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Some Theorems of the Euclidean Geometry in Pentagonal Quasigroups

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ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics (ICGG 2018)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 809))

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Abstract

Pentagonal quasigroups are IM-quasigroups in which the additional identity of pentagonality holds. Motivated by the example C(q), where q is a solution of the equation \(q^4-3q^3+4q^2-2q+1=0\), some basic geometric concepts are defined in a general pentagonal quasigroup. Such concepts are parallelogram, midpoint of a segment, regular pentagon and regular decagon with their centres. The connection between pentagonal and, much better known, GS-quasigroups is mentioned. That connection enables introduction of more geometric concepts in pentagonal quasigroups. In this article some theorems of the Euclidean geometry which use all these concepts are stated and proved in pentagonal quasigroups.

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Authors and Affiliations

  1. University of Zagreb, 10000, Zagreb, Croatia

    Stipe Vidak

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  1. Stipe Vidak
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Correspondence to Stipe Vidak .

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Editors and Affiliations

  1. Department of Architecture and Urban Studies—DASTU, School of Architecture Urban Planning and Construction Engineering—AUIC, Politecnico di Milano, Milan, Italy

    Luigi Cocchiarella

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Vidak, S. (2019). Some Theorems of the Euclidean Geometry in Pentagonal Quasigroups. In: Cocchiarella, L. (eds) ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics. ICGG 2018. Advances in Intelligent Systems and Computing, vol 809. Springer, Cham. https://doi.org/10.1007/978-3-319-95588-9_37

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  • DOI: https://doi.org/10.1007/978-3-319-95588-9_37

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-95587-2

  • Online ISBN: 978-3-319-95588-9

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