Pascal's Triangle - Like Structures,
Number Tiling Combinatorics
In Extended Dimensions;
"Pascalloids"

Updated on Sunday, January 24th, 2017
By Thomas L. Chenhall

The 3D / 4D "Pascalloid Calculator 8.9" has been updated to be able to perform r-base horizontal addition on the 3D/4D Pascalloids, color palettes added, red button fixed, and many minor errors fixed.  Third party API is now removed from the software due to minor errors with the 'w' key and on-screen button now properly toggling into and out of the full-screen mode.

The horizontal addition in r-base was originally formulated within the 2D /3D "Pascal's Canvas 4.8", my software allowing representation of large essentially 2D cross-sections of these Pascalloids. In that software, beyond its original “Introduction Mode”, there are many bitmap output modes, along with a possibly finalized self-contained help system.

On the mathematical edge of Pascalloids, I am beginning work on a Combinatoric / Algebraic Dodecahedron, assembled from ten shape-summated cubes, each of which is compressed along two opposite corners, and each rotationally translated to the correct angle. These will take yet more time to complete.

A Pascalloid Icosahedron after careful mulling over, must be treated itself as a unique structure, indivisible from its total structure, and its combinatoric solution must be a multi-stage process that does not involve shape-summation (much to my dismay), but instead the transformation of indecies to planes of numbers which must align correctly along the surfaces of a dodecahedron, incidentally forming Icosahedral corners only where five planes simultaneously intersect.

Recent: Algebra Expansion Equations are paired with Combinatoric / factorial equations. The 'N' value also indicates the exponent of the original alg. at N=1, which when taken reveals the sum total @ any N value. To generate the terms from these equations in Wolfram Mathematica for instance, we use the 'Expand' command, and pick an N value (must be whole number), then choose 'Evaluate' from the menu.

Expand[(green glowing equation)^N]

Do the results imply a geometry? Yes, but our long list of expansion results are not placed in their matching geometry yet; the geometry which is implied by the Combinatoric factorial equations. New Polar Algebraic Expansions and new Polar Combinatoric equations (+/-) are also reported in the program, specifically the 3D / 4D group of equations and their respective pascalloids, which are best viewed via the "Pascalloid Calc 8.9" application, available free in Mac Binary & Win EXE format.

This specialized 3D/4D Pascalloid generator program contains the corresponding equations that I found, except the combinatoric equation for the Deltoidalicositetrahedron is now omitted due to it's painfully large resolution at N=1.  My appologies to the exoflop people.  The spreadsheets represented by the Combinatoric EQs can be downloaded from this page, but are not linked into the program, nor are the Algebraic Expansions themselves expanded, just the expandable equations are given.  To view an algebraic expansion, pick an N value and try one out in a standard mathematics software package such as Mathematica, Maple, or the cheaper 'Sage', though I don't know how the syntax for the Expand([eq]) function will differ.  One could easily write a program to do just algebraic expansions, or even perform the expansion for any of these equations to the Nth power by hand. The summation required in the Combinatoric equations is parallel to the way the identical terms will 'sum' as they recombine.

The first in a chain of polar updates, the 6-var hexagon with three positive and three negative variables is included the 2D / 3D group. It is significant, and simple. Incidentally, I have also solved the well-known 'Fruit of Life' geometry & algebra, and the coefficient results are thus the form, though modeling it as a projection of a hyper-cube is how it forms its shape. The Pascal's Canvas 4.8 software for 2D/3D is on the back burner now, and still includes an extensive help, and has some more challenging resolutions for combinatorially proven pentagon, heptagon and nine-gon.

The second of this site in the Google Sites, will catch up soon.

Navigator:

 Pascalloids & Geometry PDF Null or 1 dimensional 1 or 2 dimensional 2 or 3 dimensional 2 or 3 D Canvas Application 3 or 4 dimensional 3 or 4 D Pascalloid Calculator

Attention teachers, students, hobbyists, programmers, engineers, inventors & mathematicians! This page contains a group of studies in number and geometry, and now (yet another programming hurdle in the distance here) algebraic terms paired with each coefficient number. The software presented is under an honorable GNU General Public License, and if requested I will send the Flash file (.fla).

Don't forget to try out the less mentioned, 2D or 3D Pascalloid Canvas program. It points out how the algorythms are constructed, making it easier to understand how scallation works at the algorithmic level. Here I mainly have been describing updates to the 3D / 4D Pascalloid Calc.

The entire library of information presented or presentable with the software is public domain, by nature. I believe it is of universal value to science, though I have limited understanding of how it applies to statistics, and perhaps an extension of meaningful story problems in Combinatorics. Even just as a 3D visualization system, its good as a mathematical example for kids >= 12 yrs old, and maybe as an entertaining animation to stare at for a few moments, it's safe for kids to view at age 4 and above. Don't be surprised if it can't compete with Xbox or Cable T.V., as there is no video-game objective, and no violence or sex involved whatsoever. This program was only designed to compute Pascalloids / to do Scallation and have a good viewable output 3D display, but I had to make it somewhat engaging to appeal to people.

