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published the first known solution with only 6 pieces (see illustration below). Nowadays, new dissections are still found (see illustration above) and the conjecture that 6 is the minimal number of necessary pieces remains unproved.
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The beauty of a dissection depends on several parameters. However, it is usual to search for solutions with the minimum number of parts. Far from being minimal, the square trisection proposed by
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needed innovative techniques to achieve their fabulous mosaics with complex geometric figures. The first solution to this problem was proposed in the 10th century AD by the
Persian mathematician
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came back to this issue and in the 19th century, solutions using 8 and 7 pieces were found, including one given by the mathematician
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Jean-Etienne
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Biography of Henry
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gave two solutions, one of which uses 8 pieces. In the late 17th century
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Cutting a square into pieces which rearrange into 3 identical squares
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into pieces that can be rearranged to form three identical squares.
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Omar
Khayyam, Mathematicians, and “conversazioni” with Artisans.
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Proofs by dissection and rearrangement of
Pythagorean theorem
391:, Association for the Improvement of Geometrical Teaching.
80:"On the geometric constructions necessary for the artisan"
411:. Bulletin d'Informatique Approfondie et Applications
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Square trisection using 6 pieces of same area (2010).
273:Towson University and The Mathematical Institute.
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66:is a geometrical problem that dates back to the
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313:On Geometric Dissections and Transformations
271:Elementary Constructions of Persian Mosaics.
86:also used his dissection to demonstrate the
334:Volume 27, Issue 2, May 2000, Pages 171-201
296:. Proceedings London Mathematical Society.
193:Hinged Dissections: Swinging and Twisting
389:Geometric Dissections and Transpositions
257:Journal of the Society of Architectural
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219:Piano-hinged Dissections: Time to Fold!
292:See appendix of L. J. Rogers (1897).
269:Reza Sarhangi, Slavik Jablan (2006).
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70:. Craftsman who mastered the art of
298:Volume s1-29, Appendix pp. 732-735.
106:uses 9 pieces. In the 14th century
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428:oai:infoscience.epfl.ch:161493
216:Frederickson, Greg N. (2006).
189:Frederickson, Greg N. (2002).
162:Frederickson, Greg N. (1997).
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315:, Messenger of Mathematics,
166:Dissections: Plane and Fancy
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259:Vol. 54, No. 1, Mar., 1995
199:Cambridge University Press
172:Cambridge University Press
78:(940-998) in his treatise
370:Récréations Mathématiques
350:Récréations mathématiques
458:Euclidean plane geometry
332:, Historia Mathematica,
403:Christian Blanvillain,
94:and published in 1875.
368:Edouard Lucas (1883).
328:Alpay Ă–zdural (2000).
253:Alpay Ă–zdural (1995).
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463:Mathematical problems
374:online (pp. 145-147).
59:of a square in three
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478:Geometric dissection
98:Search of optimality
468:History of geometry
88:Pythagorean theorem
418:2011-07-24 at the
280:2011-07-28 at the
108:Abu Bakr al-Khalil
68:Islamic Golden Age
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31:which consists of
29:dissection problem
409:Square Trisection
145:Dissection puzzle
25:square trisection
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104:Abu'l-Wafa'
84:Abu'l-Wafa'
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452:Categories
405:János Pach
393:wikisource
242:References
118:. In 1891
64:partitions
57:dissection
61:congruent
422:also at
416:Archived
407:(2010).
387:(1891).
311:(1875).
278:Archived
134:See also
21:geometry
150:Tangram
72:zellige
51:History
33:cutting
275:online
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37:square
473:Area
424:EPFL
366:(fr)
346:(fr)
228:ISBN
203:ISBN
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
