#pascals simplex

[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.

[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.