1. Stable patterns modify oscillatory pigment depositions
From “The Algorithmic Beauty of Sea Shells” © Hans Meinhardt and Springer Company

2. Many shell patterns indicate that a stable pattern exists that changes the parameters of the oscillating pigment production in a position-dependent fashion
substrate production
Time
Position
In Natica euzona, the oscillation frequency is much higher along several bands, achieved in the simulation by a higher substrate production. The thinner lines also indicate there a shorter half-life of the activator

3. In some regions, the oscillation is more rapid, explicable by a higher substrate supply; the finer pigmentation lines indicate that the activator half-life shorter there
substrate production
Time
Position
The simulations shows the expected reaction along the growing edge. A higher substrate production leads to the higher oscillation frequency.

4. If a wave enters a region of low substrate production (bb), it may terminate. A subsequent activation can take advantage of the unused substrate and trigger earlier
Time

5. Another example of wave termination and a subsequent earlier trigger

6. Nautilus: the oscillation frequency and the growth rate of the shell increases towards the periphery…
…. causing more stripes at the periphery and a curling of the growing shell. Note that the model describes the blind-ending and the bifurcating lines

7. Diffusion and saturation allows merging of traveling wavesbb
The Nautilus shell above shows bifurcating lines (arrows). This indicates that a traveling wave can become so slow that a following wave is catching up. In the model, if there is some substrate diffusion, a wave can come almost to rest. The substrate from the surrounding fuels the autocatalytic reaction. Also wave termination can occur that leads to blindly ending lines (green arrows).

8. Diffusion and saturation allows merging of traveling waves
The animated simulation at right shows that the merging of two waves occurs as follows: a wave comes to rest and splits up a new wave that spreads into the backwards direction. It merges then with the subsequent wave. This coming to a rest is also clearly to be seen on the natural pattern (arrows).

9. Saturating activator production allows slower waves without wave termination, causing the fine steep lines
Note that in this case Cypraea diluculum the juvenile pattern results from a space-time record, the later-added spot-like pattern results from a two-dimensional patterning

10. Saturating activator production allows slower waves without wave termination, causing the fine steep lines
In a region of high substrate production, bursts of activation occur nearly simultaneously. The oscillation frequency is high.
From there, activation spread into regions with low substrate production and thus lower oscillation frequencies. Restricted activator exchange leads to huge phase differences between adjacent cells, i.e., to steep lines. Due to the saturation, the activated phase is long enough to trigger the neighboring cell even under this conditions. Thus, saturation prevents wave termination.

11. Rows of patches: promotion or inhibition?
Many shells display rows of patches. This indicates a controlling stable pattern. There are two possibilities: (i) the stable pattern promotes the pigment-forming reaction; the spots appear on top of the maxima of the stable pattern (left). (ii) the stable pattern inhibits the pigmentation reaction; the spots appear between the stable pattern (center). The frequent crescent-like shapes of patches are better described with the latter model. This model accounts also for the fact that pigment-free regions between the rows are usually narrower then the rows itself…

12. The stable pattern is still regulating
The pattern on Scaphella junonia shows that the stable controlling pattern is still actively regulated. A row of large patches became spitted into two, indicating that a new pigment- suppressing maximum was inserted (arrows). At other positions, narrow rows of pigmented patches are inserted (arrowheads), indicating that the space between two inhibiting maxima became large enough to allow the onset of pigmentation (see simulation on the next page).

13. The distance between two inhibitory maxima became wide enough to allow the onset of pigmentation.
While normal patches emerge mostly in synchrony, the phase differences between the normal and the newly appearing patches are much larger. Some pulses maybe skipped.
A new maximum of the pigment-suppressing pattern were inserted: split of a broad row

14. Conclusion:
The dynamics of pigment deposition indicates that in many shells an invisible pattern exists that modifies the parameters of the oscillating pigment deposition. The oscillation frequency may be enhanced or lowered or oscillations are completely suppressed.
In the examples discussed, the inferred invisible pattern is essentially stable in time. However, it takes only a few parameter changes, and this modifying pattern would be oscillating too. As shown in chapters 7 - 9, this leads to the highly complex patterns that are more difficult to decode.