1 Horror Vacui 2 A Very Patient Turtle Who Draws Lines 3 A Taxonomy of Fractology 4 Gallery of Specimens |

Root 2 Family | Root 3 Family | Root 4 Square Grid Family | Root 4 Triangle Grid Family |

Root 5 Family | No Root 6! | Root 7 Family | Root 8 Family |

Root 9 Square Grid Family | Root 9 Triangle Grid Family | Root 10 Family | Root 12 Family |

Root 13 Square Grid Family | Root 13 Triangle Grid Family | Root 16 Square Grid Family | Root 16 Triangle Grid Family |

Root 17 and Beyond... | 5 My Brain Fillith Over | 6 References | 7 Acknowledgements |

There is a Triangular Grid root13 Family as well as a Square Grid one. Let's check out a few specimens. Here is a curve I discovered which is a partial gridfiller. I'll show it with an expanded version of the diagram scheme: |

There are a few curious-looking palindrome dragons in this family: |

Generalized Gosper Curves Discovered by Fukuda, Shimizu, and Nakamura |

Fractal explorers in Japan devised a search algorithm that finds what they call "generalized Gosper curves" (Fukuda, et al [4]). These are plane-filling curves that use the scheme I described on page 97. Basically, the Gosper Curve is drawn over a grid of seven triangles arranged in a checkerboard fashion. The picture at right shows how the interval length stretches from A to B. Each of the seven triangles maps to the larger triangle outlined with dotted lines.
This ingenious scheme permits the search algorithm to find an infinite series of generalized Gosper Curves. The diagram at right shows how a similar assembly of triangles can be used to identify a generalized Gosper curve that is a member of the root13 triangle grid family. Below I show how this curve fits within my scheme. Later on I'll show you another one of the many curves that these explorers have uncovered. |

End of chapter. |

1 Horror Vacui 2 A Very Patient Turtle Who Draws Lines 3 A Taxonomy of Fractology 4 Gallery of Specimens |

Root 2 Family | Root 3 Family | Root 4 Square Grid Family | Root 4 Triangle Grid Family |

Root 5 Family | No Root 6! | Root 7 Family | Root 8 Family |

Root 9 Square Grid Family | Root 9 Triangle Grid Family | Root 10 Family | Root 12 Family |

Root 13 Square Grid Family | Root 13 Triangle Grid Family | Root 16 Square Grid Family | Root 16 Triangle Grid Family |

Root 17 and Beyond... | 5 My Brain Fillith Over | 6 References | 7 Acknowledgements |

Brain-filling Curves - A Fractal Bestiary
by Jeffrey Ventrella Distributed by Lulu.com Cover Design by Jeffrey Ventrella |
Book web site:
BrainFillingCurves.com
ISBN 978-0-9830546-2-7 Copyright © 2012 by Jeffrey Ventrella |
eyebrainbooks.com |
FractalCurves.com |