Phase Space

This note is about the space of possibility or search space that evolution, organisms, human designers or artificial systems can explore.

Cf.

Configuration space
Decision space
Design universe
Dynamical landscape
Evolutionary space
Exploratory domain
Feasible region
Feasible set
Fitness landscape
Form space
Generative field
Geometric variation space
Idea space
Modal space
Morphological space
Optimization landscape
Outcome space
Parameter space
Phase space
Potentiality space
Search space
Shape grammar space
Solution space
State space
Strategy space
Structural possibility space
System topology
The space of possibilities
Typological field

The meanings and usage vary.

Physics and Dynamical Systems

Phase space is a multidimensional space in which all possible states of a system are represented, with each state corresponding to one unique point in the phase space. It is a mathematical framework used to describe the dynamics of a system, where each axis of the space represents one of the system's coordinates or momenta.

Cf.

classical mechanics
nonlinear dynamics

Attractors, Basins, and Thresholds

These terms apply to any dynamical system, whether a pendulum, a circuit, a climate model, an ecosystem, or an economy. Later sections cover the domain-specific elaborations.

Attractor: a state or set of states toward which a system evolves.
Basin of attraction: the set of all states from which a system settles to the same attractor if left alone. The standard picture puts a ball on a hilly surface, where valleys are basins, valley bottoms are attractors, and ridges are thresholds. The word comes from drainage basins, where every drop falling within a catchment reaches the same river whatever point it landed on.
Bifurcation: a change in a control parameter that alters the number or stability of attractors, so valleys appear, merge, or vanish.
Hysteresis: reversing a driver back to the value that triggered a flip fails to reverse the flip. Forward and backward tipping points sit at different parameter values, so the return path costs more than the outward path, and sometimes no return path exists.

Two general cautions carry into every applied use.

The picture assumes a fixed landscape. In many systems, and in all living ones, movement through the landscape reshapes it. See Niche Construction below and Niche.
Noise changes the answer. The ball-and-cup picture assumes small perturbations around a deterministic skeleton. In genuinely stochastic systems, basin depth and width predict residence time poorly, and quasi-potentials do the work instead.2

Scheffer treats this machinery across physics, ecology, climate, and society in one volume, which makes it the useful general reference.1

General form of the stability landscape and the hysteresis fold.1

Three Landscapes That Invert Each Other

The Cf. list above contains fitness landscape, optimization landscape, and dynamical landscape. These use the same drawing with opposite conventions, and conflating them produces nonsense.

Landscape	Height means	Systems move	Good regions
Fitness landscape (Wright)	Fitness or performance	Uphill	Peaks
Stability or potential landscape	Potential, or inverse stability	Downhill	Valleys
Morphospace	Nothing. Axes are form parameters and there is no height	Anywhere the constraints allow	No intrinsic ranking

Morphospace differs from both landscapes because it maps the space of forms without assigning a value to each point.5^,6

Biology and Evolutionary Theory

Cf.

Niche

This notion in the sense of total sum of possibilities does not apply or else the possibility is predefined and real innovation is impossible. Cf. Econormativity

Multiple interacting constraints (functional, developmental, historical, genetic) on high-dimensional spaces where living entities construct new evolutionary spaces rather than merely search through pre-existing landscapes, with path dependency limiting accessible trajectories.3

Morphological space describes:

Phenotypic variation
Evolutionary pathways
Developmental constraints

Morphospace in evolutionary and developmental biology.4^,5^,6

The topology of evolutionary novelty and innovation.7

The Epigenetic Landscape

Waddington drew the landscape picture for development decades before ecologists adopted it for regime shifts. A ball rolls down a branching valley system, where valleys represent developmental trajectories and ridges represent the cost of switching between them. Beneath the surface, guy-ropes and pegs stand for the genetic and molecular relations that hold the landscape in shape.8

Two features matter for later sections. Valleys here describe canalisation, so development resists perturbation and returns to its trajectory. And the drawing separates the surface from what holds its shape, which locates leverage at the pegs rather than at the ball. This anticipates accounts that treat morphogenesis as navigation through morphospace toward a target, and it explains why developmental and ecological literatures converge on the same claim about where leverage sits. See Transformation.

