Luoshu Square

The Lo Shu is part of the legacy of ancient Chinese mathematical and divinity (cf. the I Ching 易經) traditions, and is an important emblem in Feng Shui (風水)—the art of geomancy concerned with the placement of objects in relation to the flow of qi (氣), or "natural energy".

A Chinese legend concerning the pre-historic Emperor Yu (夏禹) tells of the Lo Shu, often in connection with the Yellow River Map (Hetu) and the eight trigrams. In ancient China there is a legend of a huge deluge: the people offered sacrifices to the god of one of the flooding rivers, the Luo river (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.

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.[1][2] Recent publications have provided support that the Lo Shu Magic Square was an important model for time and space. 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.[3]

The Lo Shu square on the back of a small turtle (in the center), surrounded by the signs of the Chinese zodiac and the Eight Trigrams, are all carried by a large turtle (that presumably stands for the Dragon horse that had earlier revealed the trigrams to Fu Xi). This example drawn by an anonymous Tibetan artist.

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.[notes 1] 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.

The Lo Shu is sometimes connected numerologically with the "Bagua" (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 "Later Heaven" arrangement, 1 and 9 correspond with ☵ Kǎn 坎 "Water 水" and ☲ Lí 離 "Fire 火" respectively. In the "Early Heaven" arrangement, they would correspond with ☷ Kūn 坤 "Earth 地" and ☰ Qián 乾 "Heaven 天" respectively. Like the Yellow River Map, the Lo Shu square, in conjunction with the Bagua, is sometimes used as a mandalic representation important in Feng Shui geomancy.

• Associative magic square
• Sator Square
• Tetractys
• Camunian rose

• Cammann, Schuyler (Summer 1961). "The Magic Square of Three in Old Chinese Philosophy and Religion". History of Religions. 1 (1): 37–80. doi:10.1086/462439. S2CID 162306354.
• Schinz, Alfred (1996). The Magic Square: Cities in Ancient China. Axel Menges. ISBN 9783930698028.
• Yoshio, Mikami (1913). The Development of Mathematics in China and Japan. LCCN 61-13497.

• Swetz, Frank J. (2008). The Legacy of the Luoshu (2nd Rev ed.). A. K. Peters / CRC Press. ISBN 978-1-56881-427-8.
• Berglund, Lars (1990). The Secret of the Luo Shu: Numerology in Chinese Art and Architecture. Tryckbiten. ISBN 9789162800680.

• Media related to Luoshu at Wikimedia Commons
• Lo Shu Square: Definition, Nature and History