Blud did not skip a single math class Nice vid

Holy cow he is good at transitions

I love getting detailed answers to strange and obscure questions that I never though to ask. Fascinating.

That curve is obviously the bubble. Duh.

mmmmmmmmmm, Neopolitan sorting algorithms

that was...... genius.... holllyyyyyyyyyyy

Or is it..... LMAOOOO

When I watch the bubble sort "bubble," my brain intuitively understands what's happening and why, but I can't for the life of me follow any of the math behind it. Why is my brain capable of understanding something intuitively that it can't understand mathematically?

What if you extend to complex numbers

Equal time per iteration assumption is bogus. No one would code it that way. Obvious comment but I'm not going to sort the 4-digit sock puppet comments. Sufficient to note that the first one is such.

it a bubble curve inside a bubble sort , magic

Is the purple curve wellbb defined? It seems to be "continous line, always above the vertical lines, as close as possible, always growing"? Maybe this should have been mentioned early on, for now the definition of the purple line is...but we'll come back to this".

Bubble sort makes bubble

2:20 -> 13:17

Magic 🎉🎉❤❤

Brilliant!

Wooop wood woooooooop

Let X,Y,Z and T be intregers with X<2Y, Y <3Z and Z < 4t... if t<100 what is the largest possible value of X?? Can solve my kid homework Sir ...please

Beautiful derivation! Keep up the great work :)

This is the math of pretty graphics. Real lists won't have a continuous graph even once they're fully sorted.

Your assertion that purple line has to end up on the green line eventually didn't make sense until I want thought about it. Everything else was crystal clear. The way I think of it is that the last item in the purple sublist is sorted exactly when it connects with the green line.

That nice in the end haha

VSauce reference!!!

11:27 this way of reasoning (going to a simple case where N is arbitrarily large and learning about the rest of the domain from there) is similar to how we Christians reason by going to God first, then learning about the rest from Him.

Memes

My favourite part about this video is not the bubble sort curve solution, but how harmoniously it illustrates that the *real* intellectual leap is figuring out how to formulate a problem into something one can hold on to and tackle in bits.

Wonderful! Informative! Well presented, written, and recorded! Please continue doing this, keep up the excellent work

love the bit about gamma, really great video

Watching this makes me want to replay [the] Gnorp Apologue...

HAAAAAHAAAHAAHAHAHAHAHAHHHAHA

Best video ever thanks for that !

1:40 how to prove Hx＝d/dx ln(x!)+γ

1:40 how to prove Hx＝d/dx ln(x!)+γ

Interesting video, but the music is distracting.

This thing was so dope. You sir are really cool

When handling that a>=b constraint at 16:17, you skipped the implied b!=0 constraint from dividing by b, which results in a t!=0 constraint for the curve overall. That makes sense to me though, because without the t!=0 constraint, it would be implying that unsorted, uniformly-distributed data approaches this curve.

Saw this was 20 minutes and knew it was going to be a banger haven’t seen it yet but I just know

what is that second curve formed by pure white at the bottom of the 59 minute video 0:40

oddly satisfying

11:14 why were these curves animated? It’s confusing me

Very interesting and inspiring video, blog, and draft. However, from the mathematical point of view, there is some discrepancy between what you claimed to prove and what you actually proved. First of all infinities and probabilities are tricky things together, as probability is a measure of its own kind, it needs to be carefully studied so that the events stay in controllable size when approaching infinity. Secondly, as the initial state of the list follows a probability distribution, so does the unsorted section after some iterations. From the probabilistic viewpoint, your curve does not seem to represent anything: If we have numbers from 1 to N to be sorted from their uniform random placements and we know that N-n elements are in their correct places, this is not enough information to conclude scale-invariancy nor continuity over the whole domain. If we think about the expected curve, there will be a discontinuity point at 1-t, because if the element x[(1-t)N] is in the end part of the list at the beginning, it will end up in the correct place, otherwise, the (1-t)N'st element is uniformly any element less than x[(1-t)N], therefore, the expected value is: E(X[(1-t)N]) = 0.5*(x[(1-t)N]-1)*t + x[(1-t)N]*(1-t) = x[(1-t)N] - 0.5*(x[(1-t)N]+1)*t. If I use this to scale the function, it mostly undershoots. As an upper bound, it does not make sense either because distribution P(X[k] <= x[k]) is uniform with a minimum P(X[1] <= x[1]) = t as it will be the first element only if it appears on the unsorted side. From this, I can derive feasible upper and lower bounds by setting either E(X[k] <= x[k])=0 or E(X[k] > x[k]) = 0. The upper bound is always greater than N*x/(x+t) and the lower bound surpasses the curve, so it is not representative of the Bubble-sorting behavior as the height of the bars will be always between these values, see illustration at t=0.4 if Wolfram will not fail me: www.wolframalpha.com/input?i=plot+0.4*1%2B%281-0.4%29*k%2B1%2F2+%281-k%29+%28+k+%2B+1%29*%281-0.4%29%2F1%2C+1.4*k%2F%28k%2B0.4%29%2C+%280.4%2B0.6*k%29%28k%2B1%29%2F2%2C+k%3D0..1 What your proof in the paper and blog presents is that the curve Bubble sort algorithm forms, will approach the function x/(x+t). If someone from a college background reads this, limits are usually taught wrong there. Approaching only means that we go towards something, there is no guarantee to reach the limit calculated with a reasonable amount of steps. It also does not mean that the Bubble sort curve can be approximated by its limit as it does not work as a bound.

I need this guys to write my math textbooks

I guessed that the curve is square root of something, now I feel bad about my guessing capabilities

Yo you're f'ing amazing

The line fitting was essentially the climax of this video after all the edging, while the algebraic dance was the final most intense act

euler daddy

i dont get it but the lines r cool

17:45 Just don't die before the next video, leaving us to wonder what the "different line of reasoning" was, hahaha

It's always Euler

No, just no, this did not need to happen. We apparently are running out of math in the world, for people to spend time doing math on this.