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Journals / Hagia Sophia Journal of Geometry / 2025 / Cilt: 7 Sayı: 2

Construction and Classification of Complete $(k,3)$-arcs from a Ceva 6-Figure in $PG(2,4)$

$PG(2,4)$ Projektif Düzleminde Ceva 6-Figüründen Elde Edilen Tam $(k,3)$-arkların İnşası ve Sınıflandırılması

Elif Altıntaş Kahriman(BARTIN ÜNİVERSİTESİ)0000-0002-3454-0326
Ayşe Bayar(ESKISEHIR OSMANGAZI UNIVERSITY)0000-0002-2210-5423
Pages
46–51
Publication Date
2025-12-29
DOI
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AI Summary & Key Takeaways

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

These results emphasize the combinatorial significance of Ceva-based configurations in finite projective planes and contribute to the systematic understanding of arc structures in finite geometry.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This study investigates complete $(k,3)$-arcs generated from a given Ceva 6-figure in the projective plane $PG(2,4)$.
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Key Findings & Data:The analysis reveals a unique complete $(7,3)$-arc obtained by adding the center point of the Ceva 6-figure, forming a Fano subplane, and eight distinct complete $(9,3)$-arcs constructed by adjoining three points on distinct 2-secant lines.
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Conclusion & Impact:These results emphasize the combinatorial significance of Ceva-based configurations in finite projective planes and contribute to the systematic understanding of arc structures in finite geometry.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:projektif düzlem · 3)-ark · tam ark

Abstract

This study investigates complete $(k,3)$-arcs generated from a given Ceva 6-figure in the projective plane $PG(2,4)$. The analysis reveals a unique complete $(7,3)$-arc obtained by adding the center point of the Ceva 6-figure, forming a Fano subplane, and eight distinct complete $(9,3)$-arcs constructed by adjoining three points on distinct 2-secant lines. No complete $(8,3)$-arc constructed from the given Ceva 6-figure exists. These results emphasize the combinatorial significance of Ceva-based configurations in finite projective planes and contribute to the systematic understanding of arc structures in finite geometry.

Özet

Bu çalışma, $PG(2,4)$ projektif düzleminde verilen bir Ceva 6-figüründen üretilen tam $(k,3)$-arkları araştırmaktadır. Yapılan analiz, Ceva 6-figürünün merkez noktasının eklenmesiyle bir Fano alt-düzlemi oluşturan tek bir tam $(7,3)$-arkın elde edildiğini ve farklı 2-kesen doğrular üzerinde yer alan üç noktanın eklenmesiyle sekiz farklı tam $(9,3)$-arkın oluşturulabildiğini göstermektedir. Verilen Ceva 6-figüründen türetilmiş herhangi bir tam $(8,3)$-arkın var olmadığı da belirlenmiştir. Bu sonuçlar, sonlu projektif düzlemlerde Ceva temelli konfigürasyonların kombinatoryal önemini vurgulamakta ve sonlu geometride ark yapılarının sistematik olarak anlaşılmasına katkı sağlamaktadır.

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Cite This Article

Kahriman, E. A., Bayar, A. (2025). Construction and Classification of Complete $(k,3)$-arcs from a Ceva 6-Figure in $PG(2,4)$. Hagia Sophia Journal of Geometry, Cilt: 7 Sayı: 2, 46–51. https://ajindex.com/articles/1490423

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References (16)

  1. Hirschfeld, J. W. P., & Thas, J. A. (2016). General Galois geometries. Springer Monographs in Mathematics, Springer-Verlag, London.
  2. Bayar, A., Akca, Z., Altıntaş, E., & Ekmekçi, S. (2016). On the complete arcs containing the quadrangles constructing the Fano planes of the left near field plane of order 9. New Trends in Mathematical Science, 4(4), 266–275.
  3. Ekmekçi, S., Bayar, A., Altintas, E., & Akça, Z. (2016). On the complete (k,2)-arcs of the Hall plane of order 9. International Journal of Advanced Research in Computer Science and Software Engineering, 6(10), 282–288.
  4. Altıntaş Kahriman, E., & Bayar, A.(2024). Investigating incomplete (7,3)-arcs and their extensions in PG(2,5): A study on secants and complete quadrangles. 5th Bilsel International World Scientific and Research Congress, İstanbul, (p. 536–545).
  5. Altıntaş Kahriman, E., & Bayar, A. (2024). An algorithm for constructing (k,2)-arcs containing triangle and quadrangle in PG(2,4). 5th Bilsel International Gordion Scientific Researches Congress, Ankara, (p. 987–997).
  6. Danos, V., & Regnier L. (1989). The structure of multiplicatives. Arch Math Logic, 28, 181–203.
  7. Benitez, J. (2007). A unified proof of Ceva and Menelaus’ theorems using projective geometry. Journal of Geometry and Graphics, 11(1), 39–44.
  8. Nicolae, V. (2020). On the Ceva’s and Menelaus’s theorems. Rom. J. Phys., 5(2), 43–50.
  9. Funk, B. K. (2008). Ceva and Menelaus in projective geometry. ProQuest Dissertations & Theses, University of Louisuille.
  10. Kaya, R., & Çiftçi S. (1985). On Menelaus and Ceva 6-figures in Moufang projective planes. Geom. Dedicata, 19(3), 295–296.
  11. Bayar, A., & Ekmekçi, S. (2014). On the Menelaus and Ceva 6-figures in the fibered projective planes. Abstr. Appl. Anal., 1–5.
  12. Akça, Z., Bayar, A., & Ekmekçi, S. (2020). On the intuitionistic fuzzy projective Menelaus and Ceva’s conditions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69(1), 891–899.
  13. Hirschfeld, J. W. P., & Voloch, J. F. (2015). Group-arcs of prime power order on cubic curves. Finite Geometry and Combinatorics, 191, 177–185.
  14. Hirschfeld, J. W. P., & Pichanick, E. V. D. (2016). Bounds for arcs of arbitrary degree in finite Desarguesian planes. J Comb Des, 24(4), 184–196.
  15. Ekmekçi, S., Bayar, A., & Akça, Z. (2022). On the projective planes in projective space PG(4,4). Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, 38(3), 519–524.
  16. Bayar, A., Akca, Z., Altıntaş, E., & Ekmekçi, S. (2016). On the complete arcs containing the quadrangles constructing the Fano planes of the left near field plane of order 9. New Trends in Mathematical Science, 4(4), 266–275.