SavedAdd your journal

Journals / Journal of Scientific Reports-B / 2025 / Issue: 014

Construction of complete (k,3)-Arcs containing menelaus 6-Figures in PG(2,4) using a secant-based algorithm

PG(2,4) Düzleminde Menelaus 6-figür İçeren Tam (k,3)-Arkların Sekant-Tabanlı Bir Algoritma ile Oluşturulması

Ayşe Bayar(ESKISEHIR OSMANGAZI UNIVERSITY)0000-0002-2210-5423
Elif Altıntaş Kahriman(BARTIN ÜNİVERSİTESİ)0000-0002-3454-0326
Pages
—
Publication Date
2025-12-30
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

By analyzing the extension of this configuration, one unique complete (7,3)-arc and eight distinct complete (9,3)-arcs were identified, all of which contain the Menelaus 6-figure.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This study investigates complete (k,3)-arcs containing a Menelaus 6-figure in the projective plane PG(2,4).
📊
Key Findings & Data:We developed and implemented a computational algorithm in C# to construct and classify complete arcs through a detailed analysis of the secant line distributions associated with the Menelaus 6-figure.
💡
Conclusion & Impact:By analyzing the extension of this configuration, one unique complete (7,3)-arc and eight distinct complete (9,3)-arcs were identified, all of which contain the Menelaus 6-figure.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:projective plane · 3)-arc · Menelaus 6-figure

Abstract

This study investigates complete (k,3)-arcs containing a Menelaus 6-figure in the projective plane PG(2,4). We developed and implemented a computational algorithm in C# to construct and classify complete arcs through a detailed analysis of the secant line distributions associated with the Menelaus 6-figure. The algorithm identifies seven points outside the Menelaus 6-figure, which are classified into two categories based on their incidence with 0-secant, 1-secant, and 2-secant lines. By analyzing the extension of this configuration, one unique complete (7,3)-arc and eight distinct complete (9,3)-arcs were identified, all of which contain the Menelaus 6-figure.

Özet

Bu çalışma, projektif düzlem PG(2,4)’te Menelaus 6-figürünü içeren tam (k,3)-arkları incelemektedir. C# dilinde geliştirilen ve uygulanan hesaplamalı bir algoritma aracılığıyla, Menelaus 6-figüre ait sekant doğru dağılımlarının ayrıntılı analizi yapılarak tam arkların inşası ve sınıflandırılması gerçekleştirilmiştir. Algoritma, Menelaus 6-figürün dışında bulunan yedi noktayı belirlemekte ve bu noktaları 0-sekant, 1-sekant ve 2-sekant doğrularıyla ilişkilerine göre iki ayrı kategoriye ayırmaktadır. Bu konfigürasyonun genişlemesi incelendiğinde, Menelaus 6-figürü içeren bir adet özgün tam (7,3)-ark ve sekiz farklı tam (9,3)-ark tespit edilmiştir.

🔖

Cite This Article

Bayar, A., Kahriman, E. A. (2025). Construction of complete (k,3)-Arcs containing menelaus 6-Figures in PG(2,4) using a secant-based algorithm. Journal of Scientific Reports-B, Issue: 014. https://ajindex.com/articles/1619080

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (17)

  1. J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Springer Monographs in Mathematics. Springer- Verlag London, 2016.
  2. A. Bayar, Z. Akca, E. Altintas, S. Ekmekci, “On the complete arcs containing the quadrangles constructing the Fano planes of the left near field plane of order 9,” New Trend Math. Sci., vol. 4, no.4, pp. 266-266, 2016, doi: 10.20852/ntmsci.2016.113
  3. S. Ekmekci, A. Bayar, E. Altintas, Z. Akca, “On the Complete (k,2)- Arcs of the Hall Plane of Order 9,” IJARCSSE, vol. 6, no.10, pp. 282-288, 2016, ISSN: 2277 128X.
  4. E. Altıntaş Kahriman and A. Bayar, “Investigating Incomplete (7,3)-Arcs and Their Extensions in PG(2,5): A Study On Secants And Complete Quadrangles,” 5. Bilsel International World Scientific And Research Congress, İstanbul, Türkiye, 05-06 October 2024, pp.536-545.
  5. E. Altıntaş Kahriman and A. Bayar, “An Algorithm For Constructing (k,2)-Arcs Containing Triangle And Quadrangle in PG(2,4),” 5. Bilsel International Gordion Scientific Researches Congress, Ankara/ Türkiye, 08-09 December, 2024, pp.987-997.
  6. V. Danos and L. Regnier, “The structure of multiplicatives,” Arch Math Logic, vol. 28, 1989, pp. 181-203, doi: 10.1007/BF01622878
  7. J. Benitez, “A unified proof of Ceva and Menelaus’ theorems using projective geometry,” JGG, vol. 11, no. 1, pp. 39–44, 2007. ISSN 1433-8157
  8. V. Nicolae, “On The Ceva’s And Menelaus’s Theorems.” Rom. J. Phys., [S.l.], vol. 5, no. 2, pp. 43-50, 2020. ISSN 2537-5229.
  9. B.K. Funk, Ceva and Menelaus in projective geometry, University of Louisuille, 42 p, 2008.
  10. S. Çiftçi, R. Kaya, and J.C. Ferrar, “On Menelaus and Ceva 6-figures in Moufang projective planes,” Geom. Dedicata, vol. 19, no. 3, pp. 295–296, 1985.
  11. A. Bayar and S. Ekmekçi, “On the Menelaus and Ceva 6-figures in the fibered projective planes,” Abstr. Appl. Anal.,pp. 1-5, 2014, doi: 10.1155/2014/803173
  12. Z. Akça, A. Bayar, and S. Ekmekçi, “On the intuitionistic fuzzy projective Menelaus and Ceva’s conditions,” Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 69 no.1, pp. 891-899, 2020. doi: 10.31801/cfsuasmas.567753
  13. A. Bayar and E. Altıntaş Kahriman, “Complete (11,3)-Arcs Constructed from Menelaus Configurations in PG(2,5) via Secant-Based Analysis,” Pioneer Research in Natural Science and Mathematics, All Sciences Academy, 2025, pp. 5-20.
  14. E. Altıntaş Kahriman and A. Bayar, “Complete (10,3)-Arcs Constructed from Menelaus Configurations in PG(2,5): A Secant-Based Approach,” Pioneer Research in Natural Science and Mathematics, All Sciences Academy, 2025, pp. 21-37.
  15. J.W.P. Hirschfeld and J.F. Voloch, “Group-arcs of prime order on cubic curves,” Finite Geometry and Combinatorics, vol. 191, pp. 177-185, 2015.
  16. J.W.P. Hirschfeld and E.V.D Pichanick, “Bounded for arcs of arbitrary degree in finite Desarguesian Planes,” J Comb Des, vol. 24, no. 4, pp. 184-196, 2016.
  17. S. Ekmekci, A. Bayar, Z. Akca, “On The Projective Planes In Projective Space PG(4,4),” JIST, vol. 38, no. 3, pp. 519-524, 2022.