Third, repeating a pattern under magnification is not sufficient
to guarantee fractality. |

For example, a spiral (**A** below) is symmetric under
magnification about its
center point, but about no other point. |

Iterating this process, the limiting shape is
just a single point. |

In order to produce a fractal, at each level the decomposition
must involve at least two scaled copies. |

Nested dolls (often Russian, but not in this instance, **B**) are another example
of a non-fractal involving a single scaling transformation, as is the cat bottle (**C**). |

On the other hand, the cow picture (**D**) is fractal, as would be
more obvious if the cow's left earring were turned toward us. |

Thanks to Nial Neger for finding the cat and cow pictures. |

Click on all but the spiral to magnfy in a new window. |