geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a square trisection is a type of dissection problem which consists of

cutting Cutting is the separation or opening of a physical object, into two or more portions, through the application of an acutely directed force. Implements commonly used for cutting are the knife and saw, or in medicine and science the scalpel and ...

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

into pieces that can be rearranged to form three identical squares. Trisection Blanvillain

History

The

dissection Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...

of a square in three congruent partitions is a geometrical problem that dates back to the

Islamic Golden Age The Islamic Golden Age was a period of cultural, economic, and scientific flourishing in the history of Islam, traditionally dated from the 8th century to the 14th century. This period is traditionally understood to have begun during the reign ...

. Craftsman who mastered the art of

zellige ''Zellij'' ( ar, الزليج, translit=zillīj; also spelled zillij or zellige) is a style of mosaic tilework made from individually hand-chiseled tile pieces. The pieces were typically of different colours and fitted together to form various p ...

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 Abu'l-Wafa' (940-998) in his treatise ''"On the geometric constructions necessary for the artisan"''. Abu'l-Wafa' also used his dissection to demonstrate the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...

. This geometrical proof of Pythagoras' theorem would be rediscovered in the years 1835 - 1840 by

Henry Perigal Henry Perigal, Jr. FRAS MRI (1 April 1801 – 6 June 1898) was a British stockbroker and amateur mathematician, known for his dissection-based proof of the Pythagorean theorem and for his unorthodox belief that the moon does not rotate...... Bi ...

and published in 1875.

Search of optimality

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 Abu'l-Wafa' uses 9 pieces. In the 14th century

Abu Bakr al-Khalil Abu or ABU may refer to: Places * Abu (volcano), a volcano on the island of Honshū in Japan * Abu, Yamaguchi, a town in Japan * Ahmadu Bello University, a university located in Zaria, Nigeria * Atlantic Baptist University, a Christian universit ...

gave two solutions, one of which uses 8 pieces. In the late 17th century

Jacques Ozanam Jacques Ozanam (16 June 1640, in Sainte-Olive, Ain – 3 April 1718, in Paris) was a French mathematician. Biography Jacques Ozanam was born in Sainte-Olive, Ain, France. In 1670, he published trigonometric and logarithmic tables more accura ...

came back to this issue and in the 19th century, solutions using 8 and 7 pieces were found, including one given by the mathematician

Édouard Lucas __NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him. Biography Lu ...

. In 1891

published the first known solution with only 6 pieces (see illustration below). Nowadays, new dissections are still found Christian Blanvillain,

János Pach János Pach (born May 3, 1954) is a mathematician and computer scientist working in the fields of combinatorics and discrete and computational geometry. Biography Pach was born and grew up in Hungary. He comes from a noted academic family: his ...

(2010). ''Square Trisection''. Bulletin d'Informatique Approfondie et Application
N°86 - Juin 2010
also at EPFL
oai:infoscience.epfl.ch:161493
(see illustration above) and the conjecture that 6 is the minimal number of necessary pieces remains unproved. ThreeSqrs2One

Bibliography

* * *

References

External links

Greg N. Frederickson web site
{{DEFAULTSORT:Square Trisection Euclidean plane geometry Mathematical problems History of geometry Area Geometric dissection

History

Search of optimality

See also

Bibliography

References

External links