×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Le Bellac, M., A short introduction to quantum information and quantum computation, (2006), Cambridge University Press · Zbl 1109.81002
[2] Penrose, R., Angular momentum: an approach to combinatorial space-time, (Bastin, T., Quantum Theory and Beyond, (1971), Cambridge University Press), 151-180
[3] Gauss, C. F., Werke, zweiter band, (1876), Königlichen Gesellschaft der Wissenschaften Göttingen
[4] Eastwood, M. G.; Penrose, R., Drawing with complex numbers, Math. Intelligencer, 22, 8-13, (2000) · Zbl 1052.51505
[5] Baird, P., A class of quadratic difference equations on a finite graph
[6] Baird, P.; Wehbe, M., Twistor theory on a finite graph, Comm. Math. Phys., 304, 2, 499-511, (2011) · Zbl 1216.81090
[7] R. Connelly, T. Jordan, W.J. Whiteley, Generic global rigidity of body-bar frameworks, EGRES. TR-2009-13, www.cs.elte.hu/egres/.
[8] Baird, P., Information, universality and consciousness: a relational perspective, Mind Matter, 11, 2, 21-43, (2013)
[9] Connes, A.; Rovelli, C., Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories, Classical Quantum Gravity, 11, 2899-2917, (1994) · Zbl 0821.46086
[10] Korner, J., Bounds and information theory, SIAM J. Algorithms Discrete Math., 7, 560-570, (1986) · Zbl 0603.05034
[11] Urakawa, H., A discrete analogue of the harmonic morphism, (Anand, C.; Baird, P.; Loubeau, E.; Wood, J. C., Harmonic Morphisms, Harmonic Maps and Related Topics, Research Notes in Mathematics, vol. 413, (2000), Chapman and Hall/CRC Boca Raton), 97-108 · Zbl 0982.53060
[12] Baird, P.; Tiba, M., An algorithm to prescribe the configuration of a finite graph
[13] F.R.K. Chung, Spectral Graph Theory, in: CBMS Regional Conference Series in Mathematics, vol. 92, Washington DC, 1997. · Zbl 0867.05046
[14] P. Kuchment, Quantum graphs: an introduction and a brief survey, February 2008. arXiv:0802.3442v1. · Zbl 1210.05169
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.