Learning Mathematics with Origami

Learning Mathematics with Origami, front cover
Learning Mathematics with Origami

Learning Mathematics with Origami is now available, either as a printed book with PDF download or PDF only [ISBN: 9781898611950]. You can also buy the electronic download from OrigamiUSA.

Read the review by Fintan Lynn, a primary school teacher in south-west of England, and the review by Charlene Morrow, Faculty Emerita, Mount Holyoke College and member of board of directors, OrigamiUSA.

You can interact with the dynamic geometry models here. Preview sample material from Mathematics Teaching 254. 

The mathematics behind the folds is a series of five articles that covers some of the material in Learning Mathematics with Origami from the perspective of practising paperfolders. Some material is not in the book e.g. Cube From Thirds, Starfish and eight methods for folding a square into thirds:

For some background information about one of the coauthors, read Harnessing the power of practical activity to bring mathematics to life.

Sue gave a talk to student teachers for the IMA (The Institute of Mathematics and its Applications). You can watch the recording of Using Origami in the Mathematics Classroom – Annual Student Teacher Lecture. One viewer commented “this is so inventive – great stuff!”


Updates, clarifications and corrections

Cairo Tiling made from Left Over Rectangles
Cairo Tiling made from Left Over Rectangles

The Cairo Tiling shown on page 18 was created by Dave Mitchell. See http://www.origamiheaven.com/pdfs/cairotile.pdf for more details about forming the tessellation, and more ways of folding the tile. David Bailey’s website shows that the Cairo Tile probably appeared in Cairo only a few decades ago.

The original name of Dave Mitchell’s Flipper on page 21 is Ad Infinitum. His diagrams are available at http://www.origamiheaven.com/pdfs/adinfinitum.pdf

The original name of Dave Mitchell’s Cube Tube on page 26 is Two Way Tube. This is the name of Robert Neale’s original version made with a cut.

The example lesson features models presented in Mathematical Origami by Liz Meenan (2001) (Mathematics Teaching 176, p. 23-26) http://www.atm.org.uk/Mathematics-Teaching-Journal-Archive/3875

Extensions and related models

Sonobé unit

Mike Naughton’s instructions and variations are available at amherst.edu/media/view/290032/original/oragami.pdf

For variations, you can:

Skeletal Octahedron

For variations, you could:

  • Evert interior vertices

Skeletal Octahedron with Everted Inner vertices

Nonuniform Concave Rhombicuboctahedron

  • Rotate the creases about the centre.

  • Use a different number of units, some with fewer folds. Kenneth Kawamura’s Butterfly Ball uses 12 units with only one short mirror line creased on each unit. The Faceted (Pimpled) Octahedron uses units where the waterbomb base is turned inside out into a preliminary fold – this is the limit of the eversion process mentioned above.

  • Sink central the points in each unit for ED Sullivan’s XYZ.

  • Bisect flaps with rotational or mirror symmetry for a 3D star. Versions have been made by Javier Capoblanco, Tung Ken Lam, Joe Power and others, but can be hard to assemble. Eight units make the delightful Carousel (creator unknown), sometimes called the Origami Magic Circle. For photo diagrams, go to https://snapguide.com/guides/make-3d-origami-magic-circle-medium-easy/; for a video, try https://www.youtube.com/watch?v=jQmG6kf2-bw

Related models

  • Planar units: the Skeletal Cuboctahedron consists of four regular hexagons. Transforming the regular hexagons into equilateral triangles produces WXYZ. How could you use hexagrams instead?

Skeletal CuboctahedronSkeletal Cuboctahedron with Irregular HexagonsWXYZSkeletal Cuboctahedron with Hexagrams


  • Fuse’s Belt Cube and variations have been createed/discovered by many people. If you have mountain folds at one quarter and three quarters along one edge, make valley folds at three eigths and five eigths to create pockets at the centre. You can vary the width and location of the belt.

Belt Cube

  • Dave Mitchell’s Columbus Cube and Tower. Invert a vertex so that you can stack the shapes into a tower. What else can you build?

  • Dave Mitchell’s Icarus Cube, also created/discovered by other creators. See an image at http://origamiheaven.com/modulardesigns.htm, along with pictures of the Columbus Cube and Tower.