# Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy

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## Abstract

**:**

## 1. Introduction

## 2. The Hyperelliptic Curve and the Attached KUMMER Surface from Groups ${\mathit{G}}_{\mathit{6}}$ and ${\mathbf{G}}_{\mathbf{7}}$

#### 2.1. Excerpts about Elliptic and Hyperelliptic Curves

#### 2.2. The Group ${G}_{6}:=(288,69)\cong {\mathbb{Z}}_{6}\u22ca2O$

#### 2.3. The Group ${G}_{7}:=(336,118)\cong {\mathbb{Z}}_{7}\times 2O$

## 3. The Genetic Code Revisited

## 4. Kummer Surface and Quantum Gravity

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**For the group ${G}_{5}:=(240,105)\cong {\mathbb{Z}}_{5}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 22 -dimensional Pauli group, the order of a group element in the class, the entries involved in the character and a good assignment to an amino acid according to its polar requirement value. Bold characters are for faithful representations. There is an ‘exception’ for the assignment of the sextet ‘Leu’ that is assumed to occupy two 4-dimensional slots. All characters are informationally complete except for the ones assigned to ‘Stop’, ‘Leu’, ‘Pyl’ and ‘Sec’. The notation in the entries is as follows: ${z}_{1}=-(\sqrt{5}+1)/2$, ${z}_{2}=\sqrt{5}-1$, ${z}_{3}=3(1+\sqrt{5})/2$, ${z}_{4}=\sqrt{2}$, ${z}_{5}=-2cos(\pi /15)$, compare [Table 7] of [1].

(240,105) | dimension | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{5}\u22ca({\mathbb{Z}}_{2}\xb7{S}_{4})$ | d-dit, d = 22 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ |

$\cong {\mathbb{Z}}_{5}\u22ca2O$ | amino acid | Met | Trp | Cys | Phe | Tyr | His | Gln | Asn | Lys | Glu | Asp |

order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | |

char | Cte | Cte | Cte | ${z}_{1}$ | ${z}_{1}$ | ${z}_{4}$ | ${z}_{4}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | |

polar req. | 5.3 | 5.2 | 4.8 | 5.0 | 5.4 | 8.4 | 8.6 | 10.0 | 10.1 | 12.5 | 13.0 | |

(240,105) | dimension | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 |

d-dit, d = 22 | ${d}^{2}$ | 475 | 483 | 480 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

amino acid | Ile | Stop | Leu,Pyl,Sec | Leu | Val | Pro | Thr | Ala | Gly | Ser | Arg | |

order | 10 | 15 | 15 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | |

char | Cte | Cte | Cte | ${z}_{1,2}$ | ${z}_{1,2}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{1,3}$ | ${z}_{1,3}$ | |

