## Origamics: Mathematical Explorations Through Paper FoldingThe art of origami, or paper folding, is carried out using a square piece of paper to obtain attractive figures of animals, flowers or other familiar figures. It is easy to see that origami has links with geometry. Creases and edges represent lines, intersecting creases and edges make angles, while the intersections themselves represent points. Because of its manipulative and experiential nature, origami could become an effective context for the learning and teaching of geometry.In this unique and original book, origami is an object of mathematical exploration. The activities in this book differ from ordinary origami in that no figures of objects result. Rather, they lead the reader to study the effects of the folding and seek patterns. The experimental approach that characterizes much of science activity can be recognized throughout the book, as the manipulative nature of origami allows much experimenting, comparing, visualizing, discovering and conjecturing. The reader is encouraged to fill in all the proofs, for his/her own satisfaction and for the sake of mathematical completeness. Thus, this book provides a useful, alternative approach for reinforcing and applying the theorems of high school mathematics. |

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### Contents

1 A POINT OPENS THE DOOR TO ORIGAMICS | 1 |

2 NEW FOLDS BRING OUT NEW THEOREMS | 11 |

3 EXTENSION OF THE HAGAS THEOREMS TO SILVER RATIO RECTANGLES | 21 |

4 XLINES WITH LOTS OF SURPRISES | 33 |

5 INTRASQUARESî AND ìEXTRASQUARES | 45 |

6 A PETAL PATTERN FROM HEXAGONS? | 59 |

7 HEPTAGON REGIONS EXIST? | 71 |

8 A WONDER OF ELEVEN STARS | 77 |

9 WHERE TO GO AND WHOM TO MEET | 93 |

10 INSPIRATION FROM RECTANGULAR PAPER | 107 |

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### Common terms and phrases

11-parts and 13-parts 13-parts division points 3-parts A4 paper arbitrary point bisector book fold circular regions overlap circular segment circumcenter circumcircle concentration point coordinates divide the rectangle equal parts horizontally excenter exterior angle figure firm crease fold the paper folding procedure guest Haga’s First Theorem Haga’s Second Theorem heptagon hexagon horizontal layout horizontally and vertically host intersection point left edge length long side lower right vertex lower vertex mathematical meet method midline Mother Line paper folding pattern pendulum Pentagon Region petals piece of paper point F point of intersection position of H primary crease pupils Pythagorean Pythagorean Theorem Pythagorean Triangle quadrilateral ratio of sides reference vertex result rhombus Second Theorem Fold segments short side similar triangles square paper square piece starting point Third Theorem fold topic triangle triangular flap trisection point types of polygons Unfold University of Tsukuba upper edge X-creases