This repository contains experimental code that studies what geometric constructions an observer constrained to a discrete substrate, the so-called infra-observer, would develop.
Classical geometry grew out of the measurement practices of a macro-observer, such as using a knotted rope, which implements the concepts of straightedge, compass, and unit. Using the tools available to an infra-observer, who can seemingly only walk on the graph, we try to develop similar geometric concepts and follow the same conceptual progression that led from geometric practice to Euclidean axioms and later to analytic and algebraic geometry. This approach is grounded in the assumption that our continuous abstraction of the macro-world corresponds to a limit of an expanding discrete system at macroscopic scale, as in the computational-universe model of the Wolfram Physics Project generated by hypergraph rewriting.
Constructions in the discrete world are notoriously ambiguous, so the same infra-geometric construction may branch at every step, producing several valid final infrageometric scenes. This leads to novel questions about measuring the distance of scenes and studying the stability of constructions. The infraobserver's abstraction of the construction is that of a diffuse picture where each vertex and edge of the substrate is shaded by how many candidates pass through it. For some constructions and substrate refinements diffusion sharpens into delta functions and classical Euclidean uniqueness emerges.
The capabilities of an infra-observer are determined by what the substrate lets him see.
- Unlabeled graph. He cannot tell vertices apart and only records the sequence of vertex-degrees along his walk, obtaining ideas of dimension and curvature via statistical properties.
- Ribbon graph. He additionally remembers which edge he came in along, so he can pursue trails and run a breadth- or depth-first sweep of edges up to a fixed length around himself.
- Vertex-labeled graph. He can pursue paths and breadth- or depth-first sweeps of vertices, corresponding to a mesoscale where vertices act as superpoints with internal structure that serves as the label UUID; distinct edges remain indistinguishable. This is the main playground for our infra-Euclidean geometry framework, where all geometric primitives are represented as sets of vertices.
- Vertex- and edge-labeled graph. Every step is fully resolved and he enumerates walks in the most general sense.
- defining infra-geometric primitives and postulates
- studying various notions of straightness via curvature minimizing curves
- determining information ceiling of an infra-observer on decorated graphs
- building a framework for branching constructions and diffusion abstractions
- introducing various types of coordinates, and studying the analytization and algebraization of geometry
- determine relations to projective geometry, tropical geometry, spectral properties, metric algebra, effective resistance and developing new concepts and connections
- performing an enumerative study of which postulates hold on which graphs
- performing an empirical study of whether ambiguous infra-geometric constructions converge to a unique construction at macroscopic scale as the substrate is refined
- incorporating hypergraphs and n-ary relations, or developing a new exotic infra-geometry for them
Install from the Wolfram Cloud:
PacletInstall["https://www.wolframcloud.com/obj/hajek_pavel/SyntheticInfrageometry.paclet", ForceVersionInstall -> True]
Needs["WolframInstitute`SyntheticInfrageometry`"]Or load a local checkout for development:
PacletDirectoryLoad["/path/to/SyntheticInfrageometry"]
Needs["WolframInstitute`SyntheticInfrageometry`"]MIT