_

[@pjaj43]

Sorry for a "me too" comment, but I also stumbled upon part of this method, the difference table, not constructing the polynomial, years ago when we were given IQ tests at school which contained the question of the type "What is the next number in the sequence 4, 6,12,16,?"

[@singingbanana]

That's cool. I like hearing about it.

[@bwroth355]

Question not related to this video. You did, or maybe it was Brady did a video on getting a yatzee on 6 sided dice. Ive been playing DnD and I'm curious how I would even set up the problem to figure out the odds of having all the dice land on the same number. the dice are 4 sided die, 6, 8, 10, 12, 20

[@singingbanana]

Is this what you mean: Probability they all land on 1 = (1/4)(1/6)(1/8)(1/10)(1/12)(1/20) = 1/460800; Probability they all land on 2 = 1/460800; Probability they all land on 3 = 1/460800; Probability they all land on 4 = 1/460800; So probability they all land on the same number = 1/460800 + 1/460800 + 1/460800 + 1/460800 = 1/115200

[@bwroth355]

@@singingbanana yes that's exactly what I was asking. Super helpful thank you!

[@KevinBerstene]

Hey James, something popped up here again that I found a while ago but haven't been able to get a satisfying answer to-I feel like the differences of differences method might have some secret to it but I can't quite figure it out. The differences of the sequence of square numbers (0, 1, 4, 9, 16...) is a list of the odd numbers! The way I had found it was the sum of the first N odd numbers is equal to N^2. I feel like there's got to be some derivative or something relationship here but I can't quite put my finger on it...

[@pegy6384]

Have you seen James' video with Matt Parker on the differences of squares? It's another good one to watch if you haven't seen that yet. kzbin.info/www/bejne/gpysfGucad6phbc

[@singingbanana]

Good noticing. You're right. Here's a good answer, but in particular the visual answer on this page is a nice way to understand it math.stackexchange.com/questions/136/why-are-the-differences-between-consecutive-squares-equal-to-the-sequence-of-odd

[@rmsgrey]

Rather than trying to draw ASCII art in a proportional font, here's an algebraic answer: The difference between two consecutive squares is always going to be (n+1)^2 - n^2. Expand that out and you get n^2+n+n+1 - n^2 = 2n+1. Since 2n+1 is the general formula for odd numbers, all the differences will be odd numbers, and since n can be any integer, you'll get every odd number.

[@KevinBerstene]

@@singingbanana Wow, that is exactly what my brain needed to finish putting this together! Adding a "side" to each square means that you have to increase the differences by +1 for each side and an additional 1 for the common corner. One of the other answers in that thread actually talks about the differences that this video is about and they helped me realize something else with this-the generalization is actually that the final finite difference between n^k and n^(k+1) is really just going to work down to be (k!), and the odd numbers are really just falling out of that relationship. It could have been the even numbers as easily (except that it starts at 1, obviously) but the important thing is that the differences for n^2 = 2! = 2. It's funny because that really means that differences of n^1 is...1. which is just counting.

[@singingbanana]

Nice!

[@jesusthroughmary]

The idea that unscripted would make things more relaxed and less stressful is strange to me

[@Apocalymon]

You need to set up a PayPal, cash app account so we can tip you. Also, have you read the Math Book by Clifford Pickover? There's lots of wonderful popular math books which you might love to read. Books that focus on female mathematicians, early mathematics of non-western cultures, etc.

[@singingbanana]

I haven't read that book, I'll have to look into it. Thanks.

[@francotomatillo]

Neato

[@Beatchim]

Nice! :)

[@fcolecumberri]

When a video is so unscripted you don't even finish with "if you are hapin thank for watching".

[@pegy6384]

The regular sign off is "If you have been, thanks for watching." Different video styles get different sign-offs, kind of a nice distinction.

[@williamgarnett1677]

I teach this to my Pre-Calculus students

[@phs125]

It was a terrible idea for me to try and do finite difference method on Fibonacci sequence...

[@mathsciencefancier]

Hello

[@trexpaddock]

The first time you said, "F of" I thought I heard you say something else. XD

[@mrscooby6330]

Tom Hiddleston Lite

[@frankharr9466]

Hm. This is the topic that keeps giving.

[@supermaster2012]

I watched the first few seconds without audio and I was 100% expecting an "achilles and the tortoise" clap loop lmao

[@twiddle7125]

Is it just my speakers or does it sound like he's talking through a tin can? lol

[@singingbanana]

I've checked all my devices/apps and it's all fine here. Maybe it's a KZbin problem.

[@kirkoconnell]

First post!

[@babthecookieman]

FIRST

[@babthecookieman]

I AM NEITHER PRIME NOR COMPOSITE I AM A GYPSY OF MATH

[@jamesburke2094]

If accountants were better at advanced maths, lots of financial catastrophes and misleading financial reporting would be avoided

[@gasun1274]

accountants deal with arithmetic. when it comes to high level math arithmetic isn't that important. i would argue people with advanced math training make worse accountants. math isn't clerkwork or bookkeeping.

[@flash5000tzr]

First!

[@rvsamsung77]

First

[@RobertMStahl]

Turing's wet dream?

[@HasekuraIsuna]

Hearing so much about differentials and differences reminds me of a chapter I never got in highschool. I think it was called differential equations and they were in the form like something = y-y' Still to this day I have no idea what you would solve by taking the difference between the value and its derivative. And even having watched all Numberphile and 3b1b videos, I can't remember ever seeing anything In that form... And with the number of topics on those channels you'd think something would use it! My friend said he uses it in electronics, but couldn't really explain what it represented for a layman like me.

[@singingbanana]

That's right, those are differential equations. en.wikipedia.org/wiki/Ordinary_differential_equation