Perspectives
Geometry Through a Systems Architect's Eyes
Architects make similar choices when selecting schemas, reference frames and abstractions: representations can simplify reasoning without becoming the thing…
I approach geometry with an occupational habit: I want to draw boxes and arrows. That habit is useful because architecture forces questions about state, boundaries, interfaces, resources and failure. It is dangerous because natural systems were not designed to respect our diagrams.
Geometry describes relations of position, distance, angle, curvature and transformation. A coordinate system makes some relationships convenient to express, but it does not create the underlying geometry.
Where does state live?
Geometry describes relations of position, distance, angle, curvature and transformation. A coordinate system makes some relationships convenient to express, but it does not create the underlying geometry.
Software gives us the expectation that important state should have an owner. Natural systems often distribute state across structure, concentrations, relationships and history. A snapshot can therefore tell us less than the process that produced it.
Where are the interfaces?
Architects make similar choices when selecting schemas, reference frames and abstractions: representations can simplify reasoning without becoming the thing represented.
Engineered interfaces are declarations. Natural boundaries are often material: membranes, tissues, ecological borders, channels, gradients or social conventions. They can leak, adapt and participate in the behavior they constrain.
What is the failure model?
A coordinate system is deliberately chosen. Physical spacetime and biological form impose constraints that may not fit everyday Euclidean intuition.
Failure analysis is useful because normal operation hides assumptions. A healthy component can coexist with an unhealthy whole. A local optimization can damage the larger system. Robustness at one level can create fragility at another.
History is part of the architecture
Euclidean geometry built a powerful deductive system; non-Euclidean geometries later showed that apparently obvious spatial assumptions can be alternatives rather than necessities.
In a designed system, legacy structure may be accidental baggage. In an evolved or historically accumulated system, legacy structure can be the reason the current architecture exists at all. The path is not documentation around the system; sometimes it is part of the system.
The zoom test
A good architectural description should survive zooming. Going down a level should reveal mechanisms capable of implementing the higher-level pattern. Going up should reveal regularities that justify discussing the larger entity in its own vocabulary.
Geometry is a discipline of invariant relationships: descriptions can change while structure persists.
Which properties belong to reality, and which belong to the coordinate system we chose?
Reading trail
These links are starting points for the scientific and historical ideas. The systems interpretation, analogies and conclusions here are my own.