Is there a fractal origami shape that trades volume for area to always keep a flat surface when expanded?

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I'm thinking of something like a 2.5D sierpienski type shape.

The idea is to enable an lcd type screen that could unfold to "any" size by unpacking space filling elements packed in 3d to a 2d surface as it expands by a user pulling it open.

Any resources on how to find an l-system? Or similar for creating something like this?