polar req. | 4.9 | 4.9 | 5.6 | 6.6 | 6.6 | 7.0 | 7.9 | 7.5 | 9.1 |

## References

- Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Complete quantum information in the DNA genetic code. Symmetry
**2020**, 12, 1993. [Google Scholar] [CrossRef] - Planat, M.; Gedik, Z. Magic informationally complete POVMs with permutations. R. Soc. Open sci.
**2017**, 4, 170387. [Google Scholar] [CrossRef] [PubMed][Green Version] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Informationally complete characters for quark and lepton mixings. Symmetry
**2020**, 12, 1000. [Google Scholar] [CrossRef] - Sauter, C.; Basquin, J.; Suck, D. Sm-Like proteins in eubacteria: The crystal structure of the Hfq protein from Escherichia Coli. Nucleic Acids
**2003**, 31, 4091. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lührmann, W.C.L. Spliceosome, structure and function. Cold Spring Harb. Perspect. Biol.
**2011**, 3, a003707. [Google Scholar] - Kambach, C.; Walke, S.; Young, R.; Avis, J.M.; de la Fortelle, E.; Raker, V.A.; Lührmann, R.; Li, J.; Nagai, K. Crystal structures of two Sm protein complexes and their implications for the assembly of the spliceosomal snRNPs. Call
**1999**, 96, 375–387. [Google Scholar] [CrossRef] - Zhou, L.; Zhou, Y.; Hang, J.; Wan, R.; Lu, G.; Yan, C.; Shi, Y. Crystal structure and biochemical analysis of the heptameric Lsm1-7 complex. Cell Res.
**2014**, 24, 497–500. [Google Scholar] [CrossRef][Green Version] - Kelman, Z.; Finkelstein, J.; O’Donnel, M. Why have six-fold symmetry? Curr. Biol.
**1995**, 5, 1239–1242. [Google Scholar] [CrossRef][Green Version] - Zhai, Y.; Cheng, E.; Wu, H.; Li, N.; Yung, P.Y.; Gao, N.; Tye, B.K. Open-ringed structure of the Cdt1–Mcm2–7 complex as a precursor of the MCM double hexamer. Nat. Struct. Mol. Biol.
**2017**, 24, 300–308. [Google Scholar] [CrossRef] - Kummer, E. Ueber die Flächen vierten grades mit sechszehn singulären punckten. Monatsberichte Berl. Akad.
**1864**, 6, 246–260. [Google Scholar] - Hudson, R.W.H.T. Kummer’s Quartic Surface; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Kaku, M. Strings, Conformal Fields, and M-Theory, 2nd ed.; Springer: New York, NY, USA, 2012. [Google Scholar]
- Aspinwall, P.S. K
_{3}Surfaces and String Duality. In Fields, Strings and Duality; Efthimiou, C., Greene, B., Eds.; World Scientific: Singapore, 1997; pp. 421–540. [Google Scholar] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Group geometrical axioms for magic states of quantum computing. Mathematics
**2019**, 7, 948. [Google Scholar] [CrossRef][Green Version] - LMFDB-The L-Functions and Modular Forms Database. Available online: https://www.lmfdb.org/ (accessed on 1 June 2020).
- Bosma, W.; Cannon, J.J.; Fieker, C.; Steel, A. (Eds.) Handbook of Magma Functions; Computational Algebra Group, University of Sydney: Sydney, Australia, 2019; p. 5914. [Google Scholar]
- Available online: https://sourceforge.net/projects/mathmod/ (accessed on 1 December 2020).
- Crick, F.H.C. Codon-anticodon pairing, the wobble hypothesis. J. Mol. Biol.
**1966**, 19, 548–555. [Google Scholar] [CrossRef] - Lagerkvist, U. “Two out of tree”: An alternative method for codon reading. Proc. Natl. Acad. Sci. USA
**1978**, 75, 1759–1762. [Google Scholar] [CrossRef][Green Version] - Lehmann, J.; Lichbaber, A. Degeneracy of the genetic code and stability of the base pair at the second position of the anticodon. RNA
**2008**, 14, 1264–1269. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gonzalez, D.L.; Giannerini, S.; Rosa, M. On the origin of degeneracy in the genetic code. Interface Focus
**2019**, 9, 38. [Google Scholar] [CrossRef] [PubMed] - Dolgachev, I. Kummer surfaces: 200 years of study. Not. AMS
**2020**, 67, 1527–1533. [Google Scholar] [CrossRef] - Favaro, A.; Hehl, F.W. Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points. Phys. Rev. A
**2016**, 93, 013844. [Google Scholar] [CrossRef][Green Version] - Baekler, P.; Favaro, A.; Itin, Y.; Hehl, F.W. He Kummer tensor density in electrodynamics and in gravity. Ann. Phys.
**2014**, 349, 297–324. [Google Scholar] [CrossRef][Green Version] - Klein, F. Zur theory der liniencomplexe des ersten und zwieter grades. Math. Ann.
**1870**, 2, 198–226. [Google Scholar] [CrossRef] - Klein, F. On the order-seven transformation of elliptic functions. Math. Ann.
**1878**, 14, 428–471. [Google Scholar] [CrossRef] - Jessop, C. A Treatise of the Line Complex; Cambridge University Press: Cambridge, UK, 1903. [Google Scholar]
- Clingher, A.; Malmendier, A.; Shaska, T. Six line confihurations and string dualities. Commun. Math. Phys.
**2019**, 371, 159–196. [Google Scholar] [CrossRef][Green Version] - Gompf, R.E.; Stipsicz, A.I. 4-Manifolds and Kirby Calculus; Graduate Studies in Mathematics; American Mathematical Society: Providence, RI, USA, 1999; Volume 20. [Google Scholar]
- Scorpian, A. The Wild World of 4-Manifolds; American Mathematical Society: Providence, RI, USA, 2011. [Google Scholar]
- Eguchi, T.; Ooguri, H.; Tachikawara, Y. Notes on the K
_{3}surface and the Mathieu group M_{24}. Exp. Math.**2011**, 20, 91–96. [Google Scholar] [CrossRef][Green Version] - Marrani, A.; Rios, M.; Chester, D. Monstruous M-theory. arXiv
**2020**, arXiv:2008.06742v1. [Google Scholar] - Asselmeyer-Maluga, T. Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity. In At the Frontiers of Spacetime: Scalar-Tensor Theory, Bell’s Inequality, Mach’s Principle, Exotic Smoothness; Asselmeyer-Maluga, T., Ed.; Springer: Basel, Switzerland, 2016. [Google Scholar]
- Asselmeyer-Maluga, T. Braids, 3-manifolds, elementary particles, number theory and symmetry in particle physics. Symmetry
**2019**, 10, 1297. [Google Scholar] [CrossRef][Green Version] - Planat, M.; Aschheim, R.; Amaral M., M.; Irwin, K. Quantum computation and measurements from an exotic space-time R
^{4}. Symmetry**2020**, 12, 736. [Google Scholar] [CrossRef] - Hameroff, S.; Penrose, R. Consciousness in the universe, a review of the ‘Orch OR’ theory. Phys. Life Rev.
**2014**, 11, 39–78. [Google Scholar] [CrossRef][Green Version] - Kollmann, J.K.; Polka, J.K.; Zelter, A.; Davis, T.N.; Agard, D. Microtubule nucleating γTuSC assembles structures with 13-fold microtubule-like symmetry. Nature
**2012**, 466, 879–882. [Google Scholar] [CrossRef][Green Version] - Otto, H.H. Reciprocity relation between the mass constituents of the universe and Hardy’s quantum entanglement probability. World J. Cond. Mat. Phys.
**2018**, 8, 30–35. [Google Scholar] [CrossRef][Green Version] - Chang, Y.-F. Calabi-Yau manifolds in biology and biological string-brane theory. NeuroQuantology
**2015**, 4, 465–474. [Google Scholar] [CrossRef] - Pincak, R.; Kanjamapornkul, K.; Bartos, E. A theoretical investigation of the predictability of genetic patterns. Chem. Phys.
**2020**, 535, 110764. [Google Scholar] [CrossRef] - Irwin, K.; Amaral, M.M.; Chester, D. The Self-Simulation hypothesis interpretation of quantum mechanics. Entropy
**2020**, 22, 247. [Google Scholar] [CrossRef] [PubMed][Green Version]

**Figure 2.**(

**a**) A standard plot of the Kummer surface in its 3-dimensional projection, (

**b**) a section at constant ${x}_{4}$ of the Kummer surface defined in Section 2.3.

**Table 1.**For the group ${G}_{6}:=(288,69)\cong {\mathbb{Z}}_{6}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 30 -dimensional Pauli group and the entries involved in the characters. All characters are neither faithful nor informationally complete. The notation is $I=exp(2i\pi /4)$, ${z}_{1}=-\sqrt{2}$, ${z}_{2}=I\sqrt{2}$ and ${z}_{3}=-2$$cos(\pi /9)$.

(288,69) | dimension | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{6}\u22ca({\mathbb{Z}}_{2}\xb7{S}_{4})$ | d-dit, d = 30 | 31 | 796 | 867 | 867 | 882 | 882 | 880 | 897 | 897 | 880 |

char | Cte | Cte | I | I | Cte | Cte | ${z}_{1}$ | ${z}_{2}$ | ${z}_{2}$ | ${z}_{1}$ | |

(288,69) | dimension | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |

d-dit, d = 30 | 885 | 885 | 885 | 885 | 885 | 885 | 876 | 878 | 899 | 899 | |

char | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | Cte | Cte | I | I | |

(288,69) | dimension | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 |

d-dit, d = 30 | 877 | 878 | 885 | 885 | 885 | 885 | 885 | 885 | 880 | 880 | |

char | Cte | Cte | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | ${z}_{3}$ | Cte | Cte |

**Table 2.**The structure of the addition table for the 16 singular Jacobian points of the hyperelliptic curves ${\mathcal{C}}_{6}$ and ${\mathcal{C}}_{7}$.

A | B | C | D |

B | A | D | C |

C | D | A | B |

D | C | B | A |

**Table 3.**The algebra for the character table of group ${G}_{7}:=(336,118)$. In column 1 are the characters in question. Column 2 provides the powers of the entries ${z}_{i}$, $i=1$, 2, 3 or 5. The ${z}_{i}$ are ${z}_{1}=2cos(2\pi /7)$, ${z}_{2}=2{z}_{1}$, ${z}_{3}=-6cos(\pi /7)$, ${z}_{4}=\sqrt{2}$ and ${z}_{5}=2cos(2\pi /21)$. Column 3 explains the polynomial $f\left(x\right)$ whose roots are the powers of a selected ${z}_{i}$. When $f\left(x\right)$ is an elliptic curve defined over the rationals the Cremona reference is in column 4. If $f\left(x\right)$ is a sextic polynomial it leads to a Kummer surface.

Character | ${\mathit{z}}_{\mathit{i}}$ Powers | $\mathit{f}\left(\mathit{x}\right)$ Polynomial | Cremona Ref. |
---|---|---|---|

4–6 | ${z}_{1}:[1,2,3]$ | ${x}^{3}+{x}^{2}-2x-1$ | $784{i}_{1}$ |

18–20 | ${z}_{1}:[1,2,3]$ | . | . |

. | ${z}_{2}:[1,2,3]$ | ${x}^{3}+2{x}^{2}-8x-8$ | $3136{x}_{1}$ |

. | ${z}_{1,2}$ | ${x}^{6}+3{x}^{5}-8{x}^{4}-\phantom{\rule{0ex}{0ex}}21{x}^{3}+6{x}^{2}+24x+8$ | Kummer |

27–29 | ${z}_{1}:[1,2,3]$ | . | . |

. | ${z}_{3}:[1,2,3]$ | ${x}^{3}+3{x}^{2}-18x-27$ | $1764{j}_{1}$ |

. | ${z}_{1,3}:[1,2,3]$ | ${x}^{6}+4{x}^{5}-17{x}^{4}-\phantom{\rule{0ex}{0ex}}52{x}^{3}+6{x}^{2}+72x+27$ | Kummer |

9–14 & 21–26 | ${z}_{1,2}$ | . | . |

${z}_{5}:[1,2,4,5,8,10]$ | ${x}^{6}-{x}^{5}-6{x}^{4}+\phantom{\rule{0ex}{0ex}}6{x}^{3}+8{x}^{2}-8x+1$ | Kummer |

**Table 4.**For the group ${G}_{7}:=(336,118)\cong {\mathbb{Z}}_{7}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 29 -dimensional Pauli group, the order of a group element in the class, the angles involved in the character and a good assignment to an amino acid according to its polar requirement value. Bold characters are for faithful representations. All characters are informationally complete except for the trivial character and the one assigned to ‘Met’. The entries involved in the characters are ${z}_{1}=2cos(2\pi /7)$, ${z}_{2}=2{z}_{1}$, ${z}_{3}=-6cos(\pi /7)$, ${z}_{4}=\sqrt{2}$ and ${z}_{5}=2cos(2\pi /21)$ featuring the angles $2\pi /8$ (in ${z}_{4}$), $2\pi /7$ and $2\pi /21$.

(336,118) | dimension | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{7}\u22ca({\mathbb{Z}}_{2}\xb7{S}_{4})$ | d-dit, d = 29 | 29 | 785 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ |

$\cong {\mathbb{Z}}_{7}\u22ca2O$ | amino acid | . | Met | Trp | Cys | Phe | Tyr | . | . | His | Gln |

order | 1 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | |

char | Cte | Cte | Cte | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}$ | ${z}_{4}$ | ${z}_{4}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | |

polar req. | . | 5.3 | 5.2 | 4.8 | 5.0 | 5.4 | . | . | 8.4 | 8.6 | |

(336,118) | dimension | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 |

d-dit, d = 29 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

amino acid | Asn | Lys | Glu | Asp | Ile | Stop | . | . | . | . | |

order | 14 | 14 | 14 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | |

char | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | Cte | Cte | Cte | ${z}_{1,2}$ | ${z}_{1,2}$ | ${z}_{1,2}$ | |

polar req. | 10.0 | 10.1 | 12.5 | 13.0 | 10 | 15 | . | . | . | . | |

(336,118) | dimension | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | |

d-dit, d = 29 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

amino acid | Val | Pro | Thr | Ala | Gly | . | Leu | Ser | Arg | ||

order | 28 | 28 | 28 | 42 | 42 | 42 | 42 | 42 | 42 | ||

char | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{1,3}$ | ${z}_{1,3}$ | ${z}_{1,3}$ | ||

polar req. | 5.6 | 6.6 | 6.6 | 7.0 | 7.9 | . | 4.9 | 7.5 | 9.1 |

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**MDPI and ACS Style**

Planat, M.; Chester, D.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy. *Quantum Rep.* **2021**, *3*, 68-79.
https://doi.org/10.3390/quantum3010005

**AMA Style**

Planat M, Chester D, Aschheim R, Amaral MM, Fang F, Irwin K. Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy. *Quantum Reports*. 2021; 3(1):68-79.
https://doi.org/10.3390/quantum3010005

**Chicago/Turabian Style**

Planat, Michel, David Chester, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, and Klee Irwin. 2021. "Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy" *Quantum Reports* 3, no. 1: 68-79.
https://doi.org/10.3390/quantum3010005