Editing Tangram

{{short description|Dissection puzzle}}
{{Other uses}}
{{use dmy dates|date=April 2021}}
[[Image:Tangram set 00.jpg|thumb|300px|Like most modern sets, this wooden tangram is stored in the square configuration.]]
The '''tangram''' ({{zh|c=七巧板|p=qīqiǎobǎn|l=seven boards of skill}}) 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 pieces without overlap. Alternatively the ''tans'' can be used to create original minimalist designs that are either appreciated for their inherent aesthetic merits or as the basis for challenging others to replicate its outline. It is reputed to have been invented in [[China]] sometime around the late 18th century and then carried over to [[Americas|America]] and [[Europe]] by trading ships shortly after.{{sfnp|Slocum|2003|p=21}} It became very popular in Europe for a time, and then again during [[World War I]]. It is one of the most widely recognized dissection puzzles in the world and has been used for various purposes including amusement, art, and education.<ref>{{Cite journal |last1=Campillo-Robles |first1=Jose M. |last2=Alonso |first2=Ibon |last3=Gondra |first3=Ane |last4=Gondra |first4=Nerea |date=2022-09-01 |title=Calculation and measurement of center of mass: An all-in-one activity using Tangram puzzles |url=https://aapt.scitation.org/doi/10.1119/5.0061884 |journal=American Journal of Physics |volume=90 |issue=9 |pages=652 |doi=10.1119/5.0061884 |bibcode=2022AmJPh..90..652C |s2cid=251917733 |issn=0002-9505}}</ref>{{sfnp|Slocum|2001|p=9}}<ref>{{cite book |title=Manual of Play |last=Forbrush |first=William Byron |year=1914 |publisher=Jacobs |page=315 |url=https://books.google.com/books?id=FpoWAAAAIAAJ&q=%22The+Anchor+Puzzle%22&pg=PA315 |access-date=2010-10-13}}</ref>

==Etymology==

The origin of the English word 'tangram' is unclear. One conjecture holds that it is a compound of the Greek element  '-gram' derived from ''γράμμα'' ('written character, letter, that which is drawn') with the 'tan-' element being variously conjectured to be Chinese ''t'an'' 'to extend' or Cantonese ''t'ang'' 'Chinese'.<ref>''[[Oxford English Dictionary]]'', 1910, ''[http://www.oed.com/view/Entry/197515 s.v.]''</ref> Alternatively, the word may be derivative of the archaic English 'tangram' meaning "an odd, intricately contrived thing".{{sfnp|Slocum|2003|p=23}}

In either case, the first known use of the word is believed to be found in the 1848 book ''Geometrical Puzzle for the Young'' by mathematician and future Harvard University president [[Thomas Hill (clergyman)|Thomas Hill]].<ref>{{cite book |last1=Hill |first1=Thomas |title=Puzzles to teach geometry : in seventeen cards numbered from the first to the seventeenth inclusive |date=1848 |publisher=Boston : Wm. Crosby & H.P. Nichols |url=https://archive.org/details/puzzlestoteachge00hillrich}} {{open access}}</ref> Hill likely coined the term in the same work, and vigorously promoted the word in numerous articles advocating for the puzzle's use in education, and in 1864 the word received official recognition in the English language when it was included in Noah Webster's [[American Dictionary of the English Language|''American Dictionary'']].{{sfnp|Slocum|2003|p=25}}

==History==

===Origins===
Despite its relatively recent emergence in the West, there is a much older tradition of dissection amusements in China which likely played a role in its inspiration. In particular, the modular banquet tables of the [[Song dynasty]] bear an uncanny resemblance to the playing pieces of the tangram and there were books dedicated to arranging them together to form pleasing patterns.{{sfnp|Slocum|2003|p=16}}

Several Chinese sources broadly report a well-known Song dynasty polymath Huang Bosi 黄伯思 who developed a form of entertainment for his dinner guests based on creative arrangements of six small tables called 宴几 or 燕几(''feast tables'' or ''swallow tables'' respectively). One diagram shows these as oblong rectangles, and other reports suggest a seventh table was added later, perhaps by a later inventor.

