Origami paper is the stuff that one folds origami out of. While it is true that you can fold origami out of nearly any kind of paper, they make special paper for purpose in Japan and China.

There are different kinds of origami paper. There is the stuff you see in art supply stores, that's very thin and usually got different colors on one side. There's also metallic papers, which are also very thin. Finally there are heavy, printed papers, which are quite beautiful and have stylized patterns. These are a joy to work with.

There's a kind of joy also in folding origami out of scraps of paper you find lying around or from candy bar wrappers or other strange materials. Or is that just the look on bystanders' faces?

A popular kind of paper with the origami mastahs who make their own is a three or more layer laminate paper made with two thin layers of rice paper, often patterned, with a middle layer of metal foil. The advantage to this paper is that it retains creases very easily, unlike raw rice paper which requires more force (which is difficult especially for fine details or awkward corners), but it isn't as prone to tearing or inadvertent creasing (foil shows any crease made, while rice paper hides, to a degree, small adjustments or intermediary steps) as raw foil paper.

When most people think of origami paper they probably think of the kind found in any arts and crafts shop - thin, smooth, multicoloured on one side and cut to squares. This type of paper is called 'kami'. Kami is the Japanese word for paper but is used in English in this specific sense. It was invented in the late 20th century as a mass produceable, attractive way to teach school children origami. To a serious folder it is usually not used in a finished model; it is prone to fading, may be too thick or thin, may be inaccurately cut and generally leads to an unimpressive result. However, it is in wide use since it is cheap - often selling in packs of a hundred or more and comes in a dazzling range of colours. 

A sometimes more suitable and certainly traditional paper is washi, or chiyogami if it is patterned. Washi is not made from wood pulp like office paper, instead it is made from a plant fibre called gampi. It can also be made from mulberry, wheat or rice. As such, it is quite soft and has a fluffy texture. It is often patterned with bright colours and shapes such as birds, flowers and geometric shapes. Washi is good for simple models but with complex creasing goes mushy and tears.

For complex folding, such as the many legged insects of Robert Lang foil paper is a good choice. Foil may be adhered to paper or even tissue. The foil provides stong creases while the paper stops rippage.

For wetfolding - origami with damp paper that dries into curved shapes - paper with a substance called sizing is used. This substance when damp releases the paper fibre and when dry locks it in place again. A suitable paper for this is watercolour paper, which is quite thick and can absorb a lot of water.

Other specialty paper includes tant, which is relatively thin and strong, and handmade paper, such as the kind from the Origamido studio which is made for specific projects. 

All that said, to start learning origami no special paper is needed. Origami can be made from office paper, tissues and lolly wrappers. Even things that aren't paper, such as banknotes, napkins and plastic film have origami potential.

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