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.

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.

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!