Math Papers by Ekaterina Lukasheva

Ekaterina Lukasheva is a mathematician and programmer  from Moscow who has taken to creating incredible origami and kusudama.

“Origami tessellations are complex geometrical 3-d structures. These surfaces are made using origami technique, which means only one sheet of paper is folded without stretching, cutting or gluing. These 3-d structures are indeed developable surfaces. This also means that those pieces represent the result of continuous isometric mapping of the flat surface to a 3-dimensional surface. It’s hard to believe, but they can be stretched back to a flat sheet at any time. Moreover the collapse/stretch process would be smooth. ”

Bridges Math Art Gallery

She has published three books on the art, all available on amazon.

I was particularly drawn to her recent post on Instagram where one of her designs exhibits interesting behavior…

Typically, if you compress a material in one direction, it will expand in the other, and vice-versa. This is essentially a conservation of volume effect, and is described by its Poisson Ratio. Things like rubber and most liquids are “incompressible” and will have a Poission Ratio value of 0.5, meaning a compression of a certain magnitude in one direction will turn into an extension of the exact same magnitude in the other (in 2D). Materials like metal can undergo some amount of compression in one direction before expanding in the other.

However, Lukasheva’s tessellated origami sheet above has a negative Poisson Ratio. An extension in one direction actually turns into an extension in the other. Materials that exhibit this behavior are known as auxetic materials and are quite rare (interesting fact: your tendons – in their normal operating range – are auxetic). In the case of the origami sheet, it takes advantage of the third dimension to allow for this novel behavior.

Auxetic material research is part of the larger field of metamaterials, and has applications in body armor, packing material, and high-tech shock absorbers, and beautiful paper art.

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