The epigenetic landscape and the relations that shape it.8

Ecology supplies the most developed applied vocabulary for basins, and it adds terms that do not exist in the general formalism. This section holds the domain-specific material. Transformation draws on it directly, since it defines transformation as a change in the constraints and feedbacks that hold a system in a partial and temporary equilibrium.

Regime Shifts and Alternative Stable States

Ecosystems can occupy more than one self-maintaining configuration under the same external conditions. Shallow lakes sit clear and plant-dominated or turbid and algae-dominated. Rangelands sit grassy or shrub-encroached. Reefs sit coral-dominated or algae-dominated. Gradual change in a driver produces little visible response until the system crosses a threshold and reorganises quickly.9^,13

Hysteresis makes these shifts expensive. Cutting nutrient loading back to the level that tipped a lake turbid leaves it turbid. Managers have to push much further, and some systems never return.

Catastrophic shifts and the hysteresis loop.9

Two Perspectives That Get Conflated

Beisner and colleagues separate two pictures. Both get drawn as balls in cups, and the difference matters for anyone using the metaphor carefully.10

Community perspective: the environment holds constant and the state variables move. The ball rolls across a fixed landscape.
Ecosystem perspective: the parameters change, so the landscape deforms. Valleys deepen, flatten, merge, or disappear while the ball sits still.

Most real cases involve both. Treating one as the other produces confused claims about what an intervention achieved.

Community and ecosystem perspectives on alternative stable states.10

Named Dimensions of a Basin

Resilience research decomposes the stability landscape into three measurable aspects and adds a fourth for cross-scale effects. These terms come from social-ecological systems research and have no counterpart in the general dynamical-systems vocabulary.11

Latitude (L): how far the system can move within the basin before it crosses a threshold. Basin width.
Resistance (R): how much force it takes to move the system. Basin depth.
Precariousness (Pr): how close the system currently sits to a threshold.
Panarchy (Pa): how the other three depend on basins at scales above and below. Cf. Organisational.

The same work distinguishes adaptability, the capacity to manage resilience and stay within the basin, from transformability, the capacity to create a new system when the current one becomes untenable. Transformation builds on that distinction. Walker and Salt give the plain-language version with each element labelled.12

Stability landscape with latitude, resistance and precariousness, and the same landscape after it changes shape.11

The system as a ball in a basin.12

Niche Construction

Living systems reshape the landscape they move across. Organisms alter the selective and material conditions that they and their successors inherit, so the basin structure itself becomes an output of the system rather than a fixed backdrop.16 This limits how far the ball-and-cup picture can carry, and it explains the caution given under Attractors, Basins, and Thresholds above.

Attractor and Target

A basin describes where a system ends up. It carries no claim that the system represents the destination or works toward it. Whether ecological collectives hold represented setpoints or only occupy attractors remains open, and the answer determines whether retargeting works as a design method. See Transformation and Intelligence.

Linguistics and Semiotics

Phonological space or semantic describe:

Variations in sound or meaning
Conceptual mappings

Design, Engineering, and Architecture

Morphological space and space of possibilities in:

Parametric design
Generative design
Shape grammars
Design space exploration

Oliveira collects morphological research across planning, urban design and architecture.14

Cognitive Science

Conceptual or possibility model:

Human reasoning
Decision-making
Creativity and imagination

Aronowitz locates values in the space of possibilities.15

Human/Nonhuman Society and Culture

Niche construction links biological evolution to cultural change.16

Marks and colleagues transfer fitness landscape models to collective decision-making and mark the limits of that transfer.17

Path Dependency and Lock-In

Goldstein and colleagues review lock-in and path dependency across disciplines and socio-environmental contexts.18

Méndez and colleagues explain path-dependent rigidity traps through increasing returns, power, discourses, and entrepreneurship.19

Computer Science, Artificial Intelligence, Robotics

Solution space or search in:

Algorithm design
Machine learning
Evolutionary computation
Constraint satisfaction problems

Aaron and colleagues apply bioinspired methods to analysing bioinspired robots.20

Design Potential

Definition

The full set of reachable design options forms a possibility space, also called design potential. See Redundancy and Deliberation.

Reachability matters more than size. A vast possibility space whose regions no one can reach from the current state supplies no design potential. The ecological vocabulary above therefore does useful work here, because latitude, resistance, precariousness and hysteresis give four dimensions along which a possibility space stays reachable or stops being so.

