Magic circles were invented by the

Song dynasty The Song dynasty ( ) was an Dynasties of China, imperial dynasty of China that ruled from 960 to 1279. The dynasty was founded by Emperor Taizu of Song, who usurped the throne of the Later Zhou dynasty and went on to conquer the rest of the Fiv ...

(960–1279) Chinese mathematician

Yang Hui Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial the ...

(c. 1238–1298). It is the arrangement of

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s on

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

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 centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

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'' when they share the same center. Any pair of (possibly unalike) objects with well-defined centers can be concentric, including circles, spheres, regular polygons, regular polyh ...

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 or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is 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. It is a circular arc that measures 180° (equivalently, radians, or a half-turn). It only has one line of symmetr ...

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 (1–64) are arranged in eight circles, each with eight numbers; each circle sums to 260. The total sum of all numbers is 2080 (=8×260). The circles are arranged in a 3×3 square grid with the center area open in a way that also makes the horizontal / vertical sum along the central columns and rows is 260, and the total sum of the numbers along both diagonals is 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 mathematics, especially History of mathematics, historical and 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 diago ...

: : 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 ''Suanfa tongzong'' ( zh, 算法統宗) is a mathematical text written by sixteenth century Chinese mathematician Cheng Dawei (1533–1606) and published in the year 1592. The book contains 595 problems divided into 17 chapters. The book is essent ...

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 (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

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 . # All pairs of integers distant ''n''/2 along a (major) diagonal su ...

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 Wu Wenjun ( zh, s=吴文俊; 12 May 1919 – 7 May 2017), also commonly known as Wu Wen-tsün, was a Chinese mathematician, historian, and writer. He was an academician at the Chinese Academy of Sciences (CAS), best known for Wu class, Wu fo ...

(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 figures