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 southwest 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:
 1. Folding the diagonal of a rectangle
 2. Why does “A” paper work?
 3. Why does the “60 degree fold” work?
 4. Dividing lengths into equal parts
 5. The end … or the beginning?
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
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. 2326) http://www.atm.org.uk/MathematicsTeachingJournalArchive/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:

Add extra pleats for patterns. For example, try

Minako Ishibashi’s Brocade (see http://pandacub143.deviantart.com/art/OrigamiBrocadeTutorial103380198 for some photo diagrams by pandacub143)

Lewis Simon’s “Ninja” Cube and other models.


Use an oblong and vary the angle e.g. Francis Ow’s 60 Degree Unit and other series. Tom Hull famously used the 60 degree unit in his Five Intersecting Tetrahedra, available at http://mars.wne.edu/~thull/fit.html

Alter the locations of the folds. See a video of Dave Mitchell’s Mondrian Cube at http://www.origamispirit.com/2015/05/mondriancube/
Skeletal Octahedron
For variations, you could:

Evert interior vertices

Split the diagonal creases into two pairs of creases. The extra central area can create a concave nonuniform rhombicuboctahedron. See https://en.wikipedia.org/wiki/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/make3dorigamimagiccirclemediumeasy/; for a video, try https://www.youtube.com/watch?v=jQmG6kf2bw
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?


Pictures of many more planar models are at http://www.origamee.net/gallery/planars.html



David Petty’s Planar Modular Origami gives instructions for several planar models. See http://supplies.britishorigami.info/index.php?main_page=product_info&cPath=35&products_id=463


Borromean Rhombuses

Orderly Tangles

Polypolyhedra. See http://www.langorigami.com/article/polypolyhedra
Cube

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.

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.