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I am looking for an appropriate open source software that would allow me to design/generate geometrical patterns (2d and 3d), tilings, fractals and similar. Any recommendation would be highly appreciated.

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For two dimensional work, a vector editor such as Inkscape would be useful to your goal. It supports cloning/tiling. A quick G-search shows there are many tutorials regarding creating fractals with Inkscape. Inkscape is free, opensource and multi-platform.

For three dimensional work, you may want to consider OpenSCAD, also free, opensource and multi-platform. OpenSCAD is a text based description-type program which allows one to create code to generate recursive and similar shapes by incorporating loop commands. Some two dimensional creation is supported by OpenSCAD.

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