According to Western sources, however, the tangram's historical Chinese inventor is unknown except through the pen name Yang-cho-chu-shih (Dim-witted (?) recluse, recluse = 处士). It is believed that the puzzle was originally introduced in a book titled ''Ch'i chi'iao t'u'', which was already reported as lost in 1815 by Shan-chiao in his book ''New Figures of the Tangram''. Nevertheless, it is generally believed that the puzzle was invented about 20 years earlier.{{sfnp|Slocum|2003|pp=16-19}}

The prominent third-century mathematician [[Liu Hui]] made use of construction proofs in his works and some bear a striking resemblance to the subsequently developed banquet tables which in turn seem to anticipate the tangram. While there is no reason to suspect that tangrams were used in the proof of the [[Pythagorean theorem]], as is sometimes reported, it is likely that this style of geometric reasoning went on to exert an influence on Chinese cultural life that lead directly to the puzzle.{{sfnp|Slocum|2003|p=15}}

The early years of attempting to date the Tangram were confused by the popular but fraudulently written history by famed puzzle maker [[Sam Loyd|Samuel Loyd]] in his 1908 ''The Eighth Book Of Tan''. This work contains many whimsical features that aroused both interest and suspicion amongst contemporary scholars who attempted to verify the account. By 1910 it was clear that it was a hoax. A letter dated from this year from the [[Oxford English Dictionary|Oxford Dictionary]] editor [[James Murray (lexicographer)|Sir James Murray]] on behalf of a number of Chinese scholars to the prominent puzzlist [[Henry Dudeney]] reads "The result has been to show that the man Tan, the god Tan, and the Book of Tan are entirely unknown to Chinese literature, history or tradition."{{sfnp|Slocum|2003|p=23}} Along with its many strange details ''The Eighth Book of Tan's'' date of creation for the puzzle of 4000 years in antiquity had to be regarded as entirely baseless and false.

===Reaching the Western world (1815–1820s)===
[[File:Tangram caricature France 1818.jpg|thumb|290px|left|A caricature published in France in 1818, when the tangram craze was at its peak. The caption reads: " 'Take care of yourself, you're not made of steel. The fire has almost gone out and it is winter.' 'It kept me busy all night. Excuse me, I will explain it to you. You play this game, which is said to hail from China. And I tell you that what Paris needs right now is to welcome that which comes from far away.' "]]

The earliest extant tangram was given to the Philadelphia shipping magnate and congressman Francis Waln in 1802 but it was not until over a decade later that Western audiences, at large, would be exposed to the puzzle.{{sfnp|Slocum|2003|p=21}} In 1815, American Captain M. Donnaldson was given a pair of author Sang-Hsia-koi's books on the subject (one problem and one solution book) when his ship, ''Trader'', docked there. They were then brought with the ship to Philadelphia in February 1816.  The first tangram book to be published in America was based on the pair brought by Donnaldson.{{sfnp|Slocum|2003|p=30}}

The puzzle eventually reached England, where it became very fashionable. The craze quickly spread to other European countries. This was mostly due to a pair of British tangram books, ''The Fashionable Chinese Puzzle'', and the accompanying solution book, ''Key''.{{sfnp|Slocum|2003|p=31}} Soon, tangram sets were being exported in great number from China, made of various materials, from glass, to wood, to tortoise shell.{{sfnp|Slocum|2003|p=49}}

Many of these unusual and exquisite tangram sets made their way to [[Denmark]]. Danish interest in tangrams skyrocketed around 1818, when two books on the puzzle were published, to much enthusiasm.{{sfnp|Slocum|2003|pp=99–100}}  The first of these was ''Mandarinen'' (About the Chinese Game).  This was written by a student at [[Copenhagen University]], which was a non-fictional work about the history and popularity of tangrams.  The second, ''Det nye chinesiske Gaadespil'' (The new Chinese Puzzle Game), consisted of 339 puzzles copied from ''The Eighth Book of Tan'', as well as one original.{{sfnp|Slocum|2003|pp=99–100}}

One contributing factor in the popularity of the game in Europe was that although the [[Catholic Church]] forbade many forms of recreation on the sabbath, they made no objection to puzzle games such as the tangram.{{sfnp|Slocum|2003|p=51}}

===Second craze in Germany (1891–1920s)===

Tangrams were first introduced to the German public by industrialist [[Friedrich Adolf Richter]] around 1891.<ref name="arclab">{{cite web|url=http://www.archimedes-lab.org/tangramagicus/pagetang1.html |title=Tangram the incredible timeless 'Chinese' puzzle |website=www.archimedes-lab.org}}</ref> The sets were made out of stone or false [[earthenware]],<ref>{{cite book |title=Treasury Decisions Under customs and other laws, Volume 25 |year=1890–1926 |publisher=United States Department Of The Treasury |page=1421 |url=https://books.google.com/books?id=MeUWAQAAIAAJ&q=%22The+Anchor+Puzzle%22&pg=PA1421 |access-date=September 16, 2010}}</ref> and marketed under the name "The Anchor Puzzle".<ref name="arclab"/>

