http://people.csail.mit.edu/jaffer/Marbling/Scallops

# Scallop and Clam Shells

Creating a Serpentine Marbling pattern was not much of a challenge because the folds did not branch and merge. Some patterns look as though the tines meet and separate.

One beautiful marbling pattern used for endpapers looks like an array of patterned scallop shells. With their sinews splitting and merging, it is difficult to see how such patterns are generated. Dan St. John of Chena River Marblers was kind enough to explain that this pattern is produced by sinusoidal motion of a comb having two sets of tines offset both in the direction and perpendicular to the direction of the stroke.

Splitting the offset comb into two parts and stroking each separately allows these patterns to be produced with the tools already developed.

In the progression below, the contents of the PostScript procedure Composite-map are displayed with the change from the previous contents in red. Composite-map takes an x and y coordinate pair on the stack and leaves the transformed x and y on the stack.

Here are paint circles stroked upward with a fine-toothed comb.

• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
The first line moves the center of the paint-circles to the centerline of the drawing; and down to counteract the upward drag of the combing.

Make horizontal sinusoidal displacements like the serpentine case.
• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch

Stroke two tines straight upward.
• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch
• 0 1 [ 125 325 ] 0 Line-deformation

Subtract the sinusoidal displacement; now those last two strokes are sinusoidal.
• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch
• 0 1 [ 125 325 ] 0 Line-deformation
• dup 2 mul sin -45 mul 3 2 roll add exch

Now apply the opposite horizontal sinusoidal displacement.
• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch
• 0 1 [ 125 325 ] 0 Line-deformation
• dup 2 mul sin -90 mul 3 2 roll add exch

One displacement of -90*sin replaces two displacements of -45*sin.

Stroke three tines straight upward.

• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch
• 0 1 [ 125 325 ] 0 Line-deformation
• dup 2 mul sin -90 mul 3 2 roll add exch
• 0 1 [ 25 225 425 ] 0 Line-deformation

The third tine is off-screen to the right.

Finally, subtract the sinusoidal displacement.

• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch
• 0 1 [ 125 325 ] 0 Line-deformation
• dup 2 mul sin -90 mul 3 2 roll add exch
• 0 1 [ 25 225 425 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch

[image is linked to PostScript file]

### Variations

Reversing the direction of the initial combing results in a variant looking more like giant clam shells.
• 0 -1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch
• 0 1 [ 125 325 ] 0 Line-deformation
• dup 2 mul sin -90 mul 3 2 roll add exch
• 0 1 [ 25 225 425 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch

[image is linked to PostScript file]

Reversing the direction of the final combing results in a variant like a book cover I have seen.

• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch
• 0 1 [ 125 325 ] 0 Line-deformation
• dup 2 mul sin -90 mul 3 2 roll add exch
• 0 -1 [ 25 225 425 ] 0 Line-deformation
• dup 2 mul sin 45 mul 3 2 roll add exch

[image is linked to PostScript file]

The dense crowding of contours in this pattern makes the sinusoidal channels speckled when raster-rendered.

[image is linked to PostScript file]

Oversampling by a factor of two in both directions reduces the speckling significantly.

Drawing the contours rather than filling them makes images resembling topographic maps. This image is oversampled.

Looking at a print of the entire projection of the paint circles, Carl Mikkelsen noticed that outlines of the top and bottom of the figure are similar. This naturally leads to the next chapter, which extends marbling to a two-dimensional manifold.