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


Updated on Sunday, November 20th, 2017

THOMAS L. CHENHALL

(image from corner of Tetrahedral at N=35, featuring six digits of π in yellow)

The 3D / 4D "Pascalloid Calculator 9.1" was recently updated to be able to perform r-base horizontal addition on the 3D/4D Pascalloids, and now many minor errors fixed.  A User-Defined Pascalloid mode has been added to allow anyone to formulate any geometry of 3D/4D Pascalloids.  Recent updates have finally allowed accurate Icosahedral and Dodecahedral Pascalloids, and their newfound equations are contained within the Pascalloid Calc software.  Based on the solution to the Dodecahedral, a new algebra for the Cubic was also devised.

The horizontal addition in r-base was originally formulated within the 2D /3D "Pascal's Canvas 4.8", another free 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 self-contained help system.

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.  Polarized Algebraic Expansions and Polarized 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 9.1" application, available for free.

This specialized 3D/4D Pascalloid generator program contains the corresponding equations that I found, except the combinatoric equation for the Deltoidalicositetrahedron and Rhombiccubeoctahedron are now omitted.  New spreadsheets not finalized yet are exploring usage of the Gamma function as a substitute for the factorial in Pascalloid combinatoric equations.  The spreadsheets represented by the Combinatoric EQs can be downloaded from this page, but are not linked into my software, 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 second of this site in the Google Sites, will catch up soon.

https://sites.google.com/site/pascalloids


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 algorithms 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 computational threshold.

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
Null or One Dimensional Theory in Recursion

One or Two Dimensional Recursive Adding Structure

Corners & Vars Summed
Name
2
Pascal's Triangle, etc.*

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

Two or Three Dimensional Pascalloids Synopsis



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*

Three or Four Dimensional Pascalloids Synopsis

Click here for the Combinatorics, Geometry and Pascalloids PDF

Download Circle Combinatorics to 16-sided.xls

Download All Combinatorics Experiments in .xls

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