Magic circle (mathematics)

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Magic circles were invented by the
Song dynasty The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the rest ...
(960–1279)
Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of va ...
mathematician Yang Hui (c. 1238–1298). It is the arrangement of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s on
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
s where the sum of the numbers on each circle and the sum of numbers on
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid ...
s are identical. One of his magic circles was constructed from the natural numbers from 1 to 33 arranged on four
concentric In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center poi ...
circles, with 9 at the center.

# Yang Hui magic circles

Yang Hui's magic circle series was published in his ''Xugu Zhaiqi Suanfa''《續古摘奇算法》(Sequel to Excerpts of Mathematical Wonders) of 1275. His magic circle series includes: magic 5 circles in square, 6 circles in ring, magic eight circle in square magic concentric circles, magic 9 circles in square.

## Yang Hui magic concentric circle

Yang Hui's magic concentric circle has the following properties *The sum of the numbers on four diameters = 147, ** 28 + 5 + 11 + 25 + 9 + 7 + 19 + 31 + 12 = 147 *The sum of 8 numbers plus 9 at the center = 147; **28 + 27 + 20 + 33 + 12 + 4 + 6 + 8 + 9 = 147 *The sum of eight
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also th ...
without 9 = magic number 69: such as 27 + 15 + 3 + 24 = 69 *The sum of all numbers on each circle (not including 9) = 2 × 69 *There exist 8
semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line ...
s, where the sum of numbers = magic number 69; there are 16 line segments (semicircles and radii) with magic number 69, more than a 6 order magic square with only 12 magic numbers.

## Yang Hui magic eight circles in a square

64 numbers arrange in circles of eight numbers, total sum 2080, horizontal / vertical sum = 260. From NW corner clockwise direction, the sum of 8-number circles are: : 40 + 24 + 9 + 56 + 41 + 25 + 8 + 57 = 260 : 14 + 51 + 46 + 30 + 3 + 62 + 35 + 19 = 260 : 45 + 29 + 4 + 61 + 36 + 20 + 13 + 52 = 260 : 37 + 21 + 12 + 53 + 44 + 28 + 5 + 60 = 260 : 47 + 31 + 2 + 63 + 34 + 18 + 15 + 50 = 260 : 7 + 58 + 39 + 23 + 10 + 55 + 42 + 26 = 260 : 38 + 22 + 11 + 54 + 43 + 27 + 6 + 59 = 260 : 48 + 32 + 1 + 64 + 33 + 17 + 16 + 49 = 260 Also the sum of the eight numbers along the WE/NS axis : 14 + 51 + 62 + 3 + 7 + 58 + 55 + 10 = 260 : 49 + 16 + 1 + 64 + 60 + 5 + 12 + 53 = 260 Furthermore, the sum of the 16 numbers along the two diagonals equals to 2 times 260: : 40 + 57 + 41 + 56 + 50 + 47 + 34 + 63 + 29 + 4 + 13 + 20 + 22 + 11 + 6 + 27 = 2 × 260 = 520

## Yang Hui magic nine circles in a square

72 numbers from 1 to 72, arranged in nine circles of eight numbers in a square; with neighbouring numbers forming four additional eight number circles: thus making a total of 13 eight number circles: Extra circle x1 contains numbers from circles NW, N, C, and W; x2 contains numbers from N, NE, E, and C; x3 contains numbers from W, C, S, and SW; x4 contains numbers from C, E, SE, and S. * Total sum of 72 numbers = 2628; * sum of numbers in any eight number circle = 292; * sums of three circles along horizontal lines = 876; * sum of three circles along vertical lines = 876; * sum of three circles along the diagonals = 876.

# Ding Yidong magic circles

Ding Yidong was a mathematician contemporary with Yang Hui. In his magic circle with 6 rings, the unit numbers of the 5 outer rings, combined with the unit number of the center ring, form the following
magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
: : Method of construction: :Let radial group 1 =1,11,21,31,41 :Let radial group 2=2,12,22,32,42 :Let radial group 3=3,13,23,33,43 :Let radial group 4=4,14,24,34,44 :Let radial group 6=6,16,26,36,46 :Let radial group 7=7,17,27,37,47 :Let radial group 8=8,18,28,38,48 :Let radial group 9=9,19,29,39,49 :Let center group =5,15,25,35,45 Arrange group 1,2,3,4,6,7,9 radially such that * each number occupies one position on circle * alternate the direction such that one radial has smallest number at the outside, the adjacent radial has largest number outside. * Each group occupies the radial position corresponding to the number on the Luoshu magic square, i.e., group 1 at 1 position, group 2 at 2 position etc. * Finally arrange center group at the center circle, such that :number 5 on group 1 radial :number 10 on group 2 radial :number 15 on group 3 radial :... :number 45 on group 9 radial

# Cheng Dawei magic circles

Cheng Dawei, a mathematician in the Ming dynasty, in his book Suanfa Tongzong listed several magic circles File:Suanfatongzong-792-792.jpg File:Suanfatongzong-793-793.jpg File:Suanfatongzong-795-795.jpg

# Extension to higher dimensions

In 1917, W. S. Andrews published an arrangement of numbers 1, 2, 3, and 62 in eleven circles of twelve numbers each on a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
representing the parallels and meridians of the Earth, such that each circle has 12 numbers totalling 378.W. S. Andrews, MAGIC SQUARES AND CUBES, Second Edition, Revised and Enlarged, Open Court Basic Readers (1917), page 198, fig.337

# Relationship with magic squares

A magic circle can be derived from one or more magic squares by putting a number at each intersection of a circle and a spoke. Additional spokes can be added by replicating the columns of the magic square. In the example in the figure, the following 4 × 4
most-perfect magic square A most-perfect magic square of order ''n'' is a magic square containing the numbers 1 to ''n''2 with two additional properties: # Each 2 × 2 subsquare sums to 2''s'', where ''s'' = ''n''2 + 1. # All pairs of ...
was copied into the upper part of the magic circle. Each number, with 16 added, was placed at the intersection symmetric about the centre of the circles. This results in a magic circle containing numbers 1 to 32, with each circle and diameter totalling 132.

# References

* Lam Lay Yong: A Critical Study of Hang Hui Suan Fa 《杨辉算法》 Singapore University Press 1977 * Wu Wenjun (editor in chief), Grand Series of History of Chinese Mathematics, Vol 6, Part 6 Yang Hui, section 2 Magic circle (吴文俊 主编 沈康身执笔 《中国数学史大系》 第六卷 第六篇 《杨辉》 第二节 《幻圆》) /O {{Magic polygons Chinese mathematics Song dynasty Magic shapes