Mathematical Marbling

Marbling refers to painting techniques for creating a stone-like appearance or intricate flowing designs.

Marbling originated in Asia more than 800 years ago and spread to Europe in the 1500s, where it was used for end-papers and book covers.

To the right is a detail from a marbled necktie made by Chena River Marblers. Some other sites with marbled images are:

This page is about generating marbling designs mathematically.

Jürgen Gilg (d. 2022), Luque Manuel, and I collaborated in 2018-2019 to create the pst-marble package on CTAN.org; it enables one to create mathematical marblings using LaTeX. For example, Jürgen's code to produce the marbling on the right required only three marbling actions:

	
\documentclass[pstricks]{standalone}
\usepackage{pst-marble}
\begin{document}
\begin{pspicture}(-5,-5)(5,5)
  \psMarble[
    actions={
      0 0 [5 100 0 tines][6 75 20 tines] 45 colors 65 serpentine-drops
      -90 [7 400 24 tines] 40 200 31 rake
      180 [3 400 24 tines] 40 200 31 rake
      35 150 shift
  }](10,10)
\end{pspicture}
\end{document}

http://pstricks.blogspot.com/2018/09/the-marbled-paper-with-pstricks.html shows other examples of marblings you can create using pst-marble.

You can now create pst-marble designs online! The first tutorial (about the nonpareil pattern) is Mathematical Marbling How-To.

If pictures are worth 1000 words, then mathematical marbling animations¹ can explain much.

The tunes that accompany these 3 animations are Turkish songs popular for "international folk-dancing". The songs have no particular connection to marbling, although marbling is a vibrant craft in Turkey.

Rake Patterns

Click here for the high-resolution Bouquet animation. The bouquet pattern (called "scallops" on some of my older web-pages) is a staple of Western marblers; the curved tracks are produced by 2 rakes attached together with an offset between them. This animation shows that the bouquet can also be created with a single rake drawn twice. Thanks to Roberta Jaffer for the lovely color palette.

I wrote a paper with Shufang Lu, Xiaogang Jin, Hanli Zhao, and Xiaoyang Mao explaining the mathematics used for creating bouquet and other patterns:

Lu, S.; Jaffer, A.; Jin, X.; Zhao, H.; Mao, X.; ,
"Mathematical Marbling,"
IEEE Computer Graphics and Applications
Nov.-Dec. 2012 (vol. 32 no. 6) pp 26-35
ISSN: 0272-1716
DOI:10.1109/MCG.2011.51

The video² created to accompany the article is on YouTube.

The following four pages provide a computation-oriented explanation of the mathematical marbling process:

Everyday Marbling

Click here for the high-resolution Latte animation. Marbling can be found in coffee cups around the world. This pattern is produced using the same mathematical transforms as the other marblings, but using a single stylus instead of a rake.

The fluid-dynamics of short strokes are explained in:

Aubrey Jaffer,
Oseen Flow in paint Marbling (pdf),
arXiv:1702.02106 [physics.flu-dyn]

Blake Jones has done some righteous coding, creating a GPU implementation of this algorithm which executes so fast that it renders the marbling from arbitrary stylus movements interactively in real time! The first video on his Turing clouds webpage shows the system in action.

A vortex also appears in the Latte animation; it is the subject of:

Aubrey Jaffer,
The Lamb-Oseen Vortex and Paint Marbling (pdf),
arXiv:1810.04646 [physics.flu-dyn]

Spanish Wave

Click here for the high-resolution Wave animation. The initial zigzag is pounded into a network of filaments by paint drops. The animation ends with the tank bottom's view of paper being laid on the pattern surface as the paper is jiggled back and forth. The emergence of this "Spanish Wave" pattern is usually hidden from marblers using paints in a tank.

The mathematics of Spanish wave and Turkish moire marbling are explored in:

Aubrey Jaffer, 2019,
Pigment Transport in Paint Marbling (pdf),

3-Dimensional Marbling

Shufang Lu, Xiaogang Jin, Aubrey Jaffer, Fei Gao, Xiaoyang Mao,
"Solid Mathematical Marbling",
IEEE Computer Graphics and Applications
vol. 37, no. 2, pp. 90-98, Mar.-Apr. 2017,
DOI:10.1109/MCG.2016.42

Shufang Lu, Yue Huang, Xiaogang Jin, Aubrey Jaffer, Craig S. Kaplan, and Xiaoyang Mao.
Marbling-based creative modelling.
Vis. Comput. 33, 6-8 (June 2017), 913-923.
DOI:10.1007/s00371-017-1396-3

Repeating Patterns

Fractals

The Mandelbrot set and related curves display banding, but have only a couple parameters affecting them. These couple parameters change disparate features throughout the image. Although one can affect the drawing, one cannot control it.

That is what I thought until I tried to create a Karman vortex street in a real marbling tank using less viscous sizing (bath) than usual. The mushroom patterns which emerged had smaller mushrooms inside of them! In the photograph, I have outlined mushrooms at 3 different scales (click for larger image).

Self-similarity is the hallmark of fractals; fractal designs can emerge from marbling when the viscosity of the bath is low enough that momentum of the liquid is significant. In high viscosity sizing, drawing the rake at high speeds causes bubbles from cavitation, but not vortexes.

See my blog posts about vortexes and bubbling for the quantitative analysis.

¹ The Bouquet, Latte, and Wave animations were produced from PostScript code modified from pst-marble; the tool-chain was Ghostscript "epstopdf", ImageMagick "convert", and "ffmpeg".

² The software application shown in the IEEE video was never released; it was not used for producing these three animations.

I am a guest and not a member of the MIT Computer Science and Artificial Intelligence Laboratory. My actions and comments do not reflect in any way on MIT.