Permasalahan Lukisan Geometri sebagai Suatu Lapangan
Keywords:
Membagi tiga sebuah sudut, Menggandakan volume kubusAbstract
This text discusses two famous classical problems in geometric construction, where the only permitted tools are a compass and an unmarked straightedge. These two problems are trisecting an angle and doubling the volume of a cube, both of which will be proven impossible to construct. The proof is conducted by translating geometric problems into algebraic problems. Subsequently, the concepts of fields and vector spaces are utilized to prove the constructibility of these geometric problems.
Keywords: Doubling the volume of a cube, Trisecting an angle
ABSTRAK
Tulisan ini membahas dua masalah klasik yang sangat terkenal di dalam lukisan geometri dimana alat yang diperbolehkan hanya jangka dan mistar tanpa skala. Kedua masalah tersebut adalah membagi tiga buah sudut dan menggandakan volume kubus, yang akan dibuktikan bahwa keduanya mustahil dapat dilukis. Pembuktian dilakukan dengan menerjemahkan permasalahan geometri menjadi permasalahan aljabar. Kemudian, digunakan konsep lapangan dan ruang vektor untuk membuktikan keterlukisan masalah lukisan tersebut.
References
Gallian, Joseph A. 2010. Contemporary Abstract Algebra 7th ed., Brooks/Cole,California.
Herstein, I.N. 1999. Abstract Algebra 3rd ed., John Wiley & Sons, New York.
Kalanov, T. (2010). The modern analysis of the problem of multisecting an angle. Prespacetime Journal, 1(3).
Lang, Serge. 2004. Linear Algebra 3rd ed., Springer-Verlang, New York.
Lützen, J. (2010). The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle. Centaurus, 52(1), 4-37.
Martin, G.E. 1991. Geometric Constructions 2nd ed., Springer-Verlag, New York.
Porter, A. F. (1994). Duplication of the cube. Survey Review, 32(251), 303-306.
Rediske, A. C. (2018). The Trisection of an Arbitrary Angle: A Classical Geometric Solution. Journal: Journal of Advances in Mathematics, 14(02).
Sohrab, S. H. (2012, January). A possible solution of trisection problem. Proceedings of the 6th WSEAS International Conference on Computer Engineering and Applications, and Proceedings of the 2012 American Conference on Applied Mathematics, 277-285.
Willis, L. A. (2015). Duplication of a Cube. American Journal of Applied Mathematics, 3(6), 256-258.
