Gray-Scott Nomenclature Glossary  

This is a glossary of terms used in the individual pattern descriptions in my Xmorphia gallery and associated pages.

This glossary is incomplete. Contact me with any specific requests for additions or corrections.

autosoliton : See soliton.

blue : The color generally used for areas of high u values. In monochome images the high-u regions are a darker gray or black. The blue-red color scheme was used by Pearson [2], replicating the pH indicator bromothymol blue used in physical experiments like those of Lee et al. [3]. See also red.

branch : In general, any linear feature (a stripe or negative stripe) that has a tip and which is connected at its other end to other linear features via a branch point.

branch point : a place at which three (or rarely, more than three) linear features come together.

branching structure : any pattern(s) involving linear features linked by branch points.

bullseye : A negaton encircled by a negative loop, seen in U-skate world and nearby parameters; see example at (F=0.62, k=0.061). See also target. The patterns at (F=0.42, k=0.059) are not true bullseyes because the outer ring grows as an expanding wave, and are not targets because the inner rings do not travel outwards.

clover : A closed loop with multiple (usually three to five) segments of concave curvature and an equal number of segments of convex curvature.

These shapes are seen in the upper-right of the simulation at (F=0.058, k=0.063); see also (F=0.046, k=0.063).

All "clovers" result from slight asymmetries (departures from a perfect circular shape) in closed loops when the parameter values support growth and lengthening of stripes towards the convex direction. In numerical simulations, an isolated circular loop will often produce a clover with roughly 4-fold symmetry; this is a grid effect. But even though grid effects themselves are "fake", the fact that clovers form at one parameter value and not at another shows that the true pattern dynamics are fundamentally different between those two values, and with the first parameter value, random noise would produce the same type of shape.

Many parameter values that support clovers also support Hilbert loops.

coral : A branching pattern that grows from its branch tips, making more branches as it grows. Seen at top-center near the beginning of the simulation at (F=0.058, k=0.063). Coral growth is seen in figure 1 of Lee [1] and in Pearson's pattern type kappa (κ).

dissipative soliton : See soliton.

drift : Describes the tendency of type nu solitons to repel one another, at a rate that diminishes exponentially with distance.

east : In the Gray-Scott parameter space, as seen in the Pearson paper [2] and on the xmorphia main page, "east" is the direction in which the parameter k increases.

grain : An region containing a locally regular pattern, contrasted with a neighboring region in which the pattern is different. The grains meet at a grain boundary.

grain boundary : The boundary between two grains.

grid effect : Any phenomenon that results from the use of a grid of limited resolution in a numerical simulation, which would not occur in the "ideal" (perfect) system described by the model equations, or in a real physical system. In general, grid effects produce shapes that are aligned in orthogonal and/or diagonal directions, and sometimes cause anomalous movement (such as features that move until they are aligned orthogonally or diagonally in relation to the grid). Grid effects diminish greatly in magnitude when the grid resolution is increased relative to the feature size of the pattern (by reducing Δx).

Hilbert loop : A closed loop similar to a clover, that grows while developing gradually increasing complexity. See (F=0.062, k=0.063) for an example.

A "true" Hilbert loop will grow indefinitely without ever developing branches; however most parameter values produce branches when space becomes tight.

junction : a place where three grains meet.

loop : A round feature that is like a stripe or worm bent around into a closed loop, without ends or branch points, and with similar stability and robustness. See examples at (F=0.54, k=0.063), (F=0.66, k=0.063), and (F=0.78, k=0.061).

