Yang Hui was one of the mathematicians that stood out during this time. To the right is the “Yang Hui’s Magic Concentric Circle”, another very interesting integer array. Counting the four circles and the eight semi-circles, the sum of the numbers in each semi- circle is exactly 69. There are also eight radii which, if you add the numbers up, also give 69. The second sum
is 147 – which is the sum of the numbers on four diameters and the sum of the numbers in each circle plus 9 in the center.
Talking about magic circles, the Ding Yi Dong ( 丁易 东 ) magic circle is also worth mentioning. In this model, the sum of each circle is 200. Also, taking a number on one circle and adding it to the number which is exactly on the opposite side of that circle, you always get 50 (for example, 14+36=50 and 45+5=50). Knowing some of these properties, we can deduce other properties like the sum of each diameter would always be 325 (25+6x50).
Magic squares, in turn, are constructed with similar properties. Again, the square is always related to a ‘magic’ constant which is the sum of the integers in each row, column and diagonal. In this 3 x 3 model, it is 15. This model is called the Lo Shu Square, which is the earliest magic square discovered in China. The more advanced magic squares were officially recorded in Greece in the 14 th century. These models were combined with a much
deeper meaning- linked to planets and gods. The magic squares were said to be written into medieval magic books. Ranging from the 3 x 3 model to the 9 x 9 one, they were given names of: Saturn, Jupiter, Mars, Sol (Sun), Venus, Mercury, Luna (Moon). Astronomy is very different though, but still very well connected with maths. Later people constructed the magic cube that has similar properties. Well, people never stop exploring, do they? Soon the term ‘magic hypercubes’ were introduced by mathematician John R. Hendricks.
The cubes follow the exact same rule, that is, the sum on any axis is always the same. Let’s have a look at the previously innocent-looking 3x3 model in 4d. People have been creating crazy things all the time, including the ones they can’t really see, or understand very well.
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