Impossible But Lawful: 2 Local Closure, Global Failure

Escher’s images arrest us not because they are misdrawn, but because they operate according to rules that cannot cohere into a single inhabitable world. In Post 1 we established that Escher is not about optical illusion; he is about the systemic limits of worldhood. Here, we unpack the structural pattern that makes this possible: local lawfulness and global impossibility.

The Mechanics of Local Lawfulness

Take Ascending and Descending (1960), with its endless staircase populated by figures in perpetual ascent and descent. At the level of each step and each figure:

• The movement is internally consistent: each figure climbs or descends without contradiction.

• The stairs are geometrically coherent along their own segments: no step is misaligned or impossible locally.

• Gravity is respected in each plane: each figure inhabits a logically lawful environment.

Every local relation is lawful. Nothing violates the construal of physicality within its own frame. The image is fully stable if we only follow one staircase at a time.

Yet, when we attempt to integrate all staircases into a single world, the system refuses to close. The “perpetual staircase” forms a loop that cannot exist in three-dimensional space. The global structure is non-integrable: it cannot be actualised as an inhabitable world.

This pattern is not unique to Ascending and Descending. In Relativity (1953), multiple gravitational frames coexist: figures walk upside down, sideways, or in impossible combinations. Each local interaction obeys consistent rules, but when these frames are composed, there is no overarching space in which all rules can hold simultaneously.

Rule-Following vs. Worldhood

This is the critical distinction that Escher forces us to confront:

• Rule-following: a local property. Every element obeys its own constraints perfectly.

• Worldhood: a global property. A world requires that local laws integrate coherently into an inhabitable whole.

Escher demonstrates that rule-following does not entail worldhood. The gap between these two levels is precisely where impossibility arises. Local lawfulness is necessary but not sufficient for a world.

We can make the distinction explicit with a minimal analogy in language:

• A single sentence obeys grammar perfectly.

• A paragraph of sentences may still fail to form a coherent argument.

Lawful parts do not guarantee a lawful whole; global integration is a separate achievement.

Diagramming the Impossibility

One way to visualise this is to imagine a network:

• Nodes = local transitions (steps, planes, orientations).

• Edges = lawful adjacency or interaction.

In Escher’s images, every node is consistent with its neighbors. But as we traverse the network, we encounter loops or cycles that cannot be embedded in three-dimensional space without contradiction. The network is locally valid everywhere but globally unembeddable.

This is not a trick of perspective. It is a structural phenomenon. And it is precisely why Escher’s works can be studied as systems rather than illusions: the impossibility is built into the architecture of lawful transitions.

Implications for Systems Thinking

Escher gives us a clear principle:

Local validity does not guarantee global integrability.

This principle resonates far beyond art. Any system—linguistic, bureaucratic, computational—can achieve flawless local functioning yet fail globally if the transitions do not compose into a coherent whole. The lesson is subtle: failure can be lawful, and impossibility can be systematic.

Conclusion

In Escher, the world fails not because anything goes wrong, but because nothing goes wrong at the local level. The impossibility is emergent from perfection, not defect.

In the next post, we will extend this insight: we will see how systems can over-achieve locally and, as a result, foreclose the possibility of inhabitable global coherence. Escher will serve as the model for understanding this broader class of “overachieving” systems.