Fractal Measures in Paper Marbling

Paper marbling is an old art form with major traditions in Europe, Turkey (Ebru), and Japan (Suminagashi). Creating marbled paper involves floating paints on the surface of a liquid bath, potentially using tools like combs or a stylus to stir the fluid, and transferring the resulting pattern to paper. There are a wealth of fluid dynamic concepts that marblers use to create specific effects, from modifying the Reynolds number to get a range of behaviors (Stokes to weakly turbulent) to varying the paint's surface tension to get Marangoni flows. Here we view the formation of patterns through the lens of nonlinear dynamics and, in particular, chaotic passive scalar advection. Classic work by Ott et al. showed that repeated application of a chaotic area-preserving map concentrates the gradient of the scalar on a set that is generically fractal and results in specific restrictions on the fractal dimension spectrum. Through a collaboration with professional marblers, we have numerically tested these ideas on high-resolution scans of marbled paper created by the repeated action of a combing process. The beautiful patterns that emerge clearly show the fractal measures inherent in the art of paper marbling.