#pascals triangle

[bagalois]

Hiiii

What’s your favorite math

hmmm.... i'm a pascal's triangle enjoyer. lots of fun identities, and i dont come across binomial expansions that often but i always use pascal's triangle for that. if you highlight all the odd numbers, you get sierpinski's triangle. the diagonals give you the triangular numbers, the tetrahedral numbers, and so on. the rows sum to powers of 2. the hockeystick lemma. probably some other stuff that i dont know.

this is a hard question because the word "favorite" makes me think about picking the absolute best one and having a passionate explanation for why i like it so much. pascal's triangle is not groundbreakingly cool or advanced, but it's kind of fun.

other math party tricks that i like:

the dropping sticks approximation of pi (buffon's needle)

the Long Line

the well ordering theorem

axiom of choice proofs

godel's incompleteness theorems reduces down to the halting problem

[graph100]

pascals triangle postulates that you should follow god because you have nothing to lose. critics of this mentality retort saying that this should be applied to all religions, and that the logical conclusion would be to follow all religion, pray to each god, etc. Instead, I am here to propose a new angle to see the "problem". In "Graph100's Anti-Pascal's-Triangle", one must make it their life's goal to anger each and every god, finding some way to actively denounce them, in order to secure a 100% chance of ending up in hell.

[blake447]

So here's an interesting fact I've learned while researching these things called Wolstenholme Primes. See they're related to the binomial coefficient and thus pascals triangle. We've probably all seen sierpinski's triangle in here (if not circle all the odd numbers). But here's something else interesting. If you look at the triangle, you'll see that for prime numbers, every element is divisible by that prime number. Look at 11 for example.

55, 165, 330, and 462 are all divisible by 11. This has to do with the binomial coefficient definition, in which you'll see that if unless we choose either all the elements, or none of them, the factor of our prime number is guaranteed not to cancel out from the numerator's factorial.

We can then use the fact that the sum of all the elements in the nth row of pascal's triangle adds up to 2^n. Once we remove the 1's on each end of the triangle (hence the minus 2) we know that for prime numbers p that all elements are divisible by p, their sum is as well!

Now, I don't know off the top my head if this is only true for prime numbers. It isn't immediately obvious imo. However, if this is strong enough to be an if and only if, then I think I have independently discovered perhaps the *worst* way to test for primality lmao, scaling exponentially instead of the standard logarithmically.

[art-of-mathematics]

I started to craft a visualization model for a Pascal"s 4-simplex.

Now I need to insert the 12 triangle disks. (I need to puzzle them into each other like a kind of interwebbing. )

I will describe this model and the math of it once I have finished it.

[futurebird]

Math Camp! Pascal's Triangle! Bee!

got over excited drawing this bee for my math camp students.

[superhumanoid-ai]

Multinomial coefficients and Pascal's Simplex

For binomials [ (a+b)ⁿ ] the use of Pascal's triangle is helpful. In Pascal's triangle each single row in the triangle defines the coefficients of binomials of each n-value.

For trinomials [ (a+b+c)ⁿ ] this pattern can be extended to a 3-dimensional Pascal's tetrehedron, where each level (and hence a complete triangle) in that tetrehedron defines the coefficients of trinomials of each value of n.

As for quadrinomials [ (a+b+c+d)ⁿ ] the coefficients require an own tetrahedron for each value of n.

[backslashdelta]

Math positivity!!! because it's been a while and math was amazing today: We learned how to use Pascal's triangle to expand binomials today in math and it's literally so much fun??? I love Pascal's triangle and when you use it to expand binomials it's super cool!

MATH POSITIVITY YES THANK YOU I LOVE IT HERE

I agree, Pascal's triangle is super cool!! It's math and it's a combinatorics thing and it's A TRIANGLE WHAT MORE COULD YOU POSSIBLY EVER WANT????

Thank you so much for sharing this with me.

On a related note, let's take a moment to appreciate Sierpinski's Triangle, which is what you get when you colour in all the odd numbers in Pascal's Triangle :)

[er-cryptid]

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