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

Updated on Wednesday, January 29th, 2014
By Thomas L. Chenhall


Pending addition of a help system the '2d or 3d Pascalloid Canvas' will be updated soon. I have been jealously guarding it ever since adding the ability to construct bitmaps from Pascalloids, though after some thought decided to recombine the the different edits of this program back into one program. Due this spring. All exposed Flash is mainly for tablets and smartphones, and if you have a Mac or PC you will find these programs easier to use. They can be processing intensive.

Recent: Algebra Expansion Equations are paired with Combinatoric / factorial equations. 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 8.7" application, now available in Mac Binary & Win EXE format.

This specialized program contains equations I found, except the combinatoric equation for the Deltoidalicositetrahedron I omit. Its a page long, so I just give the algebra solution to that shape instead. 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 given actually available, unless you want to 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() function will differ. One could easily write a program to do just algebraic expansions, or even do the expansion of any algebra equation 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.

I can understand how I might go about assembling a pair of equations for the dodecahedron, based on somewhat flattened cubes (along 3D diagonal); ten of them summed across each other at the correct angles to form a "Pascallated Dodecahedron", yet the Icosahedron to imagine is much more difficult, and I have grown unsure if it can be solved with the summation of several Triangular Dipyramids, and I am imagining a Tet-based solution instead.

Recently completed: a nine-gon of 27 variables @ N=1, constructed as best I could of three Equilateral Triangles, each rotationally displaced 40° from each other. This shape is computed for us, an equation procedure outlined, and results displayed in a PDF to N = 10. The edge length of each triangle is 31 units, and spreadsheet proofs are available for Triangles A, B & C. By the time the calculation delivers N=10, I notice a hint of Rhombihexagonal tiling that generates itself near the core of the shape.

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.

The second of this site in the Google Sites.

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 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 Mac Feature: Binary App + Added working menus

Special Win Feature: Full-screen at startup, compatible with XP and upwards.

Features of Raw .swf: ~221K in size, Opens in Flash-ready browser via the .html file. Actionscript 2, Flash 8 Decompilable, Ran swfsli.exe to optimize operation, used purchased copy (thanks mom) of SWF Cargo to install API, though this has been removed to rebuild the EXE and the version found available online here.

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
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'*
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 Hypercube (tiles with Squarish Pascalloid)*
~Equilateral Octagonal Edge L 17, from Hypercube* (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