Magic circle (mathematics)

Magic circles were invented by the Song Dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical. One of his magic circles was constructed from 33 natural numbers from 1 to 33 arranged on four circles , with 9 at the center.
Yang Hui magic circles
Yang Hui's magic 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 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 semicircles, where the sum of numbers = magic number 69; there are 16 line segments(semi circles and radii) with magic number 69, more than a 6 order magic square with only 12 magic numbers.
Yang Hui's magic square was published in his Xugu Zhaiqi Suanfa《續古摘奇算法》 (Sequel to Excerpts of Mathematical Wonders) of 1275.
Ding Yidong magic circles
Ding Yidong was a mathematician contemporary with Yang Hui, in his 6th order magic circle with 6 rings, the 5 out rings have connection with a 3rd order magic square: the unit number of the 8 numbers on any ring form a 3rd order magic square.
4 9 2 3 5 7 8 1 6 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
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》) ISBN 7303049266/O
Categories:  The sum of the numbers on four diameters = 147，
Wikimedia Foundation. 2010.
Look at other dictionaries:
Magic circle (disambiguation) — A magic circle is a ritually defined space in a number of magical traditions. Magic circle may also refer to: In digital media: Magic Circle (synthetic worlds), membrane enclosing synthetic worlds In mathematics: Magic circle (mathematics),… … Wikipedia
Magic (illusion) — For other varieties of magic, see Magic (disambiguation). Illusionist redirects here. For the artistic tradition, see Illusionism (art). For other uses, see The Illusionist (disambiguation). Magic The Conjurer (painting), 1475 1480, by Hieronymus … Wikipedia
Magic (paranormal) — For related ideas, see Magic (disambiguation). Magia redirects here. For other uses, see Magia (disambiguation). Magical redirects here. For the song, see Magical (song). Circe Offering the Cup to Ulysses by John William Waterhouse Magic … Wikipedia
Circle (disambiguation) — In mathematics, a circle generally is the set of all points in a plane at a fixed distance from a fixed point. Sometimes, it is used to denote the disk bounded by this set. Circle may also refer to: Social circle, a group of socially… … Wikipedia
mathematics, South Asian — Introduction the discipline of mathematics as it developed in the Indian (India) subcontinent. The mathematics of classical Indian civilization is an intriguing blend of the familiar and the strange. For the modern individual, Indian… … Universalium
List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… … Wikipedia
List of circle topics — This list of circle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or concretely in physical space. It does not include metaphors like inner circle or circular reasoning in… … Wikipedia
Chinese mathematics — Mathematics in China emerged independently by the 11th century BC.[1] The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry. Many[who?]… … Wikipedia
Areas of mathematics — Mathematics has become a vastly diverse subject over history, and there is a corresponding need to categorize the different areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities,… … Wikipedia
List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia