Article
Finite groups for the Kummer surface: the genetic
code and a quantum gravity analogy
Michel Planat 1 0000-0001-5739-546X, David Chester 2, Raymond Aschheim
3 0000-0001-6953-8202, Marcelo M. Amaral 4 0000-0002-0637-1916, Fang Fang 5and Klee Irwin
6 0000-0003-2938-3941
1 Université de Bourgogne/Franche-Comté, Institut FEMTO-ST CNRS UMR 6174, 15 B Avenue des
Montboucons, F-25044 Besançon, France.; michel.planat@femto-st.fr
2 Quantum Gravity Research, Los Angeles, CA 90290, USA; DavidC@QuantumGravityResearch.org
3 Quantum Gravity Research, Los Angeles, CA 90290, USA; raymond@QuantumGravityResearch.org
4 Quantum Gravity Research, Los Angeles, CA 90290, USA; Marcelo@quantumgravityresearch.org
5 Quantum Gravity Research, Los Angeles, CA 90290, USA; fang@quantumgravityresearch.org
6 Quantum Gravity Research, Los Angeles, CA 90290, USA;Klee@quantumgravityresearch.org
* Correspondence: michel.planat@femto-st.fr
Version January 6, 2021 submitted to Journal Not Specified
Abstract: The Kummer surface was constructed in 1864. It corresponds to the desingularisation of
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the quotient of a 4-torus by 16 complex double points. Kummer surface is kwown to play a role in
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some models of quantum gravity. Following our recent model of the DNA genetic code based on the
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irreducible characters of the finite group G5 := (240, 105) ∼= Z5 o 2O (with 2O the binary octahedral
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group), we now find that groups G6 := (288, 69) ∼= Z6 o 2O and G7 := (336, 118) ∼= Z7 o 2O can be
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used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some
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biological functions. Groups G6 and G7 are found to involve the Kummer surface in the structure of
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their character table. An analogy between quantum gravity and DNA/RNA packings is suggested.
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Keywords: Kummer surface, DNA genetic code, hexamers and pentamers, informationally complete
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characters, finite groups, hyperelliptic curve
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PACS: 02. 20.-a, 02.10.-v, 03.65.Fd, 82.39.Rt, 87.10.-e, 87.14.gk
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1. Introduction
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In a recent paper we found that the 22