Two or Three Dimensional Pascalloids Basis Overview

The basis of Pascalloids is a relative summation algorithm. Most Pascalloids do not express any predictable linear expansion, however the even numbered 'somewhat equilateral' polygon shapes do, to an extent, depending on the summation’s resolution Each point at the corner of a shape is scanned for a value and then added to the next iteration of a shape @ N+1. Not to confuse anyone, from n=0 to N-->infinity, the exact same summation is applied in a relative way across an array or series of expanding arrays, variably designed to hold the results of the summations. Most Pascalloids are irregular, though the 'equilateral' polygons try not to be. However in possible experiments to any geometry of summation available and computable across repeated arrays, using a single algorithm for a single Pascalloid. There are only a few algorithms corresponding to shapes that tile. Where there are algorithms for shapes that tile, such as triangle and hexagon, there are also Combinatoric equations that can generate the same shapes. And on top of that, where there are Combinatoric equations to generate shapes, there are also indices, such as (x, y, z), that follow along lines of exponents to algebraic equations. These equations are very similar to expansions of linear equations, such as (a+b+c+d)^{3} what is a tetrahedron drawn as points into dimensionality of unit 3. Including further variables +e+f+g incrementally increases the dimensionality of a tetrahedron. This is very hard to visualize and therefore a great impediment to making a programmed construction. If instead we examine high-resolution n-gon points, we are looking at a square, a pentagon, a hexagon, and a heptagon. Each summation has respectively four elements, five elements, six elements, and seven elements.

The algebra for a triangle is clear given that it is drawn as half a square. The third axis for exponents goes diagonally across. As one can see by the Excel files, the exponents go down in number as they file away from each side of the triangle. The total of variables is found including exponents. The algebra for a square is in fact introducing various roots to the exponents. The reason for that is that the lowest entropy regular form of four points is not a square but a tetrahedron. We can bend and mesh the four points of a tetrahedron correctly so that they form the square. The square differs from a typical (a+b+c+d)^{n} Tetrahedral 4D Pascalloid because the square does not reveal the correct expansion in its terms. As a 3D Pascalloid, the square at a specific n-value is one horizontal slice of a pyramid. ((a+b)(c+d))^{ n} is much easier to define in that the entire result has square roots applied everywhere, instead of more complex roots in the square version of (a+b+c+d)^{ n}. Also interestingly, the square of this kind has fewer terms than it’s Tetrahedral brother. And though this is made up for in the fact that the sums of just the coefficients at any given specific n-value for the square and tetrahedral versions of (a+b+c+d)^{ n} both give the same grand coefficient totals.

Up to the 5-gon again? Five corners is the simplest 2D summation alg in a 3D Pascalloid that will give some accurate results in terms of the expansions of (a+b+c+d+e)^{ n}. The accuracy of results begins to degrade with increased n-value, and depends on the resolution of the summation alg used to add 5 values in one array to each value in the next iteration’s array.

Now let’s look at the 6-gon or Hexagon. It has six corners. We can find our expansion algorithmically as well as we did for the 4-gon, and count on good results, possibly because both of these specific shapes tile. However (a+b+c+d+e+f)^{ n} cannot be expanded to get our coeficcient numbers. The 6-var Hexagonal Pascalloid is not an expansion of this equation. The expansion of this equation is actually a 5D Pascalloid + 1D if n-value is included. We would have problems trying to do an expansion of a 6D Tetrahedral Pascalloid, so instead we have our algorithmically defined terms with fractional exponents indicating roots as well. The terms are kept to unity and the polynomials also algorithmically defined. And as before, the sum total of the coefficients in the Pascalloid as a number pattern are the same at one given n-value between a 6-var hexagonal Pascalloid and a 6D Tetrahedral Pascalloid. I have not been able to prove this, but knowing the way of summation algorithms and this simple type of iteration, I strongly postulate that it is the case.

Enter the Heptagon, or 7-gon. Seven corners is the simplest 2D summation alg in a 3D Pascalloid that will give some accurate results in terms of the expansions of (a+b+c+d+e+f+g)^{ n}. The accuracy of results, like the 5-gon Pascalloid, begins to degrade with increased n-value as polynomials pack together. The heptagon also greatly depends on the resolution of the summation alg used to add 7 geometrically aligned values in one array to each value in the next iteration’s array.

The Octagonal Pascalloid is a delicate structure. What Octagon may have 16 corners, with 8 corners in a square in the middle? This describes a Combinatoric Pascalloid, but not all Octagonal Pascalloids need conform to such stipulation. It’s just that the Combinatoric Pascalloid is four dimensions (a hypercube) within two, and n-value as a third, represented by one summation of ((a+b)(c+d)(e+f)(g+h))^{ n} and the other is just (a+b+c+d+e+f+g+h)^{ n} , with fractional exponents.

As for the 9-gon, it is like the 5-gon and 7-gon, except for this anomaly at N=11, a traditionally quite accurate series of numbers indicating Pi or the length of a circumference of a circle divided by it’s diameter. One may ask, why no specified decimal point after 3? At least this part I can answer. We can substitute the initial number [1] at N=0 with instead the number [0.000001], the whole structure of the Numegon Pascalloid has its decimal point shifted to the left, causing Pi to appear redundantly at that exact set of locations @ N=11. Moreover, *why* these five digits appear in a number generated from the 9-gon Pascalloid at N=11 is still a mystery at this point.

Overall, we learn from the 9-gon Pascalloid that for all the Pascalloids, where we decide to include a decimal point is arbitrary, as long as it applies to the __whole__ individual number structure, starting with N=0.

And yet when faced with the results of the 10 variable Mayan Spiral Pascalloid, where we decide to include the decimal point is crucial to the accuracy of the Golden Mean, Pi, *e*, and the square root of 2 in this structure, though it also is not bound by a combinatoric expression.