Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling

Cemil Karaçam; Ahmet Çağdaş Girit

Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling

Abstract

This study presents a detailed derivation of a trigonometric identity governing the optimal scaling of a regular m-gon inscribed within a regular n-gon underdouble-contact constraints. Building on prior work that established containment inequalities for nested polygons in the complex plane, we focus on the symmetric configuration where rotational and vertical translation components vanish (b = 0, d = 0). In this setting, we derive a closed-form expression for the scaling factor c by equating two distinct contact conditions involving edge-vertex interactions. The resulting identity incorporates cosine and cotangent terms and reveals how geometric symmetry leads to algebraic simplification. We also provide a long-form factorization and numerical examples to illustrate the identity’s behavior across different polygon pairs. This work contributes to the broader theory of polygonal optimization and symbolic encoding in geometric configurations.

Keywords

References

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Details

Primary Language

English

Subjects

Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics), Pure Mathematics (Other)

Journal Section

Research Article

Authors

Cemil Karaçam ^*
0000-0001-7186-5114
Türkiye

Ahmet Çağdaş Girit
0009-0005-8108-9418
Türkiye

Publication Date

December 31, 2025

Submission Date

November 22, 2025

Acceptance Date

December 31, 2025

Published in Issue

Year 2025 Volume: 2 Number: 2

IZ

https://izlik.org/JA23UR97UG

Cite

RIS / Bibtex

APA

Karaçam, C., & Girit, A. Ç. (2025). Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling. Türkiye Mathematical Sciences, 2(2), 54-63. https://izlik.org/JA23UR97UG

AMA

1.Karaçam C, Girit AÇ. Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling. TMS. 2025;2(2):54-63. https://izlik.org/JA23UR97UG

Chicago

Karaçam, Cemil, and Ahmet Çağdaş Girit. 2025. “Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling”. Türkiye Mathematical Sciences 2 (2): 54-63. https://izlik.org/JA23UR97UG.

EndNote

Karaçam C, Girit AÇ (December 1, 2025) Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling. Türkiye Mathematical Sciences 2 2 54–63.

IEEE

[1]C. Karaçam and A. Ç. Girit, “Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling”, TMS, vol. 2, no. 2, pp. 54–63, Dec. 2025, [Online]. Available: https://izlik.org/JA23UR97UG

ISNAD

Karaçam, Cemil - Girit, Ahmet Çağdaş. “Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling”. Türkiye Mathematical Sciences 2/2 (December 1, 2025): 54-63. https://izlik.org/JA23UR97UG.

JAMA

1.Karaçam C, Girit AÇ. Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling. TMS. 2025;2:54–63.

MLA

Karaçam, Cemil, and Ahmet Çağdaş Girit. “Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling”. Türkiye Mathematical Sciences, vol. 2, no. 2, Dec. 2025, pp. 54-63, https://izlik.org/JA23UR97UG.

Vancouver

1.Cemil Karaçam, Ahmet Çağdaş Girit. Optimal Scaling in Double-Contact Regular Polygon Containment: Double-Contact Polygon Scaling. TMS [Internet]. 2025 Dec. 1;2(2):54-63. Available from: https://izlik.org/JA23UR97UG