http://people.csail.mit.edu/jaffer/Marbling/Scallops
	Scallop and Clam Shells

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.

• exch 180 add exch 110 add
• 0 1 [ 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 ] 0 Line-deformation

• exch 180 add exch 110 add
• 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

• exch 180 add exch 110 add
• 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

• exch 180 add exch 110 add
• 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

• exch 180 add exch 110 add
• 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.

• exch 180 add exch 110 add
• 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.

• exch 180 add exch 110 add
• 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

• exch 180 add exch 180 add
• 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.

• exch 180 add exch 110 add
• 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.