I love how you guys constantly interview people who have a genuine love of mathematics! It makes these videos so amazing.
@jmp01a245 жыл бұрын
yes they just go around Europe asking random people in the streets.
@jedizombiekiller90653 жыл бұрын
I will never forget klein bottle man
@zaneroy12553 жыл бұрын
instablaster
@Triantalex10 ай бұрын
??.
@saigonesepapaya6 ай бұрын
same!!! love this channel sm
@ImAllInNow7 жыл бұрын
Want another pascal's triangle pattern? Take the top right diagonal (1, 1, 1, ...) and treat it as if it was a decimal number with the decimal point between the first two numbers (1.11111...). Now square that number. You get 1.234567... which equals the second diagonal. Cube it and you get 1.3717421... Which is the third diagonal (you have to smush the numbers together the same was as 1,5,10,10,5,1 = 161,051). Now this works in bases other than 10. If you treat 1.11111 as a binary number it equals 2 (same way 1.9999... = 2 in decimal). Now 2^2 = 4. So we know that 1.234567... in binary equals 4. Written as an infinite sequence, 1.23456 in binary is 1 + 2/2 + 3/4 + 4/8 + 5/16 + 6/32 + ... which indeed equals 4! This pattern even has a connection to another Numberphile video, Grafting Numbers, and it's truly remarkable, although this comment section is too small to contain it.
@geri304 жыл бұрын
Lol, fermat’s last theorem reference. I like it.
@michaelhoefler51184 жыл бұрын
Jeo okay fermat
@force10guy263 жыл бұрын
It'd probably all fit now a days lol
@ExplosiveBrohoof7 жыл бұрын
For anyone interested in why these patterns emerge: *Pascal's Triangle encodes n choose k:* This arises from the fact that _C(n, k) = C(n-1, k-1) + C(n-1, k)._ This can be solved algebraically (note that _C(n, k) = n!/(k!(n-k)!)),_ but there's a combinatorial argument to it as well. Say you have _n_ different ice cream flavors and you want to choose _k_ of them for your super tall ice cream cone. Then you can consider two distinct cases: the combinations with chocolate, and the combinations without chocolate. If you include chocolate, then you have _n-1_ flavors left and you need to choose the remaining _k-1_ flavors. If you don't include chocolate, then you still have _n-1_ flavors, but you still have _k_ flavors to choose from as well. Hence, when you add them together, you should get the total _C(n, k)_ combinations. Because of this identity, you can inductively show that Pascal's Triangle encodes _n_ choose _k._ *The rows are consecutive powers of 2:* Remember that if you want to choose _k_ objects from _n_ items, you go down to the _n_ -th row in the triangle and you go over by _k._ This means that the _n_ -th row will be the numbers _C(n, 0), C(n, 1), C(n, 2), ..., C(n, n)._ Now, consider the total number of ways to choose _n_ objects, regardless of the number of items you choose. This will be the sum of all of the cases where you choose _i_ objects for _0
@Maharani19917 жыл бұрын
+
@GABRIELFILMSTUDIOS7 жыл бұрын
Arbitrary Renaissance Thanks! I'm going to try to come up with an explanation myself for the ones I did not know before actually reading your comment though.
@ExplosiveBrohoof7 жыл бұрын
When I say it takes a bit of complex induction to work out the Fibonacci pattern, I just mean that it's tough to rigorously describe. The intuition is really easy like you said: the first diagonal grabs all the left sums and the second diagonal grabs all the right sums. Regarding Fermat, that's really interesting! Thanks for taking the time to share your discoveries.
@breathless7927 жыл бұрын
I found a simpler way to describe it: Consider a string of ones and zeros as both a Pascals triangle row (in mod 2) and as a binary number. Given that take the following string 1000…0001 (between the 1’s on either side are all zeros) Each 1 creates its own mini triangle due to the zeros on either side (this partly explains Siepinski’s triangle) at a certain point the two sub triangles will meet, they will meet when the both rows are all the 1’s (i.e. like 1111….111) Proof: If you consider a row of all 1’s, 1+1=0 (mod 2) so all will be zeros in the next row apart from extra 1’s placed at either side I.e it creates 1000…0001 so the previous row to 1000…0001 will be all 1’s with 1 less of them. Now the row with the 1’s can be called row X (which means it has X terms) now if you take each sub triangle to that row (the 1000…0001 is row 1 of the mini triangle) so at this point you will at row 2X so 2X terms so when you put the two row X’s together there are no 0’s in between) Now that you’ve done 2X rows you have all 1’s again so it creates another 1000…0001 Now as binary numbers each row can be factorised as the sub triangle row (as a binary number) multiplied by the 1000…0001 and that represents multiplying that number by all previous rows Finally to prove that the numbers you multiply each past term are Fermat numbers Proof: First the number is in the form 1000…0001 (in binary) which means it’s 1 more than a power of 2 (which in binary is 1000….000) Second you take the “111…111” number and duplicate it so the number of 1’s is doubled and if you add 1 it becomes a power of 2 with the same number as zeros as the number that had 1 added had 1’s, add 1 again and it has the same number of digits, and adding the first 1 increased it digits by 1 so if the 111…111 was row X then 1000…0001 (the number obtained by adding 2) it the next row of the Triangle and the number in the middle(not the triangle the value since there is difference is 2) is a power of 2 with a number of 0’s as explained above Since you started with 1 on its own (at the top of the triangle) and each time you doubled it (1,11,1111…etc) it means that the number of 0’s in the power of two (and therefore the power) is a power of 2 so it is in the form 2^(2^N) since this number has one added to create the number you multiply by, its (2^(2^N))+1 which is the definition of a Fermat number
@hyh20126 жыл бұрын
Another way to prove the Hockey Stick Theorem: The corresponding identity is: C(n,n) + C(n+1,n) + ... + C(n+k,n) = C(n+k+1,n+1) (n,k∈N) (The other direction is just symmetry of Pascal's triangle C(n,r)=C(n,n-r)) The identity can be proved using a combinatorics argument: Number of ways to choose n + 1 items from a set of n + k + 1 elements: C(n+k+1,n+1) But this can be computed in another way by considering cases: including element 1 - C(n+k, n) excluding element 1, including element 2 - C(n+k-1, n) excluding elements 1, 2 including element 3 - C(n+k-2, n) ... Keep going with this pattern until: excluding elements 1, 2, ..., k, including element k + 1 - C(n, n) If keep going excluding more than k elements then it is impossible to choose n + 1 from n + k +1 elements. So we have covered all cases here. Equating the 2 methods of computing number of ways to choose n + 1 items from a set of n + k + 1 elements gives the identity.
