Three Dimensional Pascalloids in many cases can be used to re-generate each other's form.

The following chart shows what has been discovered through examination of the animations.

"Shape Transmutation" is what seems to be going on. Although there is self-similar properties in both the Cube-Octahedral Pascalloid and the Rhombic-Dodecahedral Pascalloid.

"Pascalloid" is a term to denote a Pascal-like structure of numbers in 2 or greater dimensions. Here we are examining the Shape Transmutations of Pascalloids in 3 dimensions.

Also, both the Icosahedron and Dodecahedron can make a Icosidodecahedron at fairly low N-values

Cube-Octahedron from cubic Pascalloid at N=2, connecting the number 2

Cubic Pascalloid at N=2, connecting the number 4, to form an Octahedron.

Cube-octahedral Pascalloid taken to N=3, connecting coefficient number 24 to form an Octahedron.

Octahedral Pascalloid at N=2, connecting the number 2, to form an Cube-octahedron.

Octahedral Pascalloid at N=5, connecting occurances of the number 180, to form a Cube.

Prism Pascalloid at N=3, connecting the number 3, to form a Hexagonal Column geometry.

Rhombic-dodecahedral Pascalloid at N=2, connecting occurances of the number 6, to form a cube.

Rhombic-dodecahedral Pascalloid at N=4, connecting the number 432, to form a Cube-octahedron.

Rhombic-dodecahedral Pascalloid at N=4, connecting the number 552, to form an Octahedron.

Tetrahedral Pascalloid at N=4, connecting the number 12, to form an Cube-octahedron.

Tetrahedral Pascalloid at N=2, connecting occurances of coefficient 2, to form an Octahedron.

Beyond and into further dimensions such as 'Four or Five' is difficult, but easy with cubic and tetrahedral geometry. Here are my explanations:

Cubic geometry, algebraic terms expansion will derive from

(A+B)^{N} ... two dimensional Pascal's Triangle Row

((A+B)(C+D))^{N} ... three (two for Row) dimensional Pascalloid

((A+B)(C+D)(E+F))^{N}... four (three for Row) dimensional Pascalloid.

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In the Combinatorics, these are all similar to the denominators..

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etc ... R dimensional Pascalloid

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Tetrahedral Geometry

Target a number of dimensions for a Hypertetrahedra R

Now the tetrahedra will have R-1 corners, and R-1 variables to the N^{th} power.

It is incredibly simple with the Hypertetrahedra, yet anything beyond four corners for a tetrahedron is impossible to visualize unless extra corners group towards the center and are invisible if the shape is considered as a solid. Hypertetrahedra will require only a very simple expression:

(A+B+C+D+....) to include R-1 variables, given as a square in 2D or a tetrahedron in 3D, additional variables falling in mid-points.^{N}