Stability-Landscape Terms Read as Design Terms

Landscape term	Design-potential reading	Design question
Latitude	Breadth of options still reachable from here	How much room remains before we lose the ability to change course?
Resistance	Effort required to move to a different option set	What does changing direction cost?
Precariousness	Proximity to a threshold that forecloses options	How close are we to losing options irreversibly?
Panarchy	Dependence of the other three on other levels	Which level's constraints bind?
Hysteresis	Asymmetry between foreclosing and restoring an option	If we lose it, what does getting it back cost, and can we?

This gives design potential a set of dimensions and connects it to the irreversibility ledger and inflection-point tests in Measuring Design-Pathway Transformation.

Nonhuman-Led Design as Expansion of Design Potential

A useful form of redundancy multiplies options within the possibility space and keeps them reachable. This supports nonhuman leadership, because nonhuman proposers reveal viable futures that humans fail to foresee, and a design that forecloses reachable options blocks that leadership.21
Read through the landscape, human-led design characteristically narrows latitude and raises precariousness. It optimises toward a single intended state and strips the slack that kept alternatives reachable. Nonhuman activity works the other way, because ecosystem engineering, niche construction and succession generate new reachable states as a by-product of ordinary living. See Niche and Transformation.
Distributed and multi-agent systems keep more futures reachable than any single agent manages, which argues for plurality on capability grounds as well as ethical ones. See Intelligence and Redundancy.
Deliberation supplies the complementary move. The propositional community expands the possibility space, and the evaluative community focuses effort so that expansion avoids becoming noise, surplus novelty, or unmanaged risk. See Deliberation.
More options do not always help. Pathway diversity counts only pathways that stay genuinely reachable, so actions that run down shared capacity or foreclose others' futures add no potential.22 Ask who gains reachable options, who loses them, and whether today's design keeps nonhuman-led pathways open. See Justice.

Defuturing as the Negation of Design Potential

Fry approaches the same phenomenon from the other side. Defuturing is design that destroys future possibility, taking away the futures of human and nonhuman beings alike.23^,24 Sustainability efforts that stay inside the industrial logic they set out to reform defuture despite their stated intent, because they leave the option-foreclosing structure intact.23

Read through the landscape, defuturing reduces latitude while deepening the basin the system already occupies. It works as the design equivalent of lock-in, and hysteresis explains why undoing it costs so much. See Path Dependency and Lock-In above, Futuring, Future, and Transformation on directionality, where transformation carries no intrinsic value and can foreclose futures as readily as open them.

Open Questions

Can latitude, resistance and precariousness be estimated for a design pathway, or do they stay qualitative outside modelled ecological systems? Cf. Measuring Design-Pathway Transformation.
Does expanding design potential have a defensible upper bound? The pathway-diversity caveat implies one exists but fails to locate it.
Whose possibility space? A space that expands for one party while contracting for another has not expanded. See Justice and Power.
Do nonhuman collectives navigate possibility spaces they represent in some sense, or only occupy basins? Transformation flags the same question.

