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Geometric Triple Systems with Base Z and Zn

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Abstract

In this paper we discuss triple systems whose elements come from an abelian group G. Every triple of elements with the same sum in G will correspond to three collinear points in the projective plane. We will show that the set of all points in such a triple system must lie on a cubic curve γ which we call the envelope of the system. We will determine the number and location of equivalent triple systems which have the same cubic envelope. These results will be used to determine exactly which finite abelian groups can be used to construct a geometric triple system.

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References

  1. Bix, R.: Conics and Cubics. Springer Science+Business Media, LLC, New York (2006)

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  2. Harris, M: Thirdpoint groupoids. Master’s thesis, Virginia State University, (2008)

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  3. Knapp, A.: Eliptic Curves. Princeton University Press, (1992)

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  4. Fletcher, R: Geometric Triple Systems with Base Z2×Zn, submitted for publication in Springer Proceedings in Mathematics and Statistics, (2016)

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Author information

Authors and Affiliations

  1. Department of Mathematics and Economics, Virginia State University, Petersburg, VA, 23806, USA

    Raymond R. Fletcher III

Authors
  1. Raymond R. Fletcher III
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Corresponding author

Correspondence to Raymond R. Fletcher III .

Editor information

Editors and Affiliations

  1. Department of Mathematics & Economics, Virginia State University, Petersburg, Virginia, USA

    Bourama Toni

Figure Appendix

Figure Appendix

Fig. 31
figure 31

(Z18,0) triple system on Type IV cubic curve with increment=2

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Fig. 32
figure 32

(Z18,1) triple system on Type IV cubic curve with increment 4

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Fig. 33
figure 33

(Z13,0) triple system on Type V cubic

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Fig. 34
figure 34

(Z14,0) triple system on Type V cubic with vertex 7 at infinity

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Fletcher, R.R. (2017). Geometric Triple Systems with Base Z and Zn . In: Toni, B. (eds) New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-319-55612-3_2

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