@3zehnutters7 жыл бұрын
You can verfy if a number n is prime by looking in the n-th row and checking if every number(beside the 1´s) in that line is 0 modulo n.
@AnastasisGrammenos7 жыл бұрын
really? wow
@Fightclub19957 жыл бұрын
So n is a prime factor of every number in the nth row (except the ones). Cool pattern.
@ZardoDhieldor7 жыл бұрын
Pascal's triangle never stops amazing me.
@3zehnutters7 жыл бұрын
i was super excited when i first discovered this pattern myself
@nuclearcoconut36647 жыл бұрын
That makes me wonder...is testing pascal's triangle as efficient as simply testing factors?
@AndersHass7 жыл бұрын
00:50 "We can go on as long as we want and for me it is when the first double digit shows up because that is when adding gets hard." lol
@jenecomprends7 жыл бұрын
Please ramble on more about this triangle!
@justinward36797 жыл бұрын
Did you know that if you put a point at every edge of Pascal's triangle you get a triangle?
@PhilBagels7 жыл бұрын
The first diagonal is all 1's. The second diagonal is the natural numbers. The third diagonal is the triangular numbers. The fourth diagonal is the tetrahedral numbers. The fifth diagonal is the pentachoronal numbers. etc. The nth diagonal is the n-1-dimensional simplex numbers.
@ZardoDhieldor7 жыл бұрын
Except you don't, because it's infinite and only has one corner and two edges.
@danielketterer96837 жыл бұрын
jenecomprends exactly. more like please don't
@TheWeepingCorpse7 жыл бұрын
Zardo Dhieldor an edge if defined by two vectors, so maybe its a vector and two rays?
@Xeroxias7 жыл бұрын
So cool to see an undergrad on numberphile
@khayanjumbe14623 жыл бұрын
I used to watch you guys when I was like 6 or 7 and now I can finally understand what you are talking about
@ravneetsingh14997 жыл бұрын
Fibonacci numbers in Pascal triangle..... JUST AWESOME
@TheWolf55757 жыл бұрын
My triangle is amazing huh?
@justinward36797 жыл бұрын
You didn't discover it you thief!
@TheWolf55757 жыл бұрын
Justin Ward but it has my name :c
@aadfg07 жыл бұрын
You stole the triangle from Chinese mathematicians. Thief!
@QualityContentX7 жыл бұрын
yee
@IceMetalPunk7 жыл бұрын
New conjecture: Pascal's triangle encodes literally everything that can be encoded.
@angelmendez-rivera3515 жыл бұрын
IceMetalPunk Sure it can. Since Pascal's triangle is of order Aleph(0), the set of things it encodes is of the size of the set of the real numbers, Aleph(1). So at the very least, it must encode almost everything.
@samuelking16243 жыл бұрын
IceMetalPunk's conjecture: "Pascal's triangle encodes literally everything that can be encoded." would be very funny to see in a paper. You should try to proove.
@ingenuity232 жыл бұрын
@@samuelking1624 a hundred years later, its a millenium prize problem, and we all were here to witness history right before our eyes
@alonkellner53757 жыл бұрын
The nth row in the Pascal triangle discribes the geometrical properties of an n dimensional simplex, for instance, the 3D simplex is a tetrahedron, a tetrahedron has 4 vertices, 6 edges, and 4 faces, which is the third row. (**to get the full row count 1 zero dimensional shape, or an 'empty' shape, and 1 3D cell, that gets you to 1,4,6,4,1)
@thanhgoal87117 жыл бұрын
I love it when mathematicians can't hide the hype while talking about numbers!!!! All of this is adorable!!!! XD
@Maharani19917 жыл бұрын
One of my favourite Numberphile videos in a long time. I knew Pascal's triangle, but I had no idea it showcased this many math phenomenons. Sierpiński's triangle is what blew my mind the most.
