The 16-var Cube-Octahedron is just a cube-octahedron, but in 3D Pascalloids and Combinatorics, there turns out to be four additions applied to the center point. In other words, four corners in its core, and 12 corners at the periphery, for a total of 16-variables to be summed (at least in the Pascalloid algorythmic approach)

If we perform a careful operation on a 6 dimensional tetrahedron containing an inverted tetrahedron at each point, we can bring a shape back from 6D to 3D in a tetrahedral sum.

and here is this Cube-Octahedral expression:



Click here for Xl doc

This calculation produces results corresponding to a 16 variable Pascalloid algorithm of a Cube-Octahedron with four additional variables added in the center. This maintains the 12 corner geometry but puts some more density in its core. Now, I have also solved an algebraic expansion with coefficients running the exact same numbers.



The geometry is still n-based, starting here at n=1.

That was a collection of views of the N = 2, 16-var Cube-Octahedral Pascalloid, what is difficult to visualize until you see many examples. As before, we can give a sum total for that iteration of N=2; 162 = 256
That was a collection of visualizations of the 16-var Cube-Octahedral Pascalloid, what is looking quite obfuscated,

At N = 3 sum total = 163 = 4,096

That was a collection of visualizations of the 16-var Cube-Octahedral Pascalloid, very hard to see the center.

At N = 4 sum total = 164 = 65,536

That was a collection of visualizations of the 16-var Cube-Octahedral Pascalloid.

At N = 5 sum total = 165 = 1,048,576

161 = 16