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Next: 4. Notes Up: 3. The Five Platonic Previous: 5. Dodecahedron

6. Other Polyhedra

In principle our algorithm should create quantum fractals for each of the regular polyhedra. The only restriction on the array of vectors ${\bf n}[i]$ is that they are all of unit length, and their sum is a zero vector. We added, for comparison with the Platonic solids configurations, two additional simple yet regular figures: double tetrahedron and icosidodecahedron. Notice that tetrahedron is self-dual, while dodecahedron and icosahedron are dual to each other. Double tetrahedron array is obtained by combining ${\bf n}[i]$ with $-{\bf n}[i]$ - that is with the inverted configuration.

Figure: Quantum Double Tetrahedron. $\alpha=0.78$
\includegraphics [width=11cm, keepaspectratio=true]{q_double_tetrahedron_07c.eps}

Icosidodecahedron has particularly simple and elegant expression for its 30 vertices: they are of the form: $(\pm 1,0,0)$ and its cyclic permutations, and $\frac{1}{2}(\pm 1,\phi ,\frac{\pm 1}{\phi})$ and its cyclic permutations, where $\phi=\frac{1+\sqrt{5}}{2}=1.61803\ldots $ is the golden ratio. All of its edges are of length $\phi$. For its 30 vertices $\alpha=0.85$ was needed to resolve the atrractor's fine structure.

Figure: Quantum Icosidodecahedron. $\alpha=0.85$
\includegraphics [width=11cm,keepaspectratio=true]{q_icosidodecahedron_085c.eps}

next up previous
Next: 4. Notes Up: 3. The Five Platonic Previous: 5. Dodecahedron

2002-04-11