Origami Heaven

A paperfolding paradise

The website of writer and paperfolding designer David Mitchell


Useful Rectangles

This page introduces various rectangles that are of proven (or potential) use as starting shapes in origami (and particularly in modular origami) and briefly discusses their most useful folding geometries.

Folding geometries are of two quite distinct kinds, natural and embedded. The difference between these two types of folding geometry is explained in Folding Geometries and Angular Systems.

Whilst all folding geometries can (at least in theory) be obtained from all paper shapes, the optimum folding solution for a particular design will usually be found by starting from the paper shape that most naturally yields the angles required.

In modular origami rectangular paper shapes are often used in double or double-strip form. A double rectangle is made by joining two rectangles along their longer edges, a double-strip rectangle by joining two rectangles along their shorter edges. Designs made from triple and triple-strip rectangles etc are also occasionally found.

  The square is undoubtedly the most versatile starting shape in origami, to the extent that some paperfolders (even some modular paperfolders) use no other (with the unfortunate result that it is quite often possible to improve the elegance of their designs by the simple expedient of switching to a more appropriate paper shape).
The natural folding geometry of the square produces angles of 90 and 45 degrees. This system of angles can be called standard folding geometry. It is easy to embed standard folding geometry within any other paper shape, no matter how unusual or irregular the shape may be.

The versatility of the square arises because it is particularly easy to embed the folding geometry of most other useful rectangles within it. The square can also easily be divided into a number of standard grids of smaller squares or silver triangles, many of which form the basis for a whole range of related designs.

The silver rectangle (also known as the pure or DIN rectangle) has sides in the proportion of 1:sqrt2 and has the unusual property that it is its own double.

The natural folding geometry of the silver rectangle yields angles of approximately 110, 70 and 55 degrees. These angles are found in the structure of the tetrahedron, the cube and the cuboctahedron as well as in the many interesting forms known as rhombic polyhedra.

A4 paper is a good approximation of a silver rectangle.

  The bronze rectangle has sides in the proportion of 1:sqrt3 and its natural folding geometry yields angles of 120, 60 and 30 degrees. The bronze rectangle is its own triple.

The bronze rectangle is, at present, rather under-used in modular origami design, perhaps because bronze rectangle geometry can so easily be embedded into any other rectangle.

The leftover rectangle is the rectangle left over when the largest possible square is removed from a silver rectangle.

The leftover rectangle has sides in the proportion of 1:1+sqrt2 and naturally yields angles of 135, 67.5 and 45 degrees.

The potential of this rectangle (which is often produced in quantity when cutting squares from A4) is still largely unexplored.

The natural folding geometry of the leftover rectangle is also natural to the silver rectangle and easily embedded in the square.

  The twin platinum rectangles (the rectangles which contain the twin golden-proportion triangles arranged apex to apex) naturally yield the 108, 72 and 36 degree angles required to model polygons and polyhedra related to the regular pentagon.

The platinum rectangles are, however, at present, little used in origami design, since it is easy to generate, or closely approximate, a similar folding geometry from the 3x1 or 2x3 rectangles and the square.

The golden rectangle has sides in the golden proportion of 1:1.618 approximately.

There are only a few modular origami designs which make use of the natural folding geometry of this rectangle.