When referring to a ringlike structure that spontaneously turns into a non-ring shape, or which does so immediately upon contact with something else, I use the term ring (See (F=0.38, k=0.063), which illustrates this contrast). See also target.

mitosis : A pattern-evolution process in which a single type lambda soliton splits into two. The process repeats as available space permits.

negative : Used to describe name features that resemble another more common feature but with high-u areas exchanged for low-u areas and vice-versa. For example, compare the network of linked loops or bubbles at (F=0.098, k=0.057) to a similar network of linked negative loops at (F=0.098, k=0.055).

negative pattern : In general, any pattern that consists mostly of high-u ("blue") space with occasional low-u ("red") features.

negative worm, negative stripe, etc. : a low-u worm surrounded by high-u space. For example, see (F=0.070, k=0.061).

negative soliton : a negaton.

negaton : A region of red (or more generally, relatively lower-u values) surrounded by blue and which resembles an inverted soliton. A true negaton is a self-sustaining feature with perfect symmetry (in absence of influence of neighboring features); such negatons can be seen at (F=0.046, k=0.0594) and at (F=0.062, k=0.061). In the more general sense, a "negaton" can be any generally small and circular red region surrounded by blue; for example see (F=0.094, k=0.057) and note the small round spots that eventually grow into larger and non-circular shapes.

north : In the Gray-Scott parameter space, as seen in the Pearson paper [2] and on the xmorphia main page, "north" is the direction in which the parameter F increases.

pacemaker : The self-sustaining, rotating central point of a spiral pattern. The pacemaker contains nearby areas that are all oscillating at the same frequency but out of phase with one another. Its period of oscillation determines the spacing of the spiral wave that radiates out from it.

parameter space : An abstract construction in which one or more parameters (ordinarily fixed values, but here treated as variable) are associated with one or more spatial dimensions. A simulation running in "parameter space" evaluates the system equation(s) as if the parameter(s) are a function of the spatial coordinate.

As an analogy, imagine that the system of equations describes the growth of bacteria in an agar. In typical models, temperature and nutrient density would be treated as parameters that are constant throughout each petri dish. In physical experiments, one would expect the bacteria growth rate to be the same at any location in a given petri dish. Typical reaction-diffusion simulations are like this — the parameter value(s) are the same for all pixels on the screen.

If instead the bacteria culture is set up so that the temperature and nutrient density vary significantly from one side of the dish to the other, the bacteria will not grow at the same rate everywhere in the dish. The results of such an experiment will be similar to a reaction-diffusion simulation in "parameter space", where the parameter(s) vary from one side of the screen to the other.

polygon : Any enclosed region surrounded by two or more stripes that join at branching points. Examples at (F=0.09, k=0.059), (F=0.11, k=0.053), and (F=0.106, k=0.053).

pulse :

noun See soliton.

verb See pulsar.

pulsar : An oscillating soliton. The oscillations have a characteristic frequency determined by the k and F parameters, with the central u and v levels moving alternately up and down. The amplitude either decays toward zero (in which case a stable soliton results) or increases until the soliton dies out.

red : The color generally used for areas of low u values. In my images it is usually more "pink" than red, and in monochome images it is a lighter gray or white. See blue for history and etymology.

ring : A round feature that is like a stripe or worm bent around into a closed loop, without ends or branch points, and that is unstable or exhibits a characteristic type of growth or movement. See examples at (F=0.34, k=0.057) and (F=0.38, k=0.059).

When referring to a ringlike structure that is as stable as a soliton I use the term loop (See (F=0.38, k=0.063), which illustrates this contrast). See also bullseye.

seed : Can refer to any small pattern that develops into something larger and more complex. I am using the term only when referring to a spiral seed. In some papers a spiral seed is considered a type of pacemaker (a class that also includes the centres of wave targets)..

soliton : A sole, self-sufficient and stable (asymptotically unchanging) island of locally low u (and high v) surrounded by locally higher u (and lower v). For example, see (F=0.054, k=0.067).

NOTE: This glossary uses "soliton" to refer to a stationary, self-sustaining stable pattern, a phenomenon that is called a dissipative soliton or autosoliton in other published work on reaction-diffusion systems. None of these terms avoid confusion with other uses — all have quite different meanings in other contexts.

In particular, a "soliton" in most other contexts a self-sustaining traveling wave (for example, see the Wolfram definition). An "autosoliton" in Gray-Scott and similar reaction-diffusion systems is usually a spot-like stationary feature (for example, see [5]) but in other fields refers to a traveling wave phenomenon (particularly in optical contexts) or a "drifting vortex" (the Rossby autosoliton, related to Rossby waves). As seen in the Wikipedia article, "dissipative soliton" refers to both stationary and moving phenomena. The term pulse is also used to refer to a non-moving stable spot (for example in [4]) but is more often used to refer to traveling phenomena.

