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{{Short description|Ancient Chinese diagram}}
{{Infobox Chinese
|pic=Lo Shu 3x3 magic square.svg
|piccap=The traditional Luoshu diagram
|pic2 = Magic Square Lo Shu.svg
|piccap2 = The equivalent [[magic square]]
|title=Luoshu Square
|name1=Luoshu
|t={{linktext|洛書}}<br>{{linktext|雒|書}}
|s={{linktext|洛书}}
|p=Luòshū
|lit=[[Luo River (Henan)|Luo]] Book
|w=Lo Shu
|hangul=낙서
|rr=Nakseo
|mr=Naksŏ
|kanji=洛書
|hiragana=らくしょ
|revhep=Rakusho
|vie=Lạc Thư
|altname=Nine Halls Diagram
|t2={{linktext|九|宮|圖}}
|s2={{linktext|九|宫|图}}
|p2=Jiǔgōngtú
|w2=Chiu-kung-t'u
}}

The '''Luoshu''' ([[pinyin]]), '''Lo Shu''' ([[Wade-Giles]]), or '''Nine Halls Diagram'''<!--Chinese in infobox per WP:MOS-ZH--> is an [[Ancient China|ancient Chinese]] diagram and named for the [[Luo River (Henan)|Luo River]] near [[Luoyang]], [[Henan]]. The Luoshu appears in [[Chinese mythology|myths]] concerning the [[Chinese inventions|invention]] of [[Chinese writing|writing]] by [[Cangjie]] and other [[culture hero]]es. It is a unique normal [[magic square]] of order three. It is usually paired with the [[Yellow River Map|River Map or Hetu]]{{mdash}}named in reference to the [[Yellow River]]{{mdash}}and used with the River Map in various contexts involving [[Chinese geomancy]], [[Chinese numerology|numerology]], [[Chinese philosophy|philosophy]], and early [[natural science]].

==Traditions==
The Lo Shu is part of the legacy of ancient Chinese mathematical and divination (cf. the [[I Ching]] {{lang|zh|易經}}) traditions, and is an important emblem in ''[[Feng Shui]]'' ({{lang|zh|風水}})—the art of [[geomancy]] concerned with the placement of objects in relation to the flow of [[qi]] ({{lang|zh|氣}}), or "natural energy".{{Citation needed|date=July 2022}}

== History ==
A Chinese legend concerning the pre-historic [[Yu the Great|Emperor Yu]] ({{lang|zh|夏禹}}) tells of the Lo Shu, often in connection with the ''Yellow River Map'' (Hetu) and the [[Bagua (concept)|eight trigrams]]. In ancient China there is a legend of a huge [[Flood myth|deluge]]: the people offered sacrifices to the god of one of the flooding rivers, the Luo river ({{lang|zh|洛河}}), to try to calm his anger.  A magical [[turtle]] emerged from the water with the curiously unnatural Lo Shu pattern on its shell: circular dots representing the integers one through nine are arranged in a three-by-three grid.<ref>{{cite journal
 | last = Cammann | first = Schuyler
 | date = April–June 1960
 | journal = Journal of the American Oriental Society
 | jstor = 595587
 | pages = 116–124
 | title = The evolution of magic squares in China
 | volume = 80| issue = 2
 | doi = 10.2307/595587
 }}</ref>

Early records dated to 650 BCE are ambiguous, referring to a "river map", but clearly start to refer to a [[magic square]] by 80 CE, and explicitly give an example of one since 570 CE.<ref name="cammann3">{{harvnb|Cammann|1961}}</ref><ref name="Swaney">{{cite web | last=Swaney |first=Mark |title=Mark Swaney on the History of Magic Squares |url=http://www.netmastersinc.com/secrets/magic_squares.htm
| archive-url=https://web.archive.org/web/20040807015853/http://www.netmastersinc.com/secrets/magic_squares.htm |archive-date=2004-08-07}}</ref>  Recent publications have provided support that the Lo Shu Magic Square was an important model for time and space.{{Cn|date=July 2024}}  It served as a basis for city planning, and tomb and temple design. The magic square was incidentally used to designate spaces of political and religious importance.<ref>{{harvnb|Schinz|1996}}{{pages needed|date=July 2019}}</ref> 

[[File:Carus-p48-Mystic-table.jpg|thumb|upright=1|The Lo Shu square on the back of a small turtle (in the center), surrounded by the signs of the [[Chinese zodiac]] and the [[Bagua|Eight Trigrams]], are all carried by a large turtle (that presumably stands for the [[Longma|Dragon horse]] that had earlier revealed the trigrams to [[Fu Xi]]).{{Citation needed|date=July 2022}}  This example drawn by an anonymous Tibetan artist.]]