More internationally, the First World War saw a great resurgence of interest in tangrams, on the homefront and trenches of both sides.  During this time, it occasionally went under the name of "The [[Sphinx]]" an alternative title for the "Anchor Puzzle" sets.<ref>{{cite web |author=Wyatt |date=26 April 2006 |title=Tangram – The Chinese Puzzle |work=h2g2 |publisher=BBC |url=https://www.bbc.co.uk/dna/h2g2/alabaster/A10423595 |access-date=3 October 2010 |url-status=dead |archive-url=https://web.archive.org/web/20111002085746/https://www.bbc.co.uk/dna/h2g2/alabaster/A10423595 |archive-date=2011-10-02}}</ref><ref>{{cite book |title=Kids Around The World Play! |last=Braman |first=Arlette |year=2002 |publisher=John Wiley and Sons |isbn=978-0-471-40984-7 |page=10 |url=https://books.google.com/books?id=fNnoxIfJg5UC&q=Kids+Around+The+World+Play! |access-date=September 5, 2010}}</ref>

==Paradoxes==
[[File:tangram_paradox_explanation.svg|thumb|Explanation of the two-monks paradox:<br />In figure 1, side lengths are labelled assuming the square has unit sides.<br />In figure 2, overlaying the bodies shows that footless body is larger by the foot's area. The change in area is often unnoticed as √2 is close to 1.5.]]
A tangram [[paradox]] is a dissection fallacy: Two figures composed with the same set of pieces, one of which seems to be a proper subset of the other.<ref name="mathematica">[http://mathworld.wolfram.com/TangramParadox.html Tangram Paradox], by Barile, Margherita, From MathWorld – A Wolfram Web Resource, created by Eric W. Weisstein.</ref> One famous paradox is that of the two [[monk]]s, attributed to [[Henry Dudeney]], which consists of two similar shapes, one with and the other missing a foot.<ref name="dudeney">{{cite book |author=Dudeney, H. |title=Amusements in Mathematics |publisher=Dover Publications |location=New York |year=1958}}</ref> In reality, the area of the foot is compensated for in the second figure by a subtly larger body.

<!--gallery-->
The two-monks paradox&nbsp;– two similar shapes but one missing a foot:
[[File:Two monks tangram paradox.svg|thumb|left]]
{{Clear}}
The Magic Dice Cup tangram paradox&nbsp;– from [[Sam Loyd]]'s book ''The 8th Book of Tan'' (1903).<ref name="eighth book 1"/> Each of these cups was composed using the same seven geometric shapes. But the first cup is whole, and the others contain vacancies of different sizes. (Notice that the one on the left is slightly shorter than the other two. The one in the middle is ever-so-slightly wider than the one on the right, and the one on the left is narrower still.)<ref>{{cite web|title=The Magic Dice Cup|date=2 April 2011 |url=https://www.futilitycloset.com/2011/04/02/the-magic-dice-cup/}}</ref>
[[File:The Magic Dice Cup tangram paradox.svg|thumb|left]]
{{Clear}}
Clipped square tangram paradox&nbsp;– from Loyd's book ''The Eighth Book of Tan'' (1903):<ref name="eighth book 1">{{cite book |url=http://www.tangram-channel.com/the-eighth-book-of-tan-by-sam-loyd-page-1/ |title=The 8th Book of Tan by Sam Loyd |year=1903 |via=Tangram Channel}}</ref>

{{blockquote|The seventh and eighth figures represent the mysterious square, built with seven pieces: then with a corner clipped off, and still the same seven pieces employed.<ref name="loyd">{{cite book |author=Loyd, Sam |title=The eighth book of Tan – 700 Tangrams by Sam Loyd with an introduction and solutions by Peter Van Note |publisher=Dover Publications |location=New York |year= 1968|page=25 }}</ref>}}
[[File:squares.GIF|thumb|left]]
<!--/gallery-->
{{Clear}}

== Number of configurations ==
[[Image:convex_tangram_shapes.svg|thumb|The 13 convex shapes matched with the tangram set]]
Over 6500 different tangram problems have been created from 19th-century texts alone, and the current number is ever-growing.{{sfn|Slocum|2001|p=37}} Fu Traing Wang and Chuan-Chih Hsiung proved in 1942 that there are only thirteen [[convex polygon|convex]] tangram configurations (segments drawn between any two points on the configuration are always completely contained inside the configuration, i.e., configurations with no recesses in the outline).<ref>
{{cite journal |author1=Fu Traing Wang |author2=Chuan-Chih Hsiung |date=November 1942 |title=A Theorem on the Tangram |journal=[[The American Mathematical Monthly]] |volume=49 |issue=9 |pages=596–599 |jstor=2303340|doi=10.2307/2303340}}</ref><ref name="isbn0-486-21483-4">{{cite book |author=Read, Ronald C. |title=Tangrams : 330 Puzzles |publisher=Dover Publications |location=New York |page=53 |isbn=0-486-21483-4 |year=1965}}</ref>