@Eltro9207 жыл бұрын
Finally, a video about Pascal's triangle, and it's amazing.
@VirtualMarmalade7 жыл бұрын
This is like magic. I didn't realize Pascal's Triangle had so much in it!
@TheWolf55757 жыл бұрын
Its magic :D
@gdash69255 жыл бұрын
This is made by 1st grader additioning
@NeemeVaino7 жыл бұрын
Reminds a childhood discovery - To calculate 5th row for example, without adding up previous ones, just multiply 1 * 5/1 * 4/2 * 3/3 * 2/4 * 1/5. Note the numerator decreasing while denominator increases. So generally, k-th number in the n-th row is n!/(k!(n-k)!)
@yosefmacgruber19205 жыл бұрын
We know that the 1st number in the 5th row is 1. So let's try and see if that is what your formula produces. 5!/(1!(4!)) = 5. Oops. 5 ≠ 1. Are you considering the 1 to be in the k=0 position and the 5 to be the k=1 position? It also appears that you are considering the top of the triangle to be of row n=0?
@angelmendez-rivera3515 жыл бұрын
Yosef MacGruber Yes. That is actually the definition of the Pascal triangle. The topmost row is the 0th row and the leftmost diagonal is the 0th diagonal. The adding recursion is not part of the definition, that's just an easy construction that works due to the properties that choose(n, k) as a function satisfies.
@angelmendez-rivera3515 жыл бұрын
Yosef MacGruber Therefore, your counterargument is incorrect.
@gavinmann41524 жыл бұрын
12:15 'there are still other things i can ramble on about....' Me: TELL ME NOWWWW
@UMosNyu7 жыл бұрын
6:26 is my favorite thing. All the mods. Had a great maths teacher who showed this pattern to us.
@austinbryan67595 жыл бұрын
I think it's pretty cool how, with Pascal mod 2, you can see how adding odds and evens work. 0 + 0 = 0 1 + 0 = 1 1 + 1 = 0 Which maps to the classic rules of adding evens and odds. What's more remarkable is that it also follows adding in single digits in binary. 1 + 1 = 10 in binary but if you make the results only the ones place then it's 0
@RodeyMcG7 жыл бұрын
11:09 Brady's little exhale when he gets what's going on is the exact same reaction I made :D Very cool!
@gdsu2347 жыл бұрын
I absolutely love this channel. Everytime the poorly thought out school curriculum kills my interest in math, one of these videos fires it up again. Thanks guys!
@adamloepker8057Ай бұрын
You explained that better than an hour of lecture in 2 minutes, well done!
@pancakeparliament7 жыл бұрын
im so happy numberphile finally did a video on pascal's triangle
@seanehle83237 жыл бұрын
I love her passion and energy.
@pyotrleflegin72557 жыл бұрын
Thank you -- your enthusiasm is infectious!
@VernePhilleas5 жыл бұрын
I love Casandra Monroe's zeal for mathematics. This video was inspiring and enriching. I love Pascal's Triangle and all the mathematical intricacies it reveals!
@NickKravitz7 жыл бұрын
Around the 8th grade in US we learn Pascal's triangle to solve n choose k problems. In high school we learn the short cut formula for each entry: n choose k equals n! / (k!(n-k!)). In university we learned binomial formula containing the n choose k term. This is by far the most practical use for Pascal's triangle.
@nathanisbored7 жыл бұрын
i did not learn pascal's triangle in 8th grade in california, or ever in school for that matter. i also never learned the choose function in school, definitely not high school. i did learn binomial theorem in calculus 2 later on tho
@tupactheory37397 жыл бұрын
I first was introduced to Pascal's Triangle in the third grade... when I was eight.
@anthonyderose64437 жыл бұрын
Nick Kravitz I learned Pascal's triangle and binomial theorem as a junior in HS to expand expressions. I always chose to use pascals in my work. Never understood binomial theorem.
@RichardDominguezTheMagicIsReal7 жыл бұрын
honestly what I really like it how excited she is to explain the triangle, her enthusiasm had me interested
@aosteklov7 жыл бұрын
she is so cool
@yvesnyfelerph.d.82975 жыл бұрын
Are you ok?
@Fer3005924 жыл бұрын
@@yvesnyfelerph.d.8297 yes
@torlachrush Жыл бұрын
Casandra Monroe's enthusiasm is inspiring.
@imveryangryitsnotbutter7 жыл бұрын
A more illustrative way to write out the part at 4:47 would be: 100,000 050,000 010,000 001,000 000,050 000,001 ------------- 161,051
@sofia.eris.bauhaus7 жыл бұрын
i once made up a "simplex operator": i generalized triangle numbers for any dimension: (triangle (2-simplex), tetrahedron (3-simplex), pentachoron (4-simplex)). the first operand was the edge length and the second operand was the number of dimension. x △ 2 was a triangle number for edge length x and so on. as i wrote down the Cayley table (like a multiplication table) i noticed quite a bit of a pattern. suddenly it struck me that i was writing down Pascal's triangle sideways. this was one of the coolest things that ocurred to me. :)
@Zeezjay7 жыл бұрын
what if you extended the triangle to a tetrahedron or any simplex of n dimensions?
