Two-dimensional integer trigonometry

• Thesis title in Czech: Dvoudimenzionální celočíselná trigonometrie
• Thesis title in English: Two-dimensional integer trigonometry
• Key words: celočíselná trigonometrie|mřížky|Euklidovská geometrie|řetězové zlomky
• English key words: integer trigonometry|lattices|Euclidean geometry|continued fractions
• Academic year of topic announcement: 2023/2024
• Thesis type: Bachelor's thesis
• Thesis language: angličtina
• Department: Department of Algebra (32-KA)
• Supervisor: doc. Mgr. Vítězslav Kala, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 18.03.2024
• Date of assignment: 20.03.2024
• Confirmed by Study dept. on: 20.03.2024
• Date of electronic submission: 09.05.2024
• Opponents: Dr. rer. nat. Siu Hang Man, Ph.D.
• Advisors: Bc. Martin Raška

Guidelines

The two-dimensional integer lattice and its integer invariants are in close connection with many number theory topics, such as continued fractions and number fields. In the thesis, the student will cover basic two-dimensional integer geometry definitions and notions, giving detailed proofs and examples [Kar]. She will further follow [BDK] in establishing known integer trigonometry identities and facts; giving detailed proofs of the propositions stated in the paper, giving examples and comparing integer identities to their Euclidean geometry analogies. The student may further potentially consider covering in detail the connection between integer tangents and continued fractions [Kar].

References

[Kar] Karpenkov, O. (2022). Geometry of Continued Fractions. Algorithms and Computation in Mathematics, vol 26. Springer Berlin, Heidelberg.
[BDK] Blackman, J., Dolan, J., Karpenkov, O. (2023). Multidimensional integer trigonometry. Communications in Mathematics, Vol. 31, Issue 2.