Lo Shu Square
Encyclopedia
Lo Shu Square or the Nine Halls Diagram , is the unique normal magic square
of order three. Lo Shu is part of the legacy of the most ancient Chinese mathematical and divinatory (Yi Jing
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) 'natural energy'.
Chinese legends concerning the pre-historic Emperor Yu
(夏禹) tell of the Lo Shu, often in connection with the Ho Tu (河圖) figure and 8 trigrams
. In ancient China there was a huge deluge: the people offered sacrifices to the god of one of the flooding rivers, the Lo river (洛水), to try to calm his anger. A magical turtle
emerged from the water with the curious and decidedly unnatural (for a turtle shell) Lo Shu pattern on its shell: circular dots giving unitary (base 1) representations (figurate numbers) of the integers one through nine are arranged in a three-by-three grid.
The odd and even numbers alternate in the periphery of the Lo Shu pattern; the 4 even numbers are at the four corners, and the 5 odd numbers (outnumbering the even numbers by one) form a cross in the center of the square. The sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15 (fifteen is the number of days in each of the 24 cycles of the Chinese solar year
). Since 5 is in the center cell, the sum of any two other cells that are directly through the 5 from each other is 10 (e.g., opposite corners add up to 10, the number of the Ho Tu (河圖)).
The Lo Shu is sometimes connected numerologically with the Ba Gua 八卦 "8 trigrams", which can be arranged in the 8 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 水 "Water" and ☲ Lí 火 "Fire" respectively. In the "Early Heaven" arrangement, they would correspond with ☷ Kūn 地 "Earth" and ☰ Qián 天 "Heaven" respectively. Like the Ho Tu (河圖), the Lo Shu square, in conjunction with the 8 trigrams
, is sometimes used as a mandalic
representation important in Feng Shui
(風水) geomancy
.
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
of order three. Lo Shu is part of the legacy of the most ancient Chinese mathematical and divinatory (Yi Jing
I Ching
The I Ching or "Yì Jīng" , also known as the Classic of Changes, Book of Changes and Zhouyi, is one of the oldest of the Chinese classic texts...
易經) traditions, and is an important emblem in Feng Shui
Feng shui
Feng shui ' is a Chinese system of geomancy believed to use the laws of both Heaven and Earth to help one improve life by receiving positive qi. The original designation for the discipline is Kan Yu ....
(風水), the art of geomancy concerned with the placement of objects in relation to the flow of qi
Qi
In traditional Chinese culture, qì is an active principle forming part of any living thing. Qi is frequently translated as life energy, lifeforce, or energy flow. Qi is the central underlying principle in traditional Chinese medicine and martial arts...
(氣) 'natural energy'.
Chinese legends concerning the pre-historic Emperor Yu
Yu the Great
Yu the Great , was a legendary ruler of Ancient China famed for his introduction of flood control, inaugurating dynastic rule in China by founding the Xia Dynasty, and for his upright moral character....
(夏禹) tell of the Lo Shu, often in connection with the Ho Tu (河圖) figure and 8 trigrams
Bagua (concept)
The bagua are eight diagrams used in Taoist cosmology to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Each consists of three lines, each line either "broken" or "unbroken," representing yin or yang, respectively...
. In ancient China there was a huge deluge: the people offered sacrifices to the god of one of the flooding rivers, the Lo river (洛水), to try to calm his anger. A magical turtle
Turtle
Turtles are reptiles of the order Testudines , characterised by a special bony or cartilaginous shell developed from their ribs that acts as a shield...
emerged from the water with the curious and decidedly unnatural (for a turtle shell) Lo Shu pattern on its shell: circular dots giving unitary (base 1) representations (figurate numbers) of the integers one through nine are arranged in a three-by-three grid.
The odd and even numbers alternate in the periphery of the Lo Shu pattern; the 4 even numbers are at the four corners, and the 5 odd numbers (outnumbering the even numbers by one) form a cross in the center of the square. The sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15 (fifteen is the number of days in each of the 24 cycles of the Chinese solar year
Chinese calendar
The Chinese calendar is a lunisolar calendar, incorporating elements of a lunar calendar with those of a solar calendar. It is not exclusive to China, but followed by many other Asian cultures as well...
). Since 5 is in the center cell, the sum of any two other cells that are directly through the 5 from each other is 10 (e.g., opposite corners add up to 10, the number of the Ho Tu (河圖)).
The Lo Shu is sometimes connected numerologically with the Ba Gua 八卦 "8 trigrams", which can be arranged in the 8 outer cells, reminiscent of circular trigram diagrams. Because north is placed at the bottom of maps in China, the 3x3 magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
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 水 "Water" and ☲ Lí 火 "Fire" respectively. In the "Early Heaven" arrangement, they would correspond with ☷ Kūn 地 "Earth" and ☰ Qián 天 "Heaven" respectively. Like the Ho Tu (河圖), the Lo Shu square, in conjunction with the 8 trigrams
Bagua (concept)
The bagua are eight diagrams used in Taoist cosmology to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Each consists of three lines, each line either "broken" or "unbroken," representing yin or yang, respectively...
, is sometimes used as a mandalic
Mandala
Maṇḍala is a Sanskrit word that means "circle". In the Buddhist and Hindu religious traditions their sacred art often takes a mandala form. The basic form of most Hindu and Buddhist mandalas is a square with four gates containing a circle with a center point...
representation important in Feng Shui
Feng shui
Feng shui ' is a Chinese system of geomancy believed to use the laws of both Heaven and Earth to help one improve life by receiving positive qi. The original designation for the discipline is Kan Yu ....
(風水) geomancy
Geomancy
Geomancy is a method of divination that interprets markings on the ground or the patterns formed by tossed handfuls of soil, rocks, or sand...
.
Further Readings
- Frank J. Swetz, The Legacy of the Luoshu, A K Peters/CRC Press; 2Rev Ed edition (16 Jun 2008), ISBN 978-1568814278.