@Sejiko7 жыл бұрын
wow this is mind blowing stuff and i have some ideas.... thank you man.
@connorp30307 жыл бұрын
I've heard they've done that for a trinomial distribution pattern, pretty neato
@Sejiko7 жыл бұрын
Imagine a programm that represent this pattern as tetrahedron in a 3d world and highlight some numbers.
@abrasivepaste7 жыл бұрын
Look up Pascal's Pyramid
@wellme33677 жыл бұрын
You get the multinomial coefficients.
@clover73597 жыл бұрын
If you go down the diagonals, you get n dimensional triangle numbers. For example, 0 dimensional triangle numbers are 1, 1, 1, 1, ... etc 1 dimensional triangle numbers would be 1, 2, 3, 4, 5, 6, 7, etc 2 dimensional triangle numbers would be 1, 3, 6, 10, 15, 21, 28, 36, etc Then the 3 dimensional triangle (pyramidal) numbers would be 1, 4, 10, 20, 35, 56, etc and so on. You can keep going in higher dimensions if you just keep going along the next diagonal.
@yosefmacgruber19205 жыл бұрын
What are dimensional triangle numbers?
@estelonb.j.eastham31957 жыл бұрын
When I saw the Fibonacci Sequence in that I actually said, "It's so beautiful!"
@pythagorasaurusrex98537 жыл бұрын
Great stuff! I knew most of the properties from the beginning of the video. But I NEVER heard about the fact that Fibonacci's row is encoded in the Pascal triangle. Mind blowing!!
@TakeWalker7 жыл бұрын
She's wearing an Autobot shirt, she is immediately the best guest you've ever had. :D
@robbes7rh2 жыл бұрын
Uncanny and completely amazing that what at first blush feels like a children's exercise contains all these other relationships and qualities,
@martinepstein98267 жыл бұрын
Here's how I heard about the relation between Pascal's triangle and Fibonacci numbers. Problem: You're climbing a ladder n rungs high and you always have a choice of climbing 1 or 2 rungs at a time (let's call these actions single steps and double steps). How many ways are there to climb the ladder? One approach to the problem is to use combinatorics: first find the number of ways to climb if you never take a double step, then the number of ways if you only take 1 double step, then 2, etc and then add it all up. This sum corresponds to adding terms along a shallow diagonal of Pascal's triangle. A second approach is to use recurrence: to climb n rungs you must first climb n-1 rungs and then take a single step OR climb n-2 rungs and then take a double step. So if f(n) is the number of ways to climb n rungs then we have f(n) = f(n-1) + f(n-2) which gives us the Fibonacci sequence.
@martinepstein98267 жыл бұрын
I remember reading that this problem was first solved by composers of Indian classical music who wanted to know how many ways there are to, say, fill a bar of music with 8th and 16th notes. Not in those western terms of course.
@dlee6457 жыл бұрын
Fascinating presentation. Do more with Casandra.
@enlongchiou7 жыл бұрын
One more interesting thing for Pascal's triangle of (x-1)^n, change sign from 3rd term on at every row turn out is trivial zero of zeta function at s=1, -2 = 1 -2 - 1, -4 = 1 - 3 -3 +1, -6 = 1 - 4 - 6 + 4 - 1, -8 = 1 - 5 - 10 + 10 - 5 + 1,etc...-2n, .nontrivial zero which are extension of trivial zero obey same rule have 2^n series, (2*n)!/(n!)^2 of moment of nontrivial zero 1,2,6,20.. right at middle line of triangle, let bottom of triangle from 0 to 1 at x-axis, 1,2,6,20.. right at x = 1/2 line as Riemann hypothesis predicted.(Euler product of (p-1)/p is 0.04875 from 2 to 99991, take 2^9632 - 1 of mod(10^10,po)/po get 34490000, 0.04875*10^10/34490000 =14.13 po is all possible combination from 2 to 99991, 2nd 0 ,21.02 = 487500000/23190000 get without 2., 487500000/19500000 = 25 3rd zero of zeta function without 2 and 3, so on...25*(1/2)(2/3)(4/5) + 1/2 - 1/6 - 5/10 + 25/30 + 1/3 - 10/15 +0/5.+3 -1 = 9 prime number counting until 25.)
@anon88577 жыл бұрын
hey enlong chiou! i remember you from g+ ! ur awesome!!
@camilogallardo43387 жыл бұрын
enlong chiou give us the nontrivial zeros!
@Israel2.3.27 жыл бұрын
A favorite of mine. If this is confusing just google 'Faulhaber's formula.' Let x/(exp(x) - 1) = a + (b/1!)x + (c/2!)x^2 + (d/3!)x^3 + (e/4!)x^4 + (f/5!)x^5 + ... [and let mx^n = (m)*(x^n)] Then 1^0 + 2^0 + 3^0 + ... + n^0 = (1/1)(an) 1^1 + 2^1 + 3^1 + ... + n^1 = (1/2)(an^2 - 2bn) 1^2 + 2^2 + 3^2 + ... + n^2 = (1/3)(an^3 - 3bn^2 + 3cn) 1^3 + 2^3 + 3^3 + ... + n^3 = (1/4)(an^4 - 4bn^3 + 6cn^2 - 4dn) 1^4 + 2^4 + 3^4 + ... + n^4 = (1/5)(an^5 - 5bn^4 + 10cn^3 - 10dn^2 + 5en) 1^5 + 2^5 + 3^5 + ... + n^5 = (1/6)(an^6 - 6bn^5 + 15cn^4 - 20dn^3 + 15en^2 - 6fn) etc. Notice the 'truncated' version of Pascal's triangle.
