A 3 by 3 magic square. Row entries: 2,7,6; 9,5,1; 4,3,8
Lo Shu magic square

Magic squares

Magic squares have fascinated mankind for thousands of years. No one really knows the origin of magic squares, but many historians think that they were invented in ancient China.

What is a magic square?

A magic square is an n x n square with a whole number written inside each cell, so that the sum of the numbers in every row, in every column and in each of the main diagonals is equal. This number is called the magic number. The main diagonals are those that stretch from corner to corner. The image above is an example of a 3 by 3 magic square. The sum of each row, each column and each of the two main diagonals is 15, so 15 is the magic number of this magic square. This is the smallest possible magic square (why?) and his been know for thousands of years. It even has a special name: the Lo Shu magic square.

The Lo Shu square

Chinese legends claim that a giant turtle, with a three-by-three magic square engraved on it’s back, came out of the Lo River to save China from a flood. The first legends that mention this were written in the fourth century B.C., but they claim that the flood occurred in the 23rd century B.C. Since then, until about a thousand years ago, the Chinese believed that magic squares really were magical. They were particularly intrigued by the Lo Shu magic square and believed that the even numbers in the square represented the “yin” – the female things in the world and that the odd numbers represented the “yang” – the male things. The numbers 1 and 6 represented water, 4 and 9 – metal, 2 and 7 – fire and 3 and 8 represented wood. The number 5 in the middle of the square represented the earth.

Cornelius Agrippa’s magic squares

Magic squares became very popular in Europe during the middle ages. One of the main characters responsible for this was Cornelius Agrippa (1486-1535), a rather extraordinary German. At university, Agrippa secretly formed a group of students who studied magic and Alchemy. Some of his friends were actually burned alive at the stake because of their dealings with black magic. Among other things, Agrippa studied law and theology at university, together with philosophy, magic and Kaballah, and wrote many important manuscripts in all of these subjects. He was sentenced to death (a few times) by the Church but somehow, always managed to escape. Apart from all of this, he was also a high-ranking officer in the army and a personal physician (doctor) to King Charles III.

Agrippa believed that every magic square was in some mystical way connected to the stars, so he associated magic squares of increasing size (3x3, 4x4, 5x5 etc…) with each of the seven so-called heavenly bodies (the sun, moon and the five naked-eye visible planets). For example, he named the Lo Shu magic square, Saturn. Here is Agrippa’s 9x9 ‘moon’ magic square:

Arippa’s moon magic square

Albrecht Dürer’s magic square

Another famous German who was also intrigued with magic squares and lived at the same time as Cornelius Agrippa was Albrecht Dürer, a famous artist and mathematician. Dürer included many mathematical and geometrical themes in his artwork, using a compass and a ruler so that his works would be very precise. He also was one of the first artists to study and use proportion in his drawings. One very unique work of art that Dürer created is an engraving called melancholy. It describes Dürer’s feelings towards depression. Dürer hid a magic square in the top right corner of melancholy. The year that Dürer engraved Melancholy, 1514, is hidden in the two centre cells of the bottom row of the magic square.

Melancholy

Magic squares today

Many other dominant people have shown interest in magic squares, including Benjamin Franklin. Today, mathematicians still study magic squares. They are interested in questions such as: is there a method by which we can construct magic squares of a given size? Methods of construction have been known since ancient times, and can be found in appropriate websites, but to date, there is no known method that can construct all possible magic squares. Mathematicians also do not even known how many magic squares there are of order n x n for n larger than 5!

Discussion

Have you seen any notable magic squares anywhere? If so, please share this with us!

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Flexagons and the Math Behind Twisted Paper

Weizmann Institute of Science