Could ancient Egyptians have done origami with papyrus?

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The papyrus is not isotropic, is anisotropic. It can be imagined as a set of elastic thin sticks all fixed in one orientation by plastic mass. It can be easily folded in one direction and is elastic and springy in the perpendicular one. So, you can do some sort of origami, especially with a wet sheet of papyrus. But its rules will be much different from those of the Japanese one and much more complicated. For example, large and small sheets will behave themselves absolutely differently.

When Egyptians wanted to make papyrus more or less isotropic, they put two layers of it, crossing the threads by right angles. That construct, though, according to my own experiments, is absolutely non-pliable. It is elastic (springy) in all directions.

But kites could be made of papyrus. But weren't....

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No. According to Wikipedia:

Codices were an improvement on the papyrus scroll, as the papyrus was not pliable enough to fold without cracking and a long roll, or scroll, was required to create large-volume texts.

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