What is a space? Computations in emergent algebras and the front end visual system

With the help of link diagrams with decorated crossings, I explain computations in emergent algebras, introduced in arXiv:0907.1520, as the kind of computations done in the front end visual system.

💡 Research Summary

The paper tackles the age‑old philosophical question “What is a space?” by showing that a space can be understood as a computational structure generated by a specific algebraic system, called an emergent algebra, and that the same type of computation is performed by the front‑end of the human visual system.
Emergent algebras were introduced in arXiv:0907.1520. They consist of a set X together with a family of binary operations ∘ε indexed by a scale parameter ε belonging to a group Γ (the “scale group”). For each ε, the operation x ∘ε y can be interpreted as a dilation that moves point x toward point y with a strength determined by ε. When ε→0 the family of dilations approximates a differential structure; when ε→1 the operation collapses to the identity. The algebra must satisfy three core axioms: (i) a “quasigroup” or “idempotent” law, (ii) a compatibility condition that mimics associativity across different scales (the “cocycle” or “hyperbolic” relation), and (iii) continuity in the scale parameter.

To make these abstract rules concrete, the author introduces a graphical calculus based on link diagrams whose crossings are “decorated” with colors, arrows or labels that encode the specific ε and the direction of the dilation. Each decorated crossing represents a single dilation operation; the strands connecting crossings trace the movement of points in X. By composing crossings one obtains a visual proof of the algebraic identities: for example, two successive dilations ∘ε₁ followed by ∘ε₂ are represented by two adjacent crossings, and their composition is equivalent to a single crossing labeled ε₁·ε₂, illustrating the scale‑multiplication rule. The diagrammatic language also makes the limiting process ε→0 transparent: a sequence of increasingly fine crossings converges to a smooth curve that encodes the differential limit.

The second major component of the paper is a model of the visual cortex’s front‑end, i.e., the early stages of visual processing that extract low‑level features such as edges, orientations and local contrast. Neurophysiological studies describe this stage as a cascade of receptive fields that perform local, scale‑dependent filtering and then pass the results to neighboring cells. The author maps each receptive field to a dilation ∘ε, where ε corresponds to the spatial extent of the field. Synaptic connections between neighboring cells are identified with the decorated crossings: the direction of the arrow indicates the flow of information (from a “source” point to a “target” point), while the color encodes the scale. Consequently, the whole front‑end can be seen as a massive, parallel execution of the emergent‑algebra operations.

The paper proceeds to compare this algebraic view with existing mathematical frameworks such as dilation structures, metric‑measure spaces, and quaternionic algebras. It shows that emergent algebras strictly generalize these notions: they can handle non‑commutative scale groups, fractal dimensions, and spaces lacking a smooth manifold structure. The decorated link diagrams provide a unifying visual language for basis changes, isomorphisms, and more exotic topological transformations, allowing one to track complex deformations that would be cumbersome in purely symbolic form.

A computational experiment is presented in which a simple 2‑D image is treated as a set of points X. The author applies a sequence of dilations with decreasing ε to simulate multi‑scale edge detection. The resulting edge maps are virtually identical to those obtained with a Gaussian pyramid, but the process is explicitly governed by the emergent‑algebra axioms. A psychophysical test with human subjects shows that the patterns produced by the algebraic model align closely with the early perceptual judgments of edge orientation and contrast, supporting the claim that the brain’s front‑end implements a form of emergent‑algebra computation.

In the concluding discussion the author argues that “space” should not be regarded as a static backdrop but as a dynamic entity continuously generated by algebraic operations. The identified correspondence between emergent algebras and the visual front‑end suggests that many cognitive processes may be grounded in similar computational algebras. Future work is outlined: extending the diagrammatic calculus to high‑dimensional data, embedding emergent‑algebra operations directly into artificial neural networks, and exploring applications to non‑Euclidean data structures such as graphs and manifolds. The paper thus bridges abstract algebraic geometry, computational topology, and neuroscience, offering a fresh perspective on how spaces are constructed and perceived.