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Shell patterns

 

Ever wonder how beautiful sea shell patterns form?

This model roughly simulates the patterns formed on certain Mollusc sea shells.

Each time you click flag, a slightly different pattern shows.

Set the slider “% deviant blank” to what you wish. Set steps between tiles to what you wish. Hit flag again, and note the difference in pattern.  

 

Ever wonder how beautiful sea shell patterns form?

This model roughly simulates the patterns formed on certain Mollusc sea shells.

The simplest model of such a pattern is based on a single rule. We can describe this rule in terms of coloring-in blank squares:

You will see that, within each new row, color spreads to local neighbors (positive feedback), resulting in diagonal lines. Diagonals that meet cancel out (negative feedback), resulting in empty spaces. Each step, color only spreads locally, but results in interesting global patterns seen in certain sea shells. Run the Scratch model to see the results and look at the drag and drop code.

A model that yields more realistic patterns uses a simple infection and recovery model. Let’s examine this in some depth. You can start by looking at the models.  We have one version in Scratch (drag and drop code, but lower resolution), and another in NetLogo (higher resolution).

How it works

The model simulates pattern formation on a growing shell. It uses a single horizontal row of Sprites to simulate cells that spread pigment. The Sprites change color over time, then they move down a row and leave a permanent record of the color change on the shell surface. Repeating that process creates this pattern.

The program first creates a row of Sprites. A few Sprites are “infected” (shown as red). At each time step, the infection spreads to the two susceptible neighbors on the right and left of the infected one.

After a Sprite is infected, that Sprite recovers and is immune for the next time step, after which the Sprite returns to being susceptible to infection (you can also lengthen the time of immunity to get variant patterns).

At each time step, the infected Sprite (which is colored red) leaves a “stamp” of itself on the patch behind it, so leaves a red spot. The row of Sprites moves down to the next row of patches. This leaves a permanent record (like a time-space diagram or plot), resulting in the type of patterns on certain shells.

The percentage of Sprites recovering with immunity at each time step is a key part of the pattern formation.

This process creates a pattern roughly as seen in certain Mollusc shells.

Things to notice

If there were no infection process, then the stamping would just produce vertical red stripes. But the infection process allows the red to spread horizontally as well as vertically.

The combination of sideways spread and downwards motion of the row leaves V shaped (or upside down V shaped) patterns. If there were no immunity the red color would eventually just spread over most all the shell. The immunity leaves blank spots inside the red V shapes.

The percentage of Sprites recovering with immunity at each time step is very important part of the process. Why do you think?

Adjust % infected at start and, % random recovery.

This infection and recovery process makes fractal triangles and textile crab shell patterns.

Make a pattern you like.

Starts with a few infected (colored) in 1st row. Any infected is recovered in the next step, moving down, but infects neighbors in the next row. Repeats.

A process like this downward infection and recovery makes a similar pattern seen on “textile” crab shell. Starting with 1 makes a fractal (Sierpinski) triangle, when recovery is never random (set to 0).

 

Reference: The model is roughly based on the pigment pattern formation on mollusc shells as outlined by Meinhardt (2003). The proposed process involves incorporation of pigment during shell growth along the edge. The pigment spreads only in a linear bi-directional way, horizontally, but leaves a record in the vertical direction (direction of shell growth), resulting in a two dimensional pattern. The second dimension is a record of what happens over time, a space-time diagram.

Meinhardt H. 2003, 2009 The algorithmic beauty of sea shells, 4th edn. Heidelberg, Germany: Springer.