next up previous
Next: 2. The algorithm Up: 2. EEQT - Quantum Previous: 2. EEQT - Quantum

1. Geometry

We have published, as an OpenSource project the algorithm implemented in Java that generates the Five Platonic Fractals - that is fractals generated by five most symmetric detector configurations. The algorithm generates self-similar patterns on a sphere of a unit radius. The points on the sphere represent (pure) states of the simplest quantum system - the spin $1/2$ rotator. This spin $1/2$ quantum system is coupled, continuously in time, to a finite number of symmetrically distributed spin-direction detectors. Thus the symmetry of the pattern reflects the symmetry of the detector directions distribution. Each spin direction is characterized by a vector ${\bf n}$ of unit length. Here we study most symmetrical configurations, therefore we chose direction vectors ${\bf n}_i$ pointing from the origin to the vertices of one of the five platonic solids. We consider the following five detectors configurations:[*]

  1. tetrahedron: 4 detectors along the directions ${\bf n}[i], i=1,\ldots ,4$
    {{0, 0, 1.}, {a[17], 0, -a[3]}, {-a[6], a[12], -a[3]}, {-a[6], -a[12], -a[3]}}
  2. octahedron: 6 detectors along the directions ${\bf n}[i], i=1,\ldots ,6$
    {{0, 0, 1.}, {1., 0, 0}, {0, 1., 0},
    {-1., 0, 0}, {0, -1., 0}, {0, 0, -1.}}
  3. cube: 8 detectors along the directions ${\bf n}[i], i=1,\ldots ,8$
    {{0, 0, 1.}, {a[17], 0, a[3]}, {-a[6], a[12], a[3]}, {-a[6], -a[12], a[3]},
    {a[6], a[12], -a[3]}, {a[6], -a[12], -a[3]}, {-a[17], 0, -a[3]}, {0, 0, -1.}}
  4. icosahedron: 12 detectors along the directions ${\bf n}[i], i=1,\ldots ,12$
    {{0, 0, 1.}, {0.a[15], 0, a[5]}, {a[2], a[13], a[5]}, {-a[10], a[7], a[5]},
    {-a[10], -a[7], a[5]}, {a[2], -a[13], a[5]}, {a[10], a[7], -a[5]},
    {a[10], -a[7], -a[5]}, {-a[2], a[13], -a[5]}, {-a[15], 0, -a[5]},
    {-a[2], -a[13], -a[5]}, {0, 0, -1.}}
  5. dodecahedron: 20 detectors along the directions ${\bf n}[i], i=1,\ldots ,20$
    {{0, 0, 1.}, {a[9], 0, a[11]}, {-a[3], a[8], a[11]}, {-a[3], -a[8], a[11]},
    {a[11], a[8], a[3]}, {a[11], -a[8], a[3]}, {-a[14], a[4], a[3]},
    {a[1], a[16], a[3]}, {a[1], -a[16], a[3]}, {-a[14], -a[4], a[3]},
    {a[14], a[4], -a[3]}, {a[14], -a[4], -a[3]}, {-a[11], a[8], -a[3]},
    {-a[1], a[16], -a[3]}, {-a[1], -a[16], -a[3]}, {-a[11], -a[8], -a[3]},
    {a[3], a[8], -a[11]}, {a[3], -a[8], -a[11]}, {-a[9], 0, -a[11]},
    {0, 0, -1.}}
where the array of real numbers $a[i],\/ i=1,\ldots ,17$ is given in the following table.
$a[1]=\frac{3-\sqrt{5}}{6}$ $a[2]=\frac{5-\sqrt{5}}{10}$ $a[3]=\frac{1}{3}$ $a[4]=\frac{\sqrt{5}-1}{2\sqrt{3}}$
$a[5]=\frac{1}{\sqrt{5}}$ $a[6]=\frac{\sqrt{2}}{3}$ $a[7]=\sqrt{\frac{5-\sqrt{5}}{10}}$ $a[8]=\frac{1}{\sqrt{3}}$
$a[9]=\frac{2}{3}$ $a[10]=\frac{5+\sqrt{5}}{10} $ $a[11]=\frac{\sqrt{5}}{3}$ $a[12]=\sqrt{\frac{2}{3}}$
$a[13]=\sqrt{\frac{5+\sqrt{5}}{10}}$ $a[14]=\frac{3+\sqrt{5}}{6}$ $a[15]=\frac{2}{\sqrt{5}}$ $a[16]=\sqrt{\frac{3+\sqrt{5}}{6}}$
$a[17]=\frac{2\sqrt{2}}{3}$      
next up previous
Next: 2. The algorithm Up: 2. EEQT - Quantum Previous: 2. EEQT - Quantum

2002-04-11