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Drawn Mathematics

Mathematics you can see, draw, and fold.

Category 03 of 05

Crease Patterns

Reading origami crease patterns, the theorems of flat folding, facts you can fold from one sheet, and traditional models, with each pattern's check stated.

3 entries

  • Woodblock print of a folded paper crane with wings spread and a much smaller crane balanced on a tall point above its back.

    Crease Patterns

    Flat Origami: Mountains and Valleys Differ by 2 at Inner Points

    Read an origami crease pattern: sort its lines into mountains and valleys, then count at each inner point. If it folds flat, they differ by 2. Crane included.

    Image credit

    "Hiden Senbazuru Orikata" by Rokiku and Rokōan — https://commons.wikimedia.org/wiki/File:Hiden_Senbazuru_Orikata.jpg — Public domain (Book published in Kyoto with a preface dated 1797 (National Diet Library record 000007300291); everyone who made a book printed in 1797 died long before 1956; PD worldwide (US: published before 1930; Japan and EU: authors died more than 70 years ago); faithful scan of a flat printed page); modified: cropped to the upper part of the page (the poem and the cutting diagram below are left out); colours not altered, not enlarged

  • Illustration of a sheet of white paper with one corner folded down, lying on green graph paper with a brass compass in front of it.

    Crease Patterns

    A4 Paper Size: Built So Each Half Keeps the Same Shape

    A4 paper is 210 × 297 mm, a √2 rectangle to within a millimetre. Fold one sheet to see the ratio, then follow the proof that only √2 survives being halved.

    Image credit

    Illustration: Drawn Mathematics. Generated scene illustration — not a photograph, historical image or researched reconstruction. Use terms: may be reused with credit to "Drawn Mathematics"; provided without any claim of historical accuracy.

  • Illustration of a sheet of cream paper with faint traces of old creases and one corner folded up, on a green drafting table between a pencil and a brass compass.

    Crease Patterns

    Kawasaki's Theorem: A Flat Fold Splits 360° Into Two 180°s

    Kawasaki's theorem: an inner point of a flat sheet folds flat only if alternate angles each total 180°. See a worked example, the ant-walk proof and the limits.

    Image credit

    Illustration: Drawn Mathematics. Generated scene illustration — not a photograph, historical image or researched reconstruction. Use terms: may be reused with credit to "Drawn Mathematics"; provided without any claim of historical accuracy.