==Pieces==
Choosing a unit of measurement so that the seven pieces can be assembled to form a square of side one unit and having area one square unit, the seven pieces are:<ref>{{Cite web|url=https://boyslife.org/hobbies-projects/projects/145561/make-a-classic-tangram-puzzle/|title=How to Make a Classic Tangram Puzzle|last=Brooks|first=David J.|date=2018-12-01|website=Boys' Life magazine|language=en|access-date=2020-03-10}}</ref>

* 2 large [[Right angle triangle|right triangles]] (hypotenuse 1, sides {{sfrac|{{sqrt|2}}|2}}, area {{sfrac|1|4}})
* 1 medium right triangle (hypotenuse {{sfrac|{{sqrt|2}}|2}}, sides {{sfrac|1|2}}, area {{sfrac|1|8}})
* 2 small right triangles (hypotenuse {{sfrac|1|2}}, sides {{sfrac|{{sqrt|2}}|4}}, area {{sfrac|1|16}})
* 1 [[Square (geometry)|square]] (sides {{sfrac|{{sqrt|2}}|4}}, area {{sfrac|1|8}})
* 1 [[parallelogram]] (sides of {{sfrac|1|2}} and {{sfrac|{{sqrt|2}}|4}}, height of {{sfrac|1|4}}, area {{sfrac|1|8}})

Of these seven pieces, the parallelogram is unique in that it has no [[reflection symmetry]] but only [[rotational symmetry]], and so its [[mirror image]] can be obtained only by flipping it over. Thus, it is the only piece that may need to be flipped when forming certain shapes.

==See also==
{{Portal|Mathematics}}
*[[Tangram (video game)|''Tangram'' (video game)]]
*[[Egg of Columbus (tangram puzzle)]]
*[[Mathematical puzzle]]
*[[Ostomachion]]
*[[Tiling puzzle]]
*[[Attribute blocks]]

== References ==
{{Reflist}}
;Sources
* {{cite book |last=Slocum |first=Jerry |title=The Tao of Tangram |year=2001 |publisher=Barnes & Noble |isbn=978-1-4351-0156-2}}
* {{cite book |last=Slocum |first=Jerry |title=The Tangram Book |year=2003 |publisher=Sterling |isbn=978-1-4027-0413-0}}

==Further reading==
* Anno, Mitsumasa. ''Anno's Math Games'' (three volumes). New York: Philomel Books, 1987. {{isbn|0-399-21151-9}} (v. 1), {{isbn|0-698-11672-0}} (v. 2), {{isbn|0-399-22274-X}} (v. 3).
* Botermans, Jack, et al. ''The World of Games: Their Origins and History, How to Play Them, and How to Make Them'' (translation of ''Wereld vol spelletjes''). New York: Facts on File, 1989. {{isbn|0-8160-2184-8}}.
* Dudeney, H. E.  ''Amusements in Mathematics''. New York: Dover Publications, 1958.
* [[Martin Gardner|Gardner, Martin]]. "Mathematical Games—on the Fanciful History and the Creative Challenges of the Puzzle Game of Tangrams", ''Scientific American'' Aug. 1974, p.&nbsp;98–103.
* Gardner, Martin. "More on Tangrams", ''Scientific American'' Sep. 1974, p.&nbsp;187–191.
* Gardner, Martin. ''The 2nd Scientific American Book of Mathematical Puzzles and Diversions''. New York: Simon & Schuster, 1961. {{isbn|0-671-24559-7}}.
* Loyd, Sam. ''Sam Loyd's Book of Tangram Puzzles (The 8th Book of Tan Part I)''. Mineola, New York: Dover Publications, 1968.
* Slocum, Jerry, et al. ''Puzzles of Old and New: How to Make and Solve Them''. De Meern, Netherlands: Plenary Publications International (Europe); Amsterdam, Netherlands: ADM International; Seattle: Distributed by University of Washington Press, 1986. {{isbn|0-295-96350-6}}.

==External links==
{{commons category|Tangrams}}
* [http://www.archimedes-lab.org/tangramagicus/pagetang1.html Past & Future: The Roots of Tangram and Its Developments]
* [http://www.archimedes-lab.org/workshoptangram.html Turning Your Set of Tangram Into A Magic Math Puzzle] by puzzle designer [[Gianni A. Sarcone|G. Sarcone]]<!--

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