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Emergence of geometry in a combinatorial universe. (English) Zbl 1278.05079
Summary: Our objective is to construct an elementary universe populated entirely by graphs from which geometry and dynamics emerge. The universe is based on a binary relation between objects: are they connected by an edge or not? It is only this relation that matters; the nature of the objects being irrelevant. Our perspective is that, out of the graphs so formed, further implicit structure is present, defined by a geometric spectrum and corresponding fields. This implicit structure comes into play when graphs correlate. Thus we propose ways in which graphs can interact and so dynamics, that is change, appears. With a suitable definition of time, this change can be ordered to give a universe endowed with local geometry and time.

05C10 Planar graphs; geometric and topological aspects of graph theory
05B25 Combinatorial aspects of finite geometries
52C99 Discrete geometry
52B11 \(n\)-dimensional polytopes
39A14 Partial difference equations
Full Text: DOI
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