Algebraic geometry on imaginary triangles

Yıl 2018, Cilt: 11 Sayı: 2, 71 - 82, 30.11.2018

Sergiy Koshkin

https://doi.org/10.36890/iejg.545133

Öz

We extend the notion of triangle to imaginary triangles with complex valued sides and angles,

and parametrize families of such triangles by plane algebraic curves.We study in detail families of

triangles with two commensurable angles, and apply the theory of plane Cremona transformations

to find “Pythagorean theorems" for these triangles, which are interpreted as the implicit equations

of their parametrizing curves.

Anahtar Kelimeler

Pythagorean triples, Pythagorean theorem, commensurable triangle, Chebyshev polynomials, rational curve, quadratic Cremona transformation, base point, exceptional curvePythagorean triples, exceptional curve

Kaynakça

• [1] Alberich-Carramiñana, M., Geometry of the plane Cremona maps. Lecture Notes in Mathematics 1769, Springer-Verlag, Berlin, 2002.
• [2] Bicknell-Johnson, M., Nearly isosceles triangles where the vertex angle is a multiple of the base angle. Applications of Fibonacci numbers, vol. 4 (Winston-Salem, NC, 1990), pp. 41-50, Kluwer, Dordrecht, 1991.
• [3] Carroll, J. and Yanosko, K., The determination of a class of primitive integral triangles. Fibonacci Quarterly, 29 (1991), no. 1, pp. 3–6.
• [4] Coble, A., Cremona transformations and applications to algebra, geometry, and modular functions. Bulletin of the American Mathematical Society, 28 (1922), no. 7, 329-364.
• [5] Coolidge, J., A treatise on algebraic plane curves. Dover Publications, New York, 1959.
• [6] Daykin, D. and Oppenheim, A., Triangles with rational sides and angle ratios. American Mathematical Monthly, 74 (1967), no. 1, pp. 45–47.
• [7] Deshpande, M., Some new triples of integers and associated triangles. Mathematical Gazette, 86 (2002), no. 543, pp. 464-466.
• [8] Fulton, W., Algebraic curves. An introduction to algebraic geometry. Advanced Book Classics, Addison-Wesley Publishing Company, Redwood City, CA, 1989.
• [9] Gibson, C., Elementary geometry of algebraic curves: an undergraduate introduction. Cambridge University Press, Cambridge, 1998.
• [10] Guy, R., Triangles with B = 3A, 2B = 3A. Bulletin of the Malayan Mathematical Society, 1 (1954) pp. 56-60.
• [11] Hoyt, J., Extending the converse of Pons Asinorum. Mathematics Magazine, 61 (1988), no. 5, pp. 308–313.
• [12] Kendig, K., Is a 2000-year-old formula still keeping some secrets? American Mathematical Monthly, 107 (2000), no. 5, 402–415.
• [13] Koshkin, S., Mixing angle trisection with Pythagorean triples. Mathematical Gazette, 100 (2016), no. 549, 492–495.
• [14] Lemmermeyer, F., Parametrization of algebraic curves from a number theorist’s point of view. American Mathematical Monthly, 119 (2012), no. 7, 573–583.
• [15] Luthar, R., Integer-sided triangles with one angle twice another. College Mathematics Journal, 15 (1984) no.3, pp. 55–56.
• [16] Nicollier, G., Triangles with two angles in the ratio 1:2. Mathematical Gazette, 98 (2014), no. 543, pp. 508–509.
• [17] Parris, R., Commensurable Triangles. College Mathematics Journal, 38 (2007), no. 5, pp. 345–355.
• [18] Reid, M., Undergraduate algebraic geometry. London Mathematical Society Student Texts 12, Cambridge University Press, Cambridge, 1988.
• [19] Rusin, D., Rational Triangles with Equal Area. New York Journal of Mathematics, 4 (1998), pp. 1-15.
• [20] Sadek, M. and Shahata, F., On rational triangles via algebraic curves. Rocky Mountain Journal of Mathematics, to appear (2017), available at https://arxiv.org/abs/1610.07971
• [21] Selkirk, K., Rational Triangles. Mathematics Magazine, 66 (1993), no. 1, 24–27.
• [22] Sendra, J., Winkler, F. and Pèrez-Dìaz, S., Rational algebraic curves. A computer algebra approach. Algorithms and Computation in Mathematics 22, Springer, Berlin, 2008.
• [23] Walker, R., Algebraic Curves. Princeton Mathematical Series 13, Princeton University Press, Princeton, 1950.
• [24] Willson, W., A generalisation of a property of the 4, 5, 6 triangle. Mathematical Gazette, 60 (1976) no. 412, pp. 130-131.