How perfect is the Dürer magic square. A 180 degrees hidden symmetry is discovered.

180 degrees symmetry.
This draw is in the title
Melencolia I, betwen
the word Melencolia
and the roman number I.

In 1514, the german engraver and great artist, Albert Dürer published his work called Melencolia I. Many mysteries surround this master work. You can see some elements as: a woman angel with a indecipherable look, a cherub, a rainbow, a comet, a flying bat and a mathematical magic square of 4*4. Here, we focus on this last element.

Dürer published in Melencolia I, one of the first magic square that are known in mathematical history.

 

Melencolia I. 1514 Dürer master work

A mathematical square have some special properties. For example a 4*4 magic square has 16 integers from 1 to 16. They are placed in a such special way that each column, row and diagonal have the same result (34) when the four integers are added. This a basic feature of a magic square. There are different degrees of perfection in a magic square. For example, if the four integers of the corner results in 34, then the magic square is more perfect, as it is the case for the Dürer square. Such squares, with more symmetries, are called most-perfect and panmagic squares. You can see in the Euler diagram, in the next section, this classification of perfection.

In this diagram, each color is a set of 4 integers with the same addition (for the Dürer square 34). An ordinary magic square, have columns, rows and the two diagonals with the same result when the set of 4 integers are added. You see that if you get more symmetries, the square becomes more perfect. For Euler, the most perfect magic square is called most-perfect magic square.

 Properties of the Dürer magic square. The hidden 180 symmetry is discovered.

The date of the engraving is depicted in the numbers placed in positions (4,2) and (4,3) (see below the magic square (row, column)). The artist was 43 years old in 1514, the 34 number inverted.

Euler classification of perfection.

Fingerprint of Dürer square. 86 patterns. Level 1,2, 4 and 5
in the Euler classification. Some elements have a  mirror symmetry.
But all the elements have a 180 degrees symmetry. Each set
of 4 number has his counterpart in order to get the 180 symmetry.

Studying the Dürer magic square, I found that (for a 4*4 square with a 34 addition) there are 86 different ways to add 34 with 4 integers set between 1 to 16. In this way 86 patterns are produced in the magic square, that you can see in the below diagram and in the next  youtube's link:

What is the classification of Dürer magic square in the Euler level of perfection?

Also in the colored image presented here, you can see these 86 different patterns that are produced by a 34 addition. Each color is a combination of 4 integers that results in 34.  Some patterns are mirror inverted, but not all.

Comparing the 86 patterns diagram with the Euler one, we can conclude that the Dürer's magic square has level 1, 2, 4 and 5 but not the 3 symmetry. Observing the 86 patterns (colored diagram) you can concludes that the Dürer's magic square has the 180 rotation symmetry. Each element has a counterpart that is 180 degree symmetrical.

This features is depicted by the S symbol in the title of melencolia I (between the word melencolia and the roman number). Also, in the engraving Melencolia you can see a polyhedron (3D mathematical figure) that also is symmetrical in a 180 degrees rotation. Mathematical symmetry depicts perfection or beauty.

180 rotation symmetry means (in the magic square) that when you rotates the integers as you can see below, the 86 patterns (colored diagram) remains invariant.

Conclusion: In this work, it was proved that the magical square is 180 symmetrical. But there also others elements with this symmetry:

1-The S in the title (between Melencolia and the I)
2-The polyhedron (faces are 180 degrees but not all).
3-And of course, the 4x4 and 34 magic square.


After 504 year from the publication of Melencolia I, here we found 3 elements with a 180 degrees symmetry (specially, the magic square that need a deep study to prove his symmetry).  

What's all means? Read the other entry of my blog to find the hidden meaning of Melencolia I, where human errors and perfection are discussed.