Sierpinski from Pascal

In this short, we shade odd entries of the Halayuda/Pascal triangle to obtain the Sierpinski triangle. Can you explain why this works?

#math #manim #fractal #fractal #sierpinski #pascaltriangle #pascal #mathshort #visualmath

[samevans]

The Sierpinski triangle is like pi. It’s pretty unremarkable until it shows up everywhere you don’t expect it to

[curtiswfranks]

I am guessing that the reason for so many entries being even is that they are binomial coefficients, meaning that they are a large factorial divided by two smaller factorials. That means that the quotient will tend to retain at least one factor of 2.

[AkariLimano]

Also those numbers are coefficients of (a+b)^n polynoms.

[viatori]

I just got done trying my hand at programming the Sierpinski triangle and now you show me this. God damn it let me be happy

[gauthamnair]

I'm just fascinated by the moiré effect when zooming out.

[gregjacksun]

Do this in 3 dimensions with spheres and you get the most efficient stacking method of cannonballs if you are on Jupiter juggling quarks while eating a PB&J when you stack them.

Strangely enough - this is also the vibrational cross section of 12 neutrinos bouncing around inside a cone.

[MintyHobGob]

Fun fact about Pascal's Triangle!
Ignoring the first set of 1s, take xⁿ-(x-1)ⁿ = yx^a - yx^a... where a is n-1 down to 0 and y is the nth row of Pascal's Triangle, alternating - and + until 1!:
2n - 1 for ^2
3n² - 3n + 1 for ^3
4n³ - 6n² + 4n - 1 for ^4

[samueldragonmaster]

I’m seeing the triforce from legends of zelda

[KienTran-bxqx]

I wish we would have a HUGH woww button now!!

[sir_dragonfly]

The sierpinski triange is the most beautiful and mathematically perfect shape in my opinion.

[YoungRichLime]

That’s crazy bro I’m getting sucked down a rabbit hole rn. Is every randomly occurring event in all of human life been just a big elaborate thought out plan for us all.

[isaacdiboss]

WHAT. That's amazing! How do you not have hundreds of thousands of subs and massive amounts of views and likes??!

[shruggzdastr-facedclown]

I wonder what pattern emerges if all primes are shaded

[kannix]

cool proof that almost all entries are even

[givrally]

It's all because of the recurrence formula. Odd + odd = even, even + even = even, odd + even = odd. All the edges are odd, and the moment you get a full row of odds you end up with a full row of evens later on, which will stay even until the odds on the edges have propagated back to the middle, in which case you're going to have alternating odds and evens, then full odds, and full evens again.

[calculus]

Looks like coefficient of a binomial coefficient

[somedudethatripsplanetinha]

looks like dalgona squid game s2 from gi huns nightmare

[MrMan-npjg]

Pascal is wack. Also, now I know exactly what a Pascal triangle is