@iustinianconstantinescu54987 жыл бұрын
enlong chiou Amazing!!!
@slipknnnot7 жыл бұрын
Says the anonymous guy called "Fish Bones"
@purplewarrior127 жыл бұрын
Definitely my favorite Numberphile!
@llamawaffles55597 жыл бұрын
Alright, i have a question if anyone is still reading the comments on this video. I was doing a problem for my Intro Quantum Mechanics class about spin-1/2 particles, and after doing a bit of math, ended up getting what is essentially a 3d square pyramid of numbers (idk if that's right, its 4 triangles, each making a side), except there is no bottom, it just goes on indefinitely as far as you want to extend it (like pascal's triangle). In fact, my professor noticed that the outer triangle of each side of the pyramid IS Pascal's triangle. Which made me curious if there was some overarching recursion relation (or other relation) to predict future rows/squares of the pyramid. If you look at one of the outside triangles, then remove it and look at the triangle beneath it, and continue doing this, this is what you find: Triangle 1: 1, 1 1, 1 2 1, 1 3 3 1, 1 4 6 4 1, 1 5 10 10 5 1, etc (Pascals Triangle) Triangle 2: 0, 1 1, 4 0 4, 9 2 2 9, 16 10 0 10 16, 25 27 5 5 27 25, 36 56 28 0 28 56 36, etc (what pattern?) Triangle 3: 4, 4 4, 1 12 1, 1 15 15 1, 16 8 40 8 16, 64 0 56 56 0 64, etc (my computer cant compute any more) Triangle 4: 0, 9 9, 36 0 36, 64 24 24 64, etc (computer cant do anything layer 10 or below) Triangle 5: 36, 36 36, etc That is all my computer can do, but as soon as you get to layer 10 of the pyramid, i go beyond the Integer limit in C# and i haven't fixed that problem yet, so my computer just gives me either 64 or null. weird bug, but yeah. if you can figure out some pattern, that would be awesome. And for anyone wondering what these numbers are here is the slightly longer story: Each layer, or square, of the pyramid corresponds to the spin of a particle, starting with zero. row zero is a spin-0 particle, row 1 is a spin-1/2 particle, row 2 is a spin-1 particle, row 3 a spin-3/2 particle etc. and the numbers each column/row correspond to the probability that the spin will be measured at that magnitude in a direction orthogonal to the currently known spin (the closes thing Quantum Mechanically to "random"). for example, row 3, column 5, in layer 7, of this pyramid corresponds to the probability that a spin-7/2 particle measured to have spin-5/2 in some direction will be measured to have spin-3/2 in an orthogonal direction. Also the probability that a spin-7/2 particle with spin 3/2 will be measured to have spin-5/2 in an orthogonal direction. However, it is easy to see that the rows do not add to unity, and that is because i have removed a normalization constant to make them all whole numbers. the normalization constant for the n-th layer is simply 2^(-n). Any more questions i would be happy to answer.
@htmlguy886 жыл бұрын
you could download pari gp and try it with high precision. though you may be relating it to pascal's simplex. en.wikipedia.org/wiki/Pascal%27s_simplex
@PatrickChavez7 жыл бұрын
Cassandra, well done. love the topic, loved the delivery!
@leobekayombo80877 жыл бұрын
Smart mathematician doing beautiful stuff!!!
@samgilson74767 жыл бұрын
My friend and I discovered the Serpenski triangle that's hidden inside in a bit of a different way. Instead of applying mod2, we found the absolute value of the difference of the two numbers above. In the same way that you add to produce Pascal's Triangle, we subtracted to create this Difference Triangle, and it turned out to be identical to the mod2 serpenski fractal thing.
@joeytje507 жыл бұрын
9:00 So... the primes' locations on the Pascal triangle are located exactly on the Fibonacci sequence's locations, ignoring 1. They occur at locations 2, then 3, then 5, and the next one (not imaged) would occur at the 8th row. So that's another way to get Fibonacci from Pascal's triangle.
@JeSuisUnKikoolol7 жыл бұрын
The 8th row is 255 and is not prime. The next prime (257) is the 9th row
@ImAllInNow7 жыл бұрын
yeah, the primes are at (assuming the top is the 0'th row) locations 1, 2, 4, 8, 16, ...
@KafshakTashtak7 жыл бұрын
They are Fermat primes, and so far we only know 5 Fermat primes. Which means the rest of the 2^n rows doesn't create a prime number as long as we know.
