Finding Roots Of Equations Numerically With Bucky Numbers

Here is some inspiration from Bucky Fuller's books Synergetics and Synergetics 2, section 260.42:

260.42  The synergetic coordinate system of nature and its finite macro-micro turnaround-limited hierarchy of primitive ascending or descending timeless-sizeless, omnisymmetrically concentric, polyhedral components provides the human mind with a rational means of resolving problems by bringing nature into a corner__a convergent terminus center, a four-dimensional corner of the four-dimensional planes of the tetrahedron. Only with the four-dimensional convergence and divergence of synergetics can the human mind reduce problems to comprehension as minimum-limit systems. The minimum polygon is a triangle; the minimum polyhedron is a tetrahedron; both of their structural behaviors are unique (see Secs. 614.00 and 621.00). By their academic training humans think only in terms of parallel and rectilinear coordination, and so they tend to hold to the unresolvable parallel interpretations of their lives’ experiences. They seek to maintain the status quo and__despite the organic and biologic manifests of birth and death__they fail to be able to take advantage of the cornerability of comprehension and the positional fixes provided by the four-dimensional, synergetic, convergent-divergent coordination.

The five rational solutions for x in the equation = B[1,1,1,1,-4] are shown in the next statement.

The B number roots of unity rotate the coordinates of their B number coefficients.

The function findaRootOne is Newton's method when iterated. The function findaRootTwo gives exactly the same results as dividing fs[x] by its symbolic first derivative at x and subtracting that from x if fs[x] is a polynomial and Length[x] is greater than half the highest exponent of x in fs[x]. The function findaRootOne is faster than findaRootTwo. Sometimes it's not easy to find a good value for delta for findaRootOne.

numbers are fields when they have an odd prime number n of coordinates if exact rational numbers are used when n is greater than 3. Floating point coordinates for numbers can get too close to certain irrational quantities so that multiplication of two non zero B numbers gives zeroes in every coordinate. But, floating point arithmetic is very fast compared to exact rational arithmetic, so lets take a chance with the N function.

The function findaRootThree is the least general findaRoot, but it goes toward a rational number solution of - x - 1 = 0 from a firstGuess of B[1,-1,-1,1,0].

The iteration is getting closer and closer to a zero of the function f, and it gets as close as you want with more iterations.

Global`gB5

The sequence of tetrahedrons in the graph above should end with an edge length zero tetrahedron. You could draw a line from each of the four vertexes of a tetrahedron to the respective vertexes of the next tetrahedron in the sequence and compute where the continuations of the lines would meet at the edge length zero tetrahedron point to figure out what the next guess should be to find a zero of the function.

Global`mz

Sometimes you only want a single number answer. The function g has been defined to use the first coordinate of the input B number, and findaRootlast, below, stops changing x when the value of the last coordinate, which is minus the edge length of the tetrahedron represented by the number, is zero.

Global`mid

Global`width

The variable firstGuess has complex numbers this time and some functions won't converge to zero without them or with numbers as the "coordinates". That is, × C or × is necessary sometimes.

The last coordinate is pretty close to zero, above.

The imaginary tetrahedron of the Bucky Numbers is shown starting with the color Hue[.5] in the graphic below.

The edge length of each tetrahedron in the graphic above is the value of - x - 1 for each guess. All of these examples are just given to show what Bucky Fuller might have meant by "bringing nature into a corner" and using "four dimensional positional fixes". It can be done without BuckyNumbers, just Synergetics coordinates. Can you "zero in" on a solution to an equation quickly by following the four lines that connect the respective vertexes of two adjacent tetrahedrons in the sequence of tetrahedrons, that have exact rational synergetics coordinates, to a volume zero tetrahedron?

Created by Wolfram Mathematica 6.0  (29 March 2008)