Bucky Number Mandelbrot

by Cliff Nelson

Abstract

The Mandelbrot set computation iterates until the distance exceeds two. Two is two times the distance of unity for complex numbers. Unity for (Subscript[B, 3])numbers is B[{1,1,-2}]. The VectorDistance function in Synergetics coordinates measures the distance going by way of nearest neighbors, similar to the manhattan distance for perpendicular coordinates. A Mandelbot like set is generated in this notebook, iterating until the VectorDistance is two times the VectorDistance of unity for (Subscript[B, 3]) numbers.

Reference

Written by Clifford J. Nelson Jan 11, 1997. Modified August 12, 1998.
Copyright Clifford J. Nelson.
See the Mathematica 3.0 notebooks SynergeticsCoordinates.nb and BuckyNumbers.nb in MathSource.

Disscusion

<< RBFields.m

<< SynCoordinates.m

B[{a, b, c}] B[{d, e, f}]

StraightLineDistance[{a, b, c}]

VectorDistance[{a, b, c}]

StraightLineDistance[{1, 1, -2}]

VectorDistance[{1, 1, -2}]

VectorDistance[b] ≥ 4 instead of StraightLineDistance[b] ≥ 2(Sqrt[3]) is a different distance function than the ordinary way of computing the Mandelbrot set , but it makes a familiar shape. The distance that must be exceeded is two times the distance of unity from the origin. Unity is 1 for complex numbers and the distance for complex numbers is (Sqrt[(Re[1]^2 + Im[1]^2)]) == (Sqrt[(1^2)]) == 1. Unity for B3 numbers is B[{1,1,-2}].

btimesb  =  
      Compile[{a, b, c}, Evaluate[timesbnum[{a, b, c}, {a, b, c}]]]

btimesb[1, 2, -3]

B[{1, 2, -3}]^2

B[%%] - %

The Mathematica Journal  Volume 7, Issue 1,  page 14 showed how to generate the Mandelbrot set with DensityPlot and complex numbers. The function mand uses B3 numbers instead, and a different distance function.

The VectorDistance[x] is the same as \(TraditionalForm
      \`\(1\/2\) 
        \(∑\+\(i = 1\)\%\(Length[x]\)Abs[
            x[[i]]]\)\) if all x are real, so VectorDistance[x] ≥ 4 can be replaced with ((Plus @@ Abs[x])≥8 ).

mand[b_]  := 
      Length[FixedPointList[(btimesb @@ #) + b &, b, 50, 
            SameTest  ->  
              ((Plus @@ Abs[#2]) ≥ 8 &)]]/
        (70.0 )

DensityPlot wouldn't be right because the x and y values would be shown in perpendicular 90 degree coordinates. StoP[{x,y,-x-y}] transforms x and y to 90 degree coordinates from what I call "Synergetics coordinates".
Descartes is said to have expected the use of oblique coordinate systems.

StoP[{a, b, c}]

It takes a long long time to generate the 40,000 points in the next statement.

g = 
      Table[{Hue[mand[{x, y, -x - y}]], 
          Point[StoP[{x, y, -x - y}]]}, {x, -2.5, 2.5, N[(5/200)]}, {
          y, -2.5, 2.5, N[(5/200)]}];

Is g the Mandelbrot set or just something close to it?

Show[Graphics[g], AspectRatio -> Automatic]