@yaseen1577 жыл бұрын
We've been learning about it's application in chemistry as part of my A-level course - In a high resolution Nuclear Magnetic Resonance imaging spectrum, the number of peaks and their sizes have relative areas described exactly by Pascal's triangle - a peak with 4 splits in it suggests 3 Hydrogen atoms are adjacent to the unique environment of hydrogen observed, and each peak has relative area 1:3:3:1
@connorp30307 жыл бұрын
Pacals triangle patterns make more sense if you think of each diaganol as a cumulative frequency of the diagonal before it 0+0+0+0+0+0+0+0+0... 0+1 +1 +1 +1 +1 +1... 0+1 +2 +3+4 +5... 0+1+3+6+10... 0+1+4+10... 0+1+5... 0+1... 0... One of the patterns I found was that the sum of squares up to n=((nC1)x(n+1C2))-(n+1C3) so for example, sum of squares up to 4=(4x10)-10 They're are patterns for individual square numbers and cube numbers too, but they're way to complicated for me to explain. It makes me wonder if they're are patterns for every exponent, and they're just too complicated to find. Also pascals triangle can be used to find the constants terms for any binomial expansion, a pascals pyramid can be used to find the constant terms for a nominal expansion. Pascals triangle is real neato.
@connorp30307 жыл бұрын
oh, also the diagonals are numbers which are needed to construct a triangle based shape, so the third row is triangular numbers, the fourth row is tetrahedral numbers, and then whatever the fourth dimentional equivalent is is the fourth row I think.
@briannalove7497 жыл бұрын
I first learnt about Pascal's Triangle because of how you can use it in the expansion of brackets with a high power. So if you had (x+a)^5, you would go to Row 5 of the triangle, and each value is a coefficient, in order, so you add that to each product, and list the powers in descending order for x and ascending order for a. Will definitely save a lot of time in exams. In January, I had a summer school for my math this year (to prepare us for senior highschool math) and we looked at other ways it can be used then as well (mainly combinations and permutations - I don't remember which). It's such an amazing mathematical tool.
@prateekgurjar16517 жыл бұрын
pascal newton and Einstein were playing hide and seek. Einstein said to newton "found you!" but Newton went and stood in a square of length one meter and said "Hey I am newton per meter square..you found pascal" HAHAHAHAHAH...ha..ha
@thatoneguy95827 жыл бұрын
Prateek Gurjar I don't get it.
@prateekgurjar16517 жыл бұрын
units of pressure is pascal..which is Newton per meter square
@sinistrolerta7 жыл бұрын
That One Guy "Newtons per meter squared" is a unit of pressure which is also known as Pascal
@prateekgurjar16517 жыл бұрын
exactly !
@soufian27337 жыл бұрын
I think in english we say "square meter" and not "meter sqaure". I could be wrong
@azdgariarada7 жыл бұрын
There's something about the way she's presented this topic that's really _transformed_ my views on Pascal's Triangle. Now I think there might be more than meets the eye.
@nachoqt7 жыл бұрын
I've never seen the Fibonacci pattern before. Mind blowing.
@randomuser7787 жыл бұрын
Fantastic episode! Bloody brilliant. Love the presenter's enthusiasm too!
@Sagitarria7 жыл бұрын
it goes waaay back into Hindu, Buddhist, and Jain mathematics as "Mount Meru" long before pascal it was known to Pingala in or before the 2nd century BC much to explore in it's relationship to cellular automata.
@Indian_Ravioli7 жыл бұрын
Such a great video.... I know about the triangle since high-school but never knew it had so many properties.
@jetlag14887 жыл бұрын
Illuminati confirmed
@hiveinsider91227 жыл бұрын
Get back to being a meme on Etho's channel! :P
@QualityContentX7 жыл бұрын
We take the Triangle and push it somewhere else
@yaseen1577 жыл бұрын
lol who'd have known you'd leave Etho's channel to find your way to a top comment here?
@H0kram7 жыл бұрын
Yes. Once again. After millions of confirmations. This confirms it again.
@joeydunn9307 жыл бұрын
Are you following me? I just came from Project Ozone #29... :)
@slebeig30357 жыл бұрын
It's great to have those kind of videos of brilliant people for free available for everyone, Thank you!
@ErnestMSaenz7 жыл бұрын
Here's one of the patterns that I've used to teach multiplication to elementary school students: 0 9, 1 8,2 7, 3 6, 4 5, 5 4, 6 3, 7 2, 8 1, 9 0 In the first column on the left, write the numbers in ascending order from zero up to nine and in the second column in descending order from nine to zero and, presto!, you've got the nine's multiplication table. I have patterns for the fours, sixes, sevens, eights and also division.
@xystem47017 жыл бұрын
I never get tired of Pascal's triangle :)
@TheKrevit7 жыл бұрын
nth
@DisDatK97 жыл бұрын
Nth + 1
@copperfield427 жыл бұрын
suc(suc(Nth))
@QweRinatrtY7 жыл бұрын
N shell!
@Frahamen7 жыл бұрын
n^mth
@ganaraminukshuk07 жыл бұрын
Zeroth
@painovoimaton4 жыл бұрын
It is really quite a remarkable thing. So many patterns within it! Pascal's triangle really exemplifies mathematical beauty to me, it has so much going on within a simple rule.
@pedroscoponi49057 жыл бұрын
I always thought pascal's triangle was kind of boring. My mistake! hehe XD Has anyone ever tried to apply the same methods to different bases?