For the more educated individual, there may be some interest in the outputs derived by utilizing the Display Range and the Modulus Function, as applied to the recursively generated data that can be copy / pasted out of the Pascalloid Calc. Algebra equations I have tested seem hold true, yet what can this imply? Everyone who uses the 3D / 4D program 'Pascalloid Calculator': be careful to keep 'N' below your computers threshold. The hardest shape is the Deltoidalicositetrahedron, which is now again sum 216 at N=1, sum 8 in polar at N=1, displayed at 33x33x33 (a nearest best-fit resolution that may require patience to compute beyond N=1). No spreadsheet is yet available to prove that shape, or its dual the Pascallated Rhombic Cuboctahedron. They require nine or seven dimensions of spreadsheet space respectively (to compute with combinatoric equations using factorials), so there is no way to make such a spread sheet yet.

This is from the perspective of current average computing capacity.

Algs Recently Repaired: Polar Octahedron, Polar Pyramid, Octahedral Pillar (unipolar and polar), many algebra equations added in (unipolar and polar).

Special Win Feature: Full-screen at startup, compatible with any PC that is Win XP or beyond.

Pascalloids / Scallation is mainly generated from a seed number of 1, and is basically the same type of simple recursive algorithm that generates Pascal's Triangle, just more possible patterns. See the 2D or 3D 'Pascalloid Canvas' app to better understand the process. That app, under same GNU license has not been updated, yet ought to work just fine, and helps to understand the Scallation process without the additional dimension that would require animation.

* indicates that equations have been identified for the geometric algorithm.

Zero or One Dimensioned Recursive Structure (hypothetical)

Corners & Vars Summed
Name
 1

One or Two Dimensional Recursive Adding Structure

Corners & Vars Summed
Name
 2

Two or Three Dimensional Recursive Adding Structures

 Corners Vars Sum
Name of Pascalloid Structure
 3 3 4 4 5 5 5 12 6 6 6 6 6 7 7 7 7 36 6 8 6 9 6 16 9 9 9 27 8 16 8 16 4 10
 Triangular; Nugent's Pyramid (phased Cube-octal tiling)* Squarish (cubic tiling)* Pentagonal (unknown tiling) at 8 X 9 Pentagonal from 5 D at 95 X 82* (PDF) Hexagonal A (irrational) Hexagonal A +/- (polarized)* Hexagonal B (irrational) Heptagonal (unknown tiling) at 20 X 19 Heptagonal from 7 D at 66 X 65* (8 MB PDF) Cube Projected into Hexagonal* Hexagonal from Projected 4 D 'Triangle within Inverse Triangle'* Hexagonal, Fruit of Life, Projected 'Geometry from Corner of Hyper-Cube'* Nine-Gon including digits for Pi at N=11 (unknown tiling) at 14 X 14 Nine-Gon Edge L 31 (Pi at N=8) from 6 D at 90 X 90* (8.6 MB PDF) Octagonal from Projected 4 D Hyper-Cube* ~Equilateral Octagonal Edge L 17, from Hyper-Cube* (28 MB (big) PDF) 10-var Basic Spiral Pascalloid from Glyph

Three or Four Dimensional Recursive Adding Structures

 Corners Vars Sum
Name of Pascalloid Structure
 4 4 8 8 6 6 5 5 5 5 6 6 14 16 12 16 16 32 12 16 8 12 12 12 20 20 26 26 24 48
 Tetrahedral* Cubic* Prismatic* Triangular Dipyramidical* Pyramidical* Octahedral from Projected 4 D Squares Within Triangle* Rhombic Dodecahedral from Projected 4 D Hypercube* Hexagonal Pillar from Projected 4 D Hypercube* Octagonal Pillar from Projected 5 D Hypercube* Cube Octahedral from Projected Tetrahedron Within Inverse Tetrahedron* Cube Octahedral Cup from Projected Tetrahedron within Inverse Triangle* Dodecahedral from Algorithm at 17 X 17 X 17 Icosahedral from Algorithm at 27 X 27 X 27 Deltoidal Icositetrahedral from Algorithm at 13 X 13 X 13 Rhombicuboctahedral from Cubic Sum of Diagonal Rotated Octahedra*

A Progression: "Combinatorics: Questions to the Answers" in .doc format

A lot of those files are on my list to update. 'Questions to the Answers' gets complex.

The following is the general summations of any given shape towards N=infinity, in terms first Pascal's Triangle, then a 3 D Bell Curve, then a 4 D cluster of density. However all the many individual numbers are gone, and we have instead one smooth shape for each dimensional space.

Sigma equals that number given twice, so the maxima of the normal function (infinity) will be precisely equal to 1. However the composition of the value of Sigma must also be a number that goes on forever for a very precise Hyper-Normal Function.

What I suppose I am to do with these functions is take what they represent and do the equivalent of glass-cutting to make my refined geometry structures. I suppose I could finally say bye bye to dealing with all those 'special' numbers if this could be done.

Here is an early statement of goals, drawn before the 3or4 D Pascallid Calculator was finished.

Have fun!

Thomas.Chenhall@gmail.com