OC Luoshu_ver.qxp:OC Luoshu rev
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11:11 AM
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The Magic Square of Order Three in Other Cultures
rotated the set of corner numbers 90 degrees clockwise to arrive at the luoshu configuration (figure 6.11c):
7
4
1
2
9
6
4
9
2
8
5
2
3
5
7
3
5
7
9
6
3
4
1
8
8
1
6
(a)
(b)
(c)
FIGURE 6.11
A later technique devised by Katahiro Takebe (1664–1739) uses an interesting transformation approach to devise the magic square of order three.28 See figure 6.12. Takebe visualizes the natural square of order three as comprised of two cruciform subsets of numbers: one formed by the two diagonals, the other by the vertical and horizontal elements— the pivotal element for both remains the number 5. Takebe first rotates the diagonal set 45 degrees counterclockwise (figure 6.12b), he then rotates the remaining set of elements 180 degrees to arrive at the luoshu configuration (figure 6.12c). 7
4
1
4
1
2
4
9
2
8
5
2
7
5
3
3
5
7
9
6
3
8
9
6
8
1
6
(a)
(b)
(c)
FIGURE 6.12
Both Ando’s and Takebe’s methods of constructing a magic square of order three stress the significance of placing even numbers at the corners of the square. In fact, if this is not done, the construction of the third-order magic square is impossible. Further, their techniques employ discrete mathematical steps focused on parts of the number configuration. Such discrete thinking seems to deviate from the continuous methods employed by other early manipulators—specifically, Indian and