To refer to similarly-sized features that are not necessarily symmetrical or stable, I use the term spot.

south : In the Gray-Scott parameter space, as seen in the Pearson paper [2] and on the xmorphia main page, "south" is the direction in which the parameter F decreases.

spiral : This term is used specifically to refer to the self-sustaining structures, reminiscent of the B-Z Belousov-Zhabotinsky reaction in a Petri dish, that are found at lower F values such as (F=0.006, k=0.031), (F=0.010, k=0.035), and (F=0.014, k=0.045).

spiral seed : see seed.

spot : Includes solitons, negatons and other similarly-sized features, but without the implication of symmetry (roundness).

stripe : Any extended linear feature, usually with stability in the direction perpendicular to its length. They are sometimes called "solitons" because of this stability. If a stripe's cross-section is stable, then its cross-section is a stable soliton in the 1-dimensional system at the same parameter values. Pearson [2] uses this term; I add the more specific worm, a stripe that is very stable laterally but lengthens or shrinks at both ends.

surface tension : a property of curved stripes at certain parameter values that causes them to move towards the concave side and away from the convex side. This causes closed loops (without side branches) to shrink.

In branched networks, branch points will generally evolve towards a 120o configuraton, and enclosed spaces (polygons) will grow or shrink according to the curvature of their sides, which will usually depend only on the number of sides of the polygon. See examples at (F=0.09, k=0.059) and (F=0.11, k=0.053).

Surface tension is also observed in negative stripes; see (F=0.106, k=0.053).

target : A spot (soliton) encircled by a ring, or the centre of a set of two or more concentric ringlike features. The more general use of the term applies to RD systems such as Belousov-Zhabotinsky, in which it is possible for a small localised solition-like structure to generate a continuous series of concentric ringlike waves (which here I would call rings), expanding ever outward.

Within Gray-Scott such patterns are rare; to avoid confusion I use the term bullseye to refer to certain negative and relatively much more stable annular patterns seen at (F=0.62, k=0.061). The patterns at (F=0.42, k=0.059) are not true bullseyes because the outer ring grows as an expanding wave, and are not targets because the inner rings do not travel outwards.

tip : The rounded end of a linear feature. For example, worms have two tips.

west : In the Gray-Scott parameter space, as seen in the Pearson paper [2] and on the xmorphia main page, "west" is the direction in which the parameter k decreases.

worm : A straight or curved but (generally roughly linear) "soliton" or stripe with two rounded ends and no branching structure. For examples, see (F=0.082, k=0.061) and (F=0.086, k=0.061). The structure is like a soliton in that the linear part does not narrow or widen, but is unstable at the ends (which either grow, as seen at (F=0.082, k=0.061), or shrink as at (F=0.086, k=0.061)).

Xmorphia : The name of a website created by Roy Wlliams in 1994, based on the Pearson paper [2] using supercomputer resources at Caltech. It is preserved by; view its December 1998 incarnation here. Xmorphia featured a clickable parameter-space image, which appears in a greatly expanded form on my site.


[1] K. J. Lee, W. D. McCormick, Q. Ouyang, and H. L. Swinney, Pattern formation by interacting chemical fronts, Science 261 (1993) 192-194.

[2] J. Pearson, Complex patterns in a simple system, Science 261 (1993) 189-192. (available at patt-sol/9304003

[3] K. J. Lee, W. D. McCormick, H. L. Swinney, and J. E. Pearson, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature 369 (1994) 215-218. (PDF at Lee et al.)

[4] A. Doelman, T. J. Kaper, and P. A. Zegeling, Pattern formation in the 1-D Gray-Scott model, Nonlinearity 10 (1997) 523-563.

[5] C. B. Muratov and V. V. Osipov, Spike autosolitons in the Gray-Scott model, CAMS Rep. 9900-10, NJIT, Newark, NJ (available at patt-sol/9804001).

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