@pedroscoponi49057 жыл бұрын
I'd love to see what those look like, out of silly curiosity
@1987Videolover7 жыл бұрын
Indicotherium its happened on every base... if u put a number n on the formula, you get (n+1)^x... x means row.. for examples.. n = 3, x = 2, u'll get 1(3^2) + 2 (3) + 1 = 9+6+1 = 16 = 4^2 n = 4, x = 3, u'll get 1(4^3) + 3(4^2) + 3(4) +1 = 64 + 48 + 12 + 1 = 125 = 5^3 and so on
@Luisa_17437 жыл бұрын
This video is absolutely amazing!!! Thank you so much for bringing these curiosities about Pascal's triangle!
@Lockirby27 жыл бұрын
It's cool to see an undergraduate on here. :) Gotta love the enthusiasm!
@harmidis7 жыл бұрын
Yea Maths is fantastic! You can always be amazed by learning new ways to interpreter Pascal's Triangle!
@mojosbigsticks7 жыл бұрын
I'm in love.
@mojosbigsticks7 жыл бұрын
A little bit of both.
@NoriMori19927 жыл бұрын
Mojos Bigstick A little bit of both what?
@mojosbigsticks7 жыл бұрын
There was previous reply asking if I was in love with Ms Monroe or with Pascal's triangle. I'm a little bit smitten with both.
@denjam24237 жыл бұрын
I was worried you fell in love with the triangle only.
@2bratsmom4 жыл бұрын
I love Pascal's Triangle and all the possibilities hidden in it. I'm from India and I am proud to say that it was originally found in the works of Pingala, the Indian Mathematician, thousands of years ago and his student brought it to Arabia and it travelled West where Pascal made it popular. This is called Meru Prasthara in Sanskrit and it means "Ladder to Mt Meru". In Vedic literature, Mt Meru is so tall that the Himalaya is like a small stone on it. :) Indeed, this triangle is infinite!
@Phalc0n13377 жыл бұрын
Can you show a Parker Triangle next?
@maikopskoy7 жыл бұрын
Wow I dont regret watching this. well done, I'm quite suprised to find this out. Thank you
@user-tq8nz5pb7n7 жыл бұрын
there is also the binomial thing
@leonardomaranon7 жыл бұрын
When you do it in mod 2, then the number of 1's in each row is always a power of two. Great video!
@lorenzocecchi68517 жыл бұрын
in Italy we call it "Tartaglia's triangle"
@Holobrine7 жыл бұрын
I love the geometric implications in Pascal's triangle. The first row is a point. The second row is a line segment. The 3rd row is a square with a diagonal line (1 point, 2 adjacent points, 1 point). 4th row is a cube with opposite triangular vertex figures (1 point, 3 adjacent points, 3 more adjacent points, 1 point). Beyond that it's hard to explain but it still works and it's BEAUTIFUL!!! What's more, these geometric figures can be used as simple diagrams to understand why Pascal's triangle works for binomial expansion. The third row, for example, is an expansion of (a+b)^2. So pick a corner and label it a^2. The two adjacent corners are ab, and the last corner is b^2.
@adrianbornabasic74997 жыл бұрын
Parker triangle - the new Parker square
@andreygutsuleac22427 жыл бұрын
I've been watching Numberphile for quite a while and I really enjoy the videos. But today I discovered I wasn't even subscribed.... It got me by surprise so I immediately clicked the button.
@numberphile7 жыл бұрын
+Andrew Gutsuleac you you also used the little bell to get notifications!
@BerMaster50007 жыл бұрын
420 Blaise it
@PleasestopcallingmeDoctorImath7 жыл бұрын
Phort won tea, Belize it
@ciarfah7 жыл бұрын
Angry Seal nice
@isabellabornberg21537 жыл бұрын
Angry Seal +
@Someone-cr8cj7 жыл бұрын
wow just wow
@AnastasisGrammenos7 жыл бұрын
Where? Why? Tell me I *need* to blaze it!
@michaelledesma13867 жыл бұрын
Casandra Monroe was amazing in this video!!! Bring her back for more please!!
@altus12537 жыл бұрын
*Claimed by Nintendo LLC
@jpphoton7 жыл бұрын
Brilliant! Keeping the bar high Brady. Tally Ho such a good show!
@HyperQbeMusic7 жыл бұрын
I wouldn't be surprised if Pi showed up in that triangle in a way… somewhere. 🤔
@harinandanrnair67687 жыл бұрын
HyperQbeMusic u just gave me something to work on... thanks
@stewheelie78743 ай бұрын
It is truly fascinating and mind boggling how this even exists.
@dQw4w9WgXc7 жыл бұрын
The Legend of Zelda reference \(°-°)/
@kateinmadison7 жыл бұрын
I've loved Pascal's Triangle since I first learned about it in 7th grade. I had so much fun playing with it in mod 2, mod 3, etc. Different pattern each time!
@zevonmxic4687 жыл бұрын
Mathimus Prime...
