Extension of Pick’s Theorem to Spherical Geometry using Girard’s Theorem

Yıl 2023, Cilt: 13 Sayı: 4, 2915 - 2925, 01.12.2023

Halil Rıdvan Öz

https://doi.org/10.21597/jist.1330915

Öz

In this article, Pick’s theorem is extended to three-dimensional bodies with two-dimensional surfaces, namely spherical geometry. The equation for the area of a polygon consisting of equilateral spherical triangles is obtained by combining Girard’s theorem used to find area of any spherical triangle and Pick’s theorem used to find area of a simple polygon with lattice point vertices in Euclidian geometry. Vertices of the polygon are represented by integer points. In this way, an equation to find area of a spherical polygon is presented. This equation could give an idea to be applied on cylindrical surfaces, hyperbolic geometry and more general surfaces. The theorem proposed in this article which is the extension of Pick’s theorem using Girard’s theorem seems to be a special case of a more general theorem.

Anahtar Kelimeler

Pick's theorem, Girard's theorem, Spherical lattice, Spherical geometry, Planar geometry

Kaynakça

• Atiyah, M. https://mathshistory.st-andrews.ac.uk/Biographies/Atiyah/quotations/.
• Bevis, M. and Cambareri, G. (1987). Computing the area of a spherical polygon of arbitrary shape. Mathematical Geology, 19(4), 335-346.
• Brooks J. B. and Strantzen, J. (2005). Spherical triangles of area π and isosceles tetrahedra. Mathematics Magazine, 78(4), 311-314.
• Gantmacher, F. R. (Ed.). (1966). The theory of matrices. New York, Chelsea Publishing Co.
• Gaskell, R. W., Klamkin, M. S. Watson, P. (1976). Triangulations and Pick’s theorem. Mathematics Magazine, 49(1), 35-37.
• Grunbaum B., Shephard, G. C. (1993). Pick’s theorem. The American Mathematical Monthly, 100(2), 150-161.
• Hadwiger Von H., Wills, J. M. (1976). Neure studien über gitterpolygone. Journal für die reine und angewandte Mathematik Magazine, 280, 61-69.
• Hilbert, D. https://www.brainyquote.com/quotes/david_hilbert_181562.
• Honsberger, R., (Ed.). (1970). Ingenuity in Mathematics. New York, Random House.
• Liu, A. C. F. (1979). Lattice points and Pick’s theorem. Mathematics Magazine, 52(4), 232-235.
• Miller, G. A. (1943). A seventh lesson in the history of mathematics. National Mathematics Magazine, 18(2), 67-76.
• Pick, G., (1899). Geometrisches zur Zahlenlehre. In: Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen "Lotos". (Neue Folge). 19, (p.311-319), Prag.
• Porta, J. M., Sarabandi S., Thomas, F. (2018). Angle-bound smoothing with application in kinematics. In: Asian MMS 2018 Conference, Paper No. Asian MMS-2018-124 (p. 1-13). Bengaluru, India.
• Reeve, J. E. (1958). A further note on the volume of lattice polyhedra. Journal London Math. Soc, 34, 57-62.
• Sarkar, J., Rashid, M., (2022). Mathematical Musings on the external anatomy of the novel corona virus, Part 4: Models of n-Cov. Resonance, 27(10), 1719-1730. DOI: https://doi.org/10.1007/s12045-022-1466-3.
• Sandeep K. G, Simon, B. N., Singh, R. Simon, S., (2016). Geometry of the generalized Bloch sphere for qutrits. Journal of Physics A: Mathematical and Theoretical, 49(16), 165203. DOI:10.1088/1751-8113/49/16/165203
• Scott, P. R. (1987). The fascination of the elementary. The American Mathematical Monthly, 94(8), 759-768.
• Todhunter, I. (Ed.). (1886). Spherical trigonometry for the use of colleges and schools. London, MacMillan and Co.
• Varberg, D. E. (1985). Pick’s theorem revisited. The American Mathematical Monthly, 92(8), 584-587.
• Zandi, R., Reguera, D., Bruinsma, R. F., Gelbart, W. M., Rudnick, J., (2004). Origin of icosahedral symmetry in viruses. PNAS, 101(44), 15556-15560. https://doi.org/10.1073/pnas.0405844101.