Pascal Triangle

[Anonymous No.8753350]

This is my new favorite mathematical thing.
This triangle is M-A-G-I-C-A-L

Share any patterns you find in the pascal triangle in this thread.

https://www.youtube.com/watch?v=0iMtlus-afo

[Anonymous No.8753365]

Geometric numbers in Pascal's triangle:

1st diagonal = 1 1 1 1 1 (the sequence is the number of dots you can put in one dimension)

2nd diagonal = 1 2 3 4 5 (the sequence is the number of dots you can put in two dimensions, with the line being one dot longer at each time)

3rd diagonal = 1, 3, 6, 10, 15 the number of dots you need to create "triangles" from the said dots. also known as triangular number

4th diagonal - 1, 4, 10, 20, 35 - the number of dots you need to create tetrahedron, with the tetrahedron becoming one stage larger at each number. also known as the Tetrahedral number

5th diagonal - same concept but for a 4th dimensional version of a tetrahedron called pentatope

[Anonymous No.8753370]

Geometric numbers in Pascal's triangle:

1st diagonal = 1 1 1 1 1 (the sequence is the number of dots you can put in zero dimensions)

2nd diagonal = 1 2 3 4 5 (the sequence is the number of dots you can put in one dimension (aka just a line) with the line being one dot longer at each time)

3rd diagonal = 1, 3, 6, 10, 15 the number of dots you need to create "triangles" from the said dots. also known as triangular number

4th diagonal - 1, 4, 10, 20, 35 - the number of dots you need to create tetrahedron, with the tetrahedron becoming one stage larger at each number. also known as the Tetrahedral number

5th diagonal - same concept but for a 4th dimensional version of a tetrahedron called pentatope

[Anonymous No.8753378]

Also, the last row in any size of Pascal triangle is actually the instructions on how to turn the triangle into Pascal's pyramid.

https://en.wikipedia.org/wiki/Pascal's_pyramid

"It is well known that the numbers along the three outside edges of the nth Layer of the tetrahedron are the same numbers as the nth Line of Pascal's triangle. However, the connection is actually much more extensive than just one row of numbers. This relationship is best illustrated by comparing Pascal's triangle down to Line 4 with Layer 4 of the tetrahedron."

"Multiplying the numbers of each line of Pascal's triangle down to the nth Line by the numbers of the nth Line generates the nth Layer of the Tetrahedron. In the following example, the lines of Pascal's triangle are in italic font and the rows of the tetrahedron are in bold font.[1]"

[Anonymous No.8753382]

Sum of the n-th horizontal line is 2^n for n<8 pattern returns for some certain higher "n"

[Anonymous No.8753410]

>>8753350

Not only is it a fun meme, but it is actually /useful/.

Often, when you're doing anything with polynomials or combinations, or even special sub-cases of powers of 2, it is downright /helpful/ to jot down the first few rows just to recall what the pattern is supposed to be, generally, and to check your work briefly.

In this respect, the triangle rivals Euler's identity for "meme that is actually useful when you get down to it" status. Just because mathematicians roll their eyes when some normie references the latter, does not mean that they (the mathematicians) don't make regular use of the tool, in the appropriate contexts.