Square Trisection
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In
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 ...
, a square trisection is a type of
dissection problem In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a ...
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 wikt:cut, cutting are the knife and saw, or in medicine and science the scal ...
a
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.


History

The dissection of a square in three
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
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 pa ...
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. This geometrical proof of Pythagoras' theorem would be rediscovered in the years 1835 - 1840 by
Henry Perigal Henry Perigal, Jr. Royal Astronomical Society, FRAS Royal Institution, MRI (1 April 1801 – 6 June 1898) was a British stockbroker and amateur mathematician, known for his dissection problem, dissection-based proof of the Pythagorean theorem an ...
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 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 accu ...
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 Lucas ...
. In 1891
Henry Perigal Henry Perigal, Jr. Royal Astronomical Society, FRAS Royal Institution, MRI (1 April 1801 – 6 June 1898) was a British stockbroker and amateur mathematician, known for his dissection problem, dissection-based proof of the Pythagorean theorem an ...
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.


See also

* Proofs by dissection and rearrangement of Pythagorean theorem *
Dissection puzzle A dissection puzzle, also called a transformation puzzle or ''Richter Puzzle'', is a tiling puzzle where a set of pieces can be assembled in different ways to produce two or more distinct geometric shapes. The creation of new dissection puzzles ...
*
Tangram The tangram () is a dissection puzzle consisting of seven flat polygons, called ''tans'', which are put together to form shapes. The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pi ...


Bibliography

* * *


References


External links


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