Kvadrat trisektsiya

Geometriyada kvadrat trisektsiya — bu uchta bir xil kvadrat hosil qilish uchun kvadratni boʻlaklarga boʻlishdan iborat boʻlgan kesish holati hisoblanadi.

Tarixi

Kvadratning uchta mos keladigan boʻlinmalarga boʻlinishi islomning oltin davriga borib taqaladigan geometrik muammo hisoblanadi. Zellige sanʼatini puxta egallagan hunarmandga murakkab geometrik figuralar bilan ajoyib mozaikaga erishish uchun innovatsion usullar kerak edi. Bu muammoning birinchi yechimi milodiy 10-asrda fors matematigi Abul-Vafo (940-998) tomonidan "Hunarmand uchun zarur boʻlgan geometrik konstruktsiyalar toʻgʻrisida" risolasida taklif qilingan^[1] .Abul-Vafo ham Pifagor teoremasini koʻrsatish uchun oʻz parchasidan foydalangan .Pifagor teoremasining bu geometrik isboti 1835-1840^[2]-yillarda Genri Perigal tomonidan qayta kashf etilgan va 1875-yilda nashr etilgan^[3].

Optimallikni izlash

Dissektsiyaning goʻzalligi bir nechta parametrlarga bogʻliq. Biroq, minimal qismlar soni bilan echimlarni izlash odatiy holdir. Abul-Vafo tomonidan taklif qilingan kvadrat trisektsiya minimal boʻlishdan uzoqda, 9 ta qismdan foydalanadi. 14-asrda Abu Bakr al-Xalil ikkita yechim berdi, ulardan biri 8 ta boʻlakka boʻlgan^[4]. 17-asr oxirida Jak Ozanam bu masalaga qaytdi^[5] va 19-asrda 8 va 7 boʻlaklardan foydalangan holda echimlarini topildi, shu jumladan matematik Eduard Lukas tomonidan berilgan^[6] 1891-yilda Genri Perigal atigi 6 dona^[7] bilan birinchi maʼlum boʻlgan yechimni nashr etdi (quyidagi rasmga qarang). Hozirgi vaqtda yangi boʻlaklar hali ham topilmoqda (yuqoridagi rasmga qarang) va 6 ta zarur boʻlaklarning minimal soni degan taxmin hali ham tasdiqlanmagan.

Yana qarang

Pifagor teoremasini parchalash va qayta tartibga solish orqali isbotlash
Dissektsiya boshqotirmasi
Tangram

Bibliografiya

Frederickson, Greg N.. Dissections: Plane and Fancy. Cambridge University Press, 1997. ISBN 0-521-57197-9.
Frederickson, Greg N.. Hinged Dissections: Swinging and Twisting. Cambridge University Press, 2002. ISBN 0-521-81192-9.
Frederickson, Greg N.. Piano-hinged Dissections: Time to Fold!. en:A K Peters, 2006. ISBN 1-56881-299-X.

Manbalar

↑ Alpay Özdural (1995). Omar Khayyam, Mathematicians, and „conversazioni“ with Artisans. Journal of the Society of Architectural Vol. 54, No. 1, Mar., 1995
↑ See appendix of L. J. Rogers (1897). Biography of Henry Perigal: On certain Regular Polygons in Modular Network. Proceedings London Mathematical Society. Volume s1-29, Appendix pp. 732-735.
↑ Henry Perigal (1875). On Geometric Dissections and Transformations, Messenger of Mathematics, No 19, 1875.
↑ Alpay Özdural (2000). Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World, Historia Mathematica, Volume 27, Issue 2, May 2000, Pages 171-201.
↑ (fr) Jean-Etienne Montucla (1778), completed and re-edited by Jacques Ozanam (1640-1717) Récréations mathématiques, Tome 1 (1694), p. 297 Pl.15.
↑ (fr) Edouard Lucas (1883). Récréations Mathématiques, Volume 2. Paris, Gauthier-Villars. Second of four volumes. Second edition (1893) reprinted by Blanchard in 1960. See pp. 151 and 152 in Volume 2 of this edition. online (pp. 145-147).
↑ Henry Perigal (1891). Geometric Dissections and Transpositions, Association for the Improvement of Geometrical Teaching. wikisource

Havolalar

Greg N. Frederikson veb-sayti

[1] Alpay Özdural (1995). Omar Khayyam, Mathematicians, and „conversazioni“ with Artisans. Journal of the Society of Architectural Vol. 54, No. 1, Mar., 1995

[2] See appendix of L. J. Rogers (1897). Biography of Henry Perigal: On certain Regular Polygons in Modular Network. Proceedings London Mathematical Society. Volume s1-29, Appendix pp. 732-735.

[3] Henry Perigal (1875). On Geometric Dissections and Transformations, Messenger of Mathematics, No 19, 1875.

[4] Alpay Özdural (2000). Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World, Historia Mathematica, Volume 27, Issue 2, May 2000, Pages 171-201.

[5] (fr) Jean-Etienne Montucla (1778), completed and re-edited by Jacques Ozanam (1640-1717) Récréations mathématiques, Tome 1 (1694), p. 297 Pl.15.

[6] (fr) Edouard Lucas (1883). Récréations Mathématiques, Volume 2. Paris, Gauthier-Villars. Second of four volumes. Second edition (1893) reprinted by Blanchard in 1960. See pp. 151 and 152 in Volume 2 of this edition. online (pp. 145-147).

[7] Henry Perigal (1891). Geometric Dissections and Transpositions, Association for the Improvement of Geometrical Teaching. wikisource

[1]

[2]

[3]

[4]

[5]

[6]

[7]