The Mathematics of Origami

If you have ever held a piece of origami in your hand you have in all probability been at least tempted to open it just to see how the folding was done. The geometry involved in the piece is something you could easily see in the creases displayed on the opened paper.  

Scientists and artists have studied these geometric aspects as well as origamists and mathematicians. Mathematicians throughout time have developed ways to use geometry to define origami; they have designed highly sophisticated models using fundamental theorems. They have studied and found amazing similarities between tessellations and origami (tessellations is the name for a figure comprised of a shape that is repeated over and over again with no gaps or overlap when fitted to a flat surface).  Teachers around the world have used origami to teach different concepts in chemistry, physics and architecture as well as math. 

Origami construction is defined as the folding of paper using the raw edges, points of the paper and any creases or points subsequently created by those folds.  The folded paper is seen as both an art piece and a geometric form.  The folds produce varying sizes of triangles, rectangles and other shapes.  A single fold can bisect and angle twice or as in the case of a reverse fold, make 4 triangles at once.  

 

When the first steps to making a figure are applied to other figures, resulting in a number of figures having common shapes, the common shapes are called bases.  There are several established bases such as the bird, the kite, the windmill and the water-bomb to name a few.   Modern origami relies heavily on these existing bases alone and in combination when designing new figures.  As an example the kite base is used to make quite a few of the different zoo animals.   Studying the creases of existing models has led to the creation of many new models.  These creases show definite patterns of triangles, rectangles and other shapes.  The geometric study of the crease lines over the last twenty-five years has paved the way for the discovery of new bases.  Not all designs are combinations or parts of other bases; some like the box pleat are completely original.   

Some origamists saw the base as a set of areas each independent of the other differing only in their length and arrangement.  With this in mind they went on to develop computer programs that are capable of doing all the math necessary to generate crease patterns for any base from a given length and area arrangement.   With the aid of computer programs using intricate mathematical theorems origami has become as much a puzzle as a piece of art.  Mathematical origamists are now designing more and more complex, realistic models still sticking to the simple rule of one sheet of paper with no cuts.  These programs are also used to solve problems involving getting large pieces of paper folded to fit a specific sized flat surface.