==The layout==
The odd and even numbers alternate in the periphery of the Lo Shu pattern; the four even numbers are at the four corners, and the five odd numbers (which outnumber the even numbers by one) form a cross in the center of the square. The sums in each of the three rows, in each of the three columns, and in both diagonals, are all 15.<ref group=notes name="b">15 is the number of days in each of the 24 cycles of the [[Chinese calendar|Chinese solar year]].</ref>  Since "5" is in the center cell, the sum of any two other cells that are directly through the five from each other must be 10; e.g., opposite squares and corners add up to 10, the number of the ''Yellow River Map''.{{Citation needed|date=July 2022}}

==In numerology==
The Lo Shu is sometimes connected numerologically with the "[[Bagua]]" ({{lang|zh|八卦}} or eight [[trigrams]]"), that can be arranged in the eight outer cells, reminiscent of circular trigram diagrams. Because north is placed at the bottom of maps in China, the 3x3 [[magic square]] having number 1 at the bottom and 9 at the top is used in preference to the other rotations/reflections. As seen in the [[Bagua#King_Wen's_"Later_Heaven"|"Later Heaven"]] arrangement, 1 and 9 correspond with ☵ Kǎn {{lang|zh|坎}} "Water {{lang|zh|水}}" and ☲ Lí {{lang|zh|離}} "Fire {{Lang|zh|火}}" respectively. In the [[Bagua#Fu_Xi's_"Earlier_Heaven"|"Early Heaven"]] arrangement, they would correspond with ☷ Kūn {{lang|zh|坤}} "Earth {{lang|zh|地}}" and ☰ Qián {{lang|zh|乾}} "Heaven {{lang|zh|天}}" respectively. Like the ''Yellow River Map'', the Lo Shu square, in conjunction with the Bagua, is sometimes used as a [[mandala|mandalic]] representation important in Feng Shui geomancy.{{Citation needed|date=July 2022}}

==See also==
* [[Associative magic square]]
* [[Camunian rose]]
* [[Sator Square]]
* [[Tetractys]]
* [[Yellow River Map]]

==References==
===Notes===
{{Reflist|group=notes}} 

===Citations===
{{Reflist}}

==Bibliography==
*{{Cite journal | authorlink=Schuyler Cammann |last=Cammann |first=Schuyler  |title=The Magic Square of Three in Old Chinese Philosophy and Religion |journal=History of Religions |volume= 1 |number=1 |date=Summer 1961 |pages=37–80 |doi=10.1086/462439 |s2cid=162306354 }}
*{{Cite book | first=Alfred |last=Schinz |title=The Magic Square: Cities in Ancient China |publisher=Axel Menges |date=1996 |isbn=9783930698028 }}
*{{Cite book | authorlink=Yoshio Mikami |last=Yoshio |first=Mikami |title=The Development of Mathematics in China and Japan |date=1913 |lccn=61-13497 }}

==Further reading==
*{{Cite book | first=Frank J. |last=Swetz |title=The Legacy of the Luoshu |publisher=A. K. Peters / CRC Press |edition=2nd Rev |date=2008 |isbn=978-1-56881-427-8 }}
*{{Cite book | first=Lars |last=Berglund |title=The Secret of the Luo Shu: Numerology in Chinese Art and Architecture |publisher=Tryckbiten |date=1990 |isbn=9789162800680 }}

==External links==
*{{commons category inline|Luoshu}}
*[http://www.taliscope.com/Collection_en.html Lo Shu Square: Definition, Nature and History]

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[[Category:Chinese folk religion]]
[[Category:Geomancy]]
[[Category:Magic squares]]
[[Category:Numerology]]