Date of Publication
2026
Document Type
Bachelor's Thesis
Degree Name
Bachelor of Science in Mathematics with Specialization in Computer Applications
Subject Categories
Mathematics
College
College of Science
Department/Unit
Mathematics and Statistics Department
Thesis Advisor
April Lynne D. Say-awen
Defense Panel Chair
Francis Joseph H. Campeña
Defense Panel Member
Rigor B. Ponsones
Abstract (English)
Quasicrystals reveal that long-range order in matter can exist without translational symmetry, exhibiting symmetries such as fivefold or eightfold that are not possible in conventional crystals. Mathematical models of these structures, known as aperiodic or quasiperiodic tilings, provide a geometric framework for studying nonperiodic order. Octagonal tilings are of particular significance, as eightfold symmetry has been experimentally observed in a variety of quasicrystalline systems, including colloidal, oxide, and molecular assemblies. This work studies the combinatorial and graph-theoretic properties of the (1, 1)-octagonal tiling introduced by Say-awen and Coates (2025). In particular, we extend the constructive method of Singh et al. (2024) to obtain Hamiltonian cycles in arbitrary patches of the (1, 1)-octagonal tiling. As additional results, we apply Euler’s coordination number relation as a consistency check and examine vertex colorings.
Abstract Format
html
Abstract (Filipino)
Ipinapakita ng mga quasicrystal na ang mahabang-saklaw na kaayusan (long-range order) sa materya ay maaaring umiral kahit walang simetriyang translasyonal (translational symmetry), at maaari itong magpakita ng mga simetriyang gaya ng limang-tupi (fivefold) at walong-tupi (eightfold) na hindi matatagpuan sa mga karaniwang kristal. Ang mga matematikal na modelo ng mga estrukturang ito, na tinatawag na aperiodic o quasiperiodic tilings, ay nagbibigay ng isang geometrikong balangkas sa pag-aaral ng di-peryodikong kaayusan (nonperiodic order). Ang mga oktagonal na tiling ay may mahalagang papel sapagkat ang walong-tuping simetriya ay napatunayang umiiral sa iba’t ibang uri ng sistemang quasicrystalline, kabilang ang mga sistemang koloydal, oksido, at mga molekular na assemblage. Sinusuri sa pag-aaral na ito ang mga kombinatoriyal at teoretikong-graph na katangian ng (1,1)-oktagonal na tiling na ipinakilala nina Say- awen at Coates (2025). Partikular na inaangkop ang konstruktibong pamamaraan nina Singh et al. (2024) upang makuha ang mga Hamilton cycle sa mga arbitraryong bahagi (patches) ng (1,1) oktagonal na tiling. Bukod dito, inilalapat ang ilang kilalang pormula—tulad ng ugnayan ng koordinasyon ni Euler (Euler’s coordination number relation)—bilang pagsusuri ng konsistensi, at sinisiyasat din ang mga vertex coloring.
Abstract Format
html
Language
English
Format
Electronic
Keywords
Graph theory; Combinatorial analysis; Tiling (Mathematics)
Recommended Citation
Ramos, C. C. (2026). Combinatorial and graph-theoretic properties of the (1,1)-octagonal tilings. Retrieved from https://animorepository.dlsu.edu.ph/etdb_math/68
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Embargo Period
4-14-2027