@rohitgulati40747 жыл бұрын
I don't know but I was really happy that this was a lady doing the video. it was fantastic. And even better was that she fucking loves Maths and the mysteries that it holds. What a video man. Kudos to you
@stevethecatcouch65327 жыл бұрын
Cassandra, it wasn't nice to tease us with three or four Fermat primes without mentioning that the run of primes is interrupted after 2^2^4 + 1.
@skyenet2 жыл бұрын
I've always loved Pascal's triangle, and I love that it still contains mysteries. Has anyone tried to see what happens if you add another axis of 1's? Will we get something that informs us about trinomials, with for example, rows that sum to powers of 3?
@MarcoBeri7 жыл бұрын
The right name is Tartaglia's Triangle :-)
@claudiuacsinte47577 жыл бұрын
Facciamoci sentire ahaha
@matt-vi2pn7 жыл бұрын
Marco Beri quindi non sono l'unico italiano qui
@ChristianJiang7 жыл бұрын
Ehi, Triangolo di Yang Hui, ci siamo arrivati prima noi :-)
@tgvv29807 жыл бұрын
stavo pensando l'esatta stessa cosa... perché questi tizi lo stanno chiamando "triangolo di pascal".....?
@hdman5117 жыл бұрын
Volevo proprio scriverlo..
@vijayetc7 жыл бұрын
Extremely interesting. I got curious and searched online for PI and e, and no surprises, they are hidden in this triangle as well. Amazing!. Thanks for this.
@fyukfy23667 жыл бұрын
I hate when people say "upside down triangle" just because a triangle isn't on its point doesn't make it upside down, it's still just a triangle
@JaredFT7 жыл бұрын
fyukfy gaming But the base is on top, so yeah, it's upside down. Your logic only applies to horizontally symmetric shapes like squares, circles, etc.
@lowlize7 жыл бұрын
Every side of a triangle can be a base, but a vertex cannot. So when a vertex is on the bottom and a side is on the top, you can legitimately say the triangle is upside down.
@shorterneilisbored70787 жыл бұрын
If you flip a table over, it's still a table, but upside down.
@fyukfy23667 жыл бұрын
lowlize but you can't legitimately call it "upside down" or "rightside up" or anything like that because those aren't terms nor are the definitions in geometry. Just because in human society when a triangle is on its base it looks "correct" doesn't mean it is
@lowlize7 жыл бұрын
I know, but if a side can be called a base there must be a reason, right? Even in abstract geometry we work using our physical intuition of a space with a privileged direction (that defined by gravity), so if it can be useful to construct an immediate visual representation (upside down triangle) without ambiguity, why not use it?
@Minecraftster1487907 жыл бұрын
If you take a row, put each number in binary and write each number vertically each ending at the same line then the shape of that will be a parabola which gets better and better the further down the triangle you go. There's a nice animation on wikipedia
@Someone-cr8cj7 жыл бұрын
damnnn Casandra Monroe back at it again with a transformers shirt... .rly?
@pressplayhomie7 жыл бұрын
Someone My exact sentiments.
@CastelDawn7 жыл бұрын
Would be appropriate if it was a video about primes.
@Someone-cr8cj7 жыл бұрын
yeah not the best shirt for a mathematician to wear... it takes your focus of the actual video
@dermathze7007 жыл бұрын
I didn't even notice it until the 8 minute mark or so.
@Someone-cr8cj7 жыл бұрын
pure mathematics joke...
@OrangeC77 жыл бұрын
I think that the reason why there are so many things in pascal's triangle is because it encompasses the one thing almost all of our math is based on, addition. This means that since most of our math has it's roots at addition, we'll find many things inside of the summitive nature of this triangle.
@TheTruthSentMe7 жыл бұрын
It's inevitable to find patterns if you look for them.
@Sejiko7 жыл бұрын
search for pattern is like a hidden background door in maths.you could find something useful for physiks or other topics.
@ramiel5557 жыл бұрын
especially when they're actually there
@Nixitur7 жыл бұрын
There is a world of difference between seeing patterns from random noise and being able to _prove_ there's patterns in something like this.
@brokenwave61257 жыл бұрын
TheTruthSentMe Sure you can choose to select and omit things at your choosing and then find any pattern. but when ever single number in a sequence is used, in a systematic way...it's not a made up pattern. it's really encoded in the numbers. just like everything in this video.
@procactus91097 жыл бұрын
Sometimes you don't even have to focus on finding a pattern. People find patterns that don't really exist everyday almost.
@Madsy97 жыл бұрын
Enjoyed both the topic and the new speaker. Hope we'll get more episodes with Casandra in the future :)
@robertvdhill3677 жыл бұрын
I love so many things about this video!
@gordonfrohman2507 жыл бұрын
Thanks to you i instantly solved a circuits problem involving infinite resistances organized as a pascal triangle. Thanks!
@JHashcroft5 жыл бұрын
Pascal's Triangle is one of those mathematical concepts that's named after a European, but it's been around for millennia before they were born and could have been discovered anywhere, or even discovered independently on different continents.
@J7Handle7 жыл бұрын
Pascal's triangle describes the coefficients of the resulting polynomial from (a + b)^n A 3 dimensional Pascal's pyramid made to represent the coefficients of the resulting polynomial from (a + b + c)^n is related in really cool ways to the standard Pascal's triangle and I would have liked some more details on that.