The magic square flexagon

Not all book flex flexagons are non-cyclic. Here is an example of a magic square tetra-tetra-flexagon that is cyclic and flexed using the book flex. First, though, we’ll briefly introduce 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.

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.

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 (3×3, 4×4, 5×5 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 9×9 ‘moon’ magic square:

Folding the magic square flexagon

If you want to learn more about, and make, this flexagon, download the template here. This is the only template that is not available as a direct download or in the template booklet. To make the flexagon, watch the video and follow the instructions. This flexagon features four famous magic squares created by Josep Maria Subirachs (the Sagrada family church magic square, Benjamin Franklin, Albrech Dürer and Cornelius Agrippa.

Discussion

Try and make other versions of magic square flexagons – and let us know if you succeed! Just take a picture of them and upload the pictures to your social media with the hashtag #FLflexagonsgalore. Then, post a link to it in the comments below.