Marbled torus

Marbling the Torus

The entire R×R plane is affected by comb homeomorphisms. How would we marble some other two-dimensional manifold, or create a marbled wallpaper design which matches itself at the edges?

The Torus

Imagine a rectangular sheet rolled so that one edge meets its opposite edge. In order for a pattern drawn on the sheet to blend invisibly, the colors and boundary-line slopes must match at the joined edges.

Now imagine that tube stretched and rolled so that its top and bottom circles meet. As before, to look continuous the colors and boundary lines must match.

This is the torus. To a 2-dimensional observer in the torus surface, the universe looks infinite, but repetitive; looking directly north or south multiple copies of each feature are evenly spaced; looking directly east or west the copies of each feature are evenly spaced -- but not necessarily with the same spacing as appears north-south.


If we unbend the torus sheet and tile the plane with it (all with the same orientation), the 2-dimensional perception of the universe is unchanged; features repeat at uniform intervals both horizontally and vertically. The seams between copies of the sheet will be invisible. Thus the creation of a marbled torus and marbling wallpaper are equivalent.

Many of the deformations discussed so far are not well-defined on the torus.

Since each feature appears infinitely many times, the total displacement will be an infinite sum. A given point always lies in one half-open interval between a straight stroke and one of its (parallel) images. Without loss of generality, assume that a point at (x,y) lies to the right of a vertical stroke (at xL). The sum of displacements due to the stroke and its images (spaced by rx) to the left side is:

z·uxxL+j·rx = z·uxxL

(urx)j = z·uxxL
The sum of displacements due to the right-side images is:

z·urx+xLx+j·rx = z·urx+xLx

(urx)j = z·urx+xLx
The total displacement is then:
z  uxxL + urx+xLx
1 − urx

Because all strokes will be subject to the constant 1/(1−urx) deepening factor, we don't bother to compute it and let the artist control it through z. Thus we need sum only the contributions from the two closest lines. Because the contributions are symmetrical to both sides of the stroke, it doesn't matter which side of the interval is half-open when the point lies on the stroke.


Creating homeomorphisms from the toroid patch onto itself requires discipline:

The resulting (oversampled) bouquet image can tile the plane with perfect continuity across seams. A monochrome version of this pattern is the background for these pages.

[image is linked to bouquet tiling]

The images below are from an earlier version of mathematical marbling (hyperbolic vs. exponential); I can't find a VRML viewer which does image mapping (2010-12).
cylinder.scm generates a 3d marbled cylinder, shown here from the four compass directions.
Marbled Cylinder Marbled Cylinder Marbled Cylinder Marbled Cylinder

torus.scm generates a 3d marbled torus
Marbled Torus

The next chapter explores application for these designs.

Copyright © 2004, 2007, 2010 Aubrey Jaffer

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.
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