References

Footnotes

Nolting, Ben C., and Karen C. Abbott. “Balls, Cups, and Quasi-Potentials: Quantifying Stability in Stochastic Systems.” Ecology 97, no. 4 (2016): 850–64. https://doi.org/10.1890/15-1047.1.˄
Scheffer, Marten. Critical Transitions in Nature and Society. Princeton Studies in Complexity. Princeton: Princeton University Press, 2009.˄
Mitteroecker, Philipp, and Simon M. Huttegger. “The Concept of Morphospaces in Evolutionary and Developmental Biology: Mathematics and Metaphors.” Biological Theory 4, no. 1 (2009): 54–67. doi:10/cx74fj.˄
McGhee, George R. The Geometry of Evolution: Adaptive Landscapes and Theoretical Morphospaces. Cambridge: Cambridge University Press, 2006.˄
Longo, Giuseppe. “How Future Depends on Past and Rare Events in Systems of Life.” Foundations of Science 23, no. 3 (2018): 443–74. https://doi.org/10/g8q6q7.˄
Lamsdell, James C. “The Conquest of Spaces: Exploring Drivers of Morphological Shifts through Phylogenetic Palaeoecology.” Palaeogeography, Palaeoclimatology, Palaeoecology 583 (2021): 110672. doi:10.1016/j.palaeo.2021.110672.˄
Erwin, Douglas H. “The Topology of Evolutionary Novelty and Innovation in Macroevolution.” Philosophical Transactions of the Royal Society B: Biological Sciences 372, no. 1735 (2017): 20160422. doi:10.1098/rstb.2016.0422.˄
Waddington, Conrad Hal. The Principles of Embryology. London: George Allen & Unwin, 1956.˄
Scheffer, Marten, Steve Carpenter, Jonathan A. Foley, Carl Folke, and Brian Walker. “Catastrophic Shifts in Ecosystems.” Nature 413, no. 6856 (2001): 591–96. https://doi.org/10.1038/35098000.˄
Folke, Carl, Steve Carpenter, Brian Walker, Marten Scheffer, Thomas Elmqvist, Lance Gunderson, and C. S. Holling. “Regime Shifts, Resilience, and Biodiversity in Ecosystem Management.” Annual Review of Ecology, Evolution, and Systematics 35 (2004): 557–81. https://doi.org/10.1146/annurev.ecolsys.35.021103.105711.˄
Beisner, Beatrix E., Daniel T. Haydon, and Kim Cuddington. “Alternative Stable States in Ecology.” Frontiers in Ecology and the Environment 1, no. 7 (2003): 376–82. https://doi.org/10.1890/1540-9295(2003)001[0376:ASSIE]2.0.CO;2.˄
Walker, Brian, Crawford S. Holling, Stephen Carpenter, and Ann Kinzig. “Resilience, Adaptability and Transformability in Social–Ecological Systems.” Ecology and Society 9, no. 2 (2004): 1–9. https://doi.org/10.5751/es-00650-090205.˄
Walker, Brian, and David Salt. Resilience Thinking: Sustaining Ecosystems and People in a Changing World. Washington, DC: Island Press, 2006.˄
Laland, Kevin N., John Odling-Smee, and Marcus W. Feldman. “Niche Construction, Biological Evolution, and Cultural Change.” The Behavioral and Brain Sciences 23, no. 1 (2000): 131–46. doi:10/c58hwb.˄
Oliveira, Vítor, ed. Morphological Research in Planning, Urban Design and Architecture. Cham: Springer, 2021.˄
Aronowitz, Sara. “Locating Values in the Space of Possibilities.” Philosophy of Science 92, no. 1 (2025): 121–40. doi:10/g99d6v.˄
Marks, Peter, Lasse Gerrits, and Johannes Marx. “How to Use Fitness Landscape Models for the Analysis of Collective Decision-Making: A Case of Theory-Transfer and Its Limitations.” Biology & Philosophy 34, no. 1 (2019): 7. doi:10/gjpt4h.˄
Goldstein, Jenny E., Benjamin Neimark, Brian Garvey, and Jacob Phelps. “Unlocking ‘Lock-in’ and Path Dependency: A Review across Disciplines and Socio-Environmental Contexts.” World Development 161 (2023): 106116. doi:10/g83j86.˄
Méndez, Pablo, Jaime Amezaga, and Luis Santamaría. “Explaining Path-Dependent Rigidity Traps: Increasing Returns, Power, Discourses, and Entrepreneurship Intertwined in Social-Ecological Systems.” Ecology and Society 24, no. 2 (2019). doi:10/gkshvh.˄
Aaron, Eric, Joshua Hawthorne-Madell, Ken Livingston, and John H. Long. “Morphological Evolution: Bioinspired Methods for Analyzing Bioinspired Robots.” Frontiers in Robotics and AI 8 (2022): 717214. doi:10/g9pv2h.˄
Roudavski, Stanislav, and Alexander Holland. “Are You Blocking Nonhuman Leadership? Five Questions to Find Out.” PDC ’26: Proceedings of the 19th Participatory Design Conference (New York) 2 (2026): 372–81. https://doi.org/10.1145/3789492.3796416.˄
Lade, Steven, Brian Walker, and L. Jamila Haider. “Resilience as Pathway Diversity: Linking Systems, Individual, and Temporal Perspectives on Resilience.” Ecology and Society 25, no. 3 (2020). https://doi.org/10.5751/ES-11760-250319.˄
Fry, Tony. Design Futuring: Sustainability, Ethics and New Practice. Sydney: University of New South Wales Press, 2009.˄
Fry, Tony. Defuturing: A New Design Philosophy. London: Bloomsbury Visual Arts, 2020.˄

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