On the (magic) Square

Richard A. O'Keefe
25 October 2011

In an earlier note, I examined something that Dan Brown had his character Langdon say in The Da Vinci Code In this note, I examine something that he has Langdon say in The Lost Symbol.

In Chapter 70, Langdon uses Albrecht Dürer's magic square as a decoder. Let's see some actual text.

... Primarily, though, modern man had relegated magic squares to the category of “recreational mathematics,” some people still deriving pleasure from the quest to discover new “magical” configurations. Sudoku for geniuses.

Katherine quickly analyzed Dürer's square, adding up the numbers in several rows and columns.

 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1

“Thirty-four,” she said. “Every direction adds up to thirty-four.”

“Exactly,” Langdon said. “But did you know that this magic square is famous because Dürer accomplished the seemingly impossible?” He quickly showed Katherine that in addition to making the rows, columns, and diagonals add up to thirty-four, Dürer had also found a way to make the four quadrants, the four center squares, and even the four corner squares add up to that number. “Most amazing, though, was Dürer's ability to position the numbers 15 and 14 together in the bottom row as an indication of the year in which he accomplished this incredible feat!”

Katherine scanned the numbers, amazed by all the combinations.

Langdon's tone grew more excited now. “Extraordinarily, Melancholia I represents the very first time in history that a magic square appeared in European art. Some historians believe that this was Dürer's encoded way of indicating that the Ancient Mysteries had traveled outside the Egyptian Mystery Schools and were now held by the European secret societies.” ...

Who are these historians? Dürer drew his square using the numerals we call “Arabic”—these had been commonly known throughout Europe since about the middle of the previous century, and took nearly another half century afterwards to become commonly used, though it took even longer for such numerals to be commonly used for dates. Dürer was interested in numbers. In 1525 he published the first known book on mathematics for adults to appear in German.

If by “the Ancient Mysteries” Dan Brown means Indian mathematics, then yes, they had travelled to Europe by that time, and every literate person in Europe knew it. As for anything esoteric, I think we need better evidence.

The “Arabic” numerals made calculation much easier. Finding magic squares doesn't take any deep mathematics, but it does help to be able to add fast. Could it be that Dürer was flaunting his ability at calculation? Nah, that's far too prosaic for Brown!

As a teaser for what's coming, I remark that if there is at least one magic square satisfying some strange property, then there are at least 8, all of them essentially the same but rotated and reflected so they look different. There is only one 3x3 magic square, for example, if you take rotations and reflections into account.

Before we ask just how “impossible” Dürer's accomplishment was, let us note a rather strange specific technical incompetence on Katherine's part. Recall that Katherine is supposed to be a scientist. In fact, she is supposed to be an experimental scientist. More than that, she is supposed to be an experimental scientist in a highly contentious area closely related to parapsychology. How contentious? Well, Brown writes at the very beginning of the book that “All rituals, science, artworks, and monuments in this novel are real.” Others would say that none of the science in the book is real. That's contentious. This is an area where it is not enough to have interesting results; not enough to be scrupulously honest; not enough to have superb experimental technique precisely and abundantly recorded. No, it is necessary to have rigorous experiment design as well. Indeed, much of the work in parapsychlology is better designed, carried out, and analysed, and less believed, than much published medical work.

In short, anyone expecting to get a hearing for “Noetic Science” must know a great deal about the formal design of experiments. And that means knowing about Latin squares. Asking me to believe in an experimental scientist working in such a field who is ignorant of Latin squares is like asking me to believe in a Western politician who doesn't know what a vote is! I can't do it, and I won't do it, and I resent an author being so contemptuous of the intelligence of his readers that he expects me to believe it.

Now for some numbers. By exhaustive enumeration, there are

• 7,040 arrangements of the numbers 1–16 in a 4×4 square such that all the rows, all the columns, and both diagonals add up to the same total (which has to be 34).
• 3,456 of which arrangements have the four quadrants summing to 34. Note that if one quadrant sums to 34, the others must also sum to 34.
• 3,456 of these also have the four outermost and four innermost squares summing to 34. That's all of them.
• 192 of these, or 3 out of every 110 4×4 squares, has [15,14,x,y] or [x,15,14,y] or [x,y,15,14] in some row.

Here, for example, is a Dürer square I find a bit more impressive. It puts the date in the top row. But see how the top left quadrant and the bottom right quadrant are identical in the unit digits? And how the third row contains 10 11 which is as close as I could get to 2011? Amazing, no?

 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16

Well, no. It's actually one I picked at random.

It gets better. According to the Schools edition of the Wikipedia, entry “Magic Square”

In about 1510 Heinrich Cornelius Agrippa wrote De Occulta Philosophia, drawing on the Hermetic and magical works of Marsilio Ficino and Pico della Mirandola, and in it he expounded on the magical virtues of seven magical squares of orders 3 to 9, each associated with one of the astrological planets. This book was very influential throughout Europe until the counter-reformation, and Agrippa's magic squares, sometimes called Kameas, continue to be used within modern ceremonial magic in much the same way as he first prescribed.

The order 4 square, the square of Jupiter, is

 4 14 15 1 9 7 6 12 5 11 10 8 16 2 3 13

This square, published 4 years before Dürer's picture, already has all the cited properties of Dürer's square except one: it does not contain 1514 in the bottom row. We can fix that with two mirrors.

First let's turn the Square of Jupiter upside down. We get

 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

Now let's flip left and right. We get

 13 3 2 16 8 10 11 5 12 6 7 9 1 15 14 4

This is very nearly Düer's square. It shows that Dürer could have found a “seemingly impossible” square with all the properties Langdon is so impressed by with no labour at all, just trivial flipping.

To get the square he actually used takes a bit more, but it doesn't take much playing with this square to discover that you can swap 13-and-16 and 1-and-4 and get a square that is not quite so obviously derived from the square of Jupiter.

Now we have another specific technical incompetence: Langdon. Anyone with even a nodding aquaintance with Western esotericism must have heard of Cornelius Agrippa; someone with Langdon's purported skills really ought to have read De Occulta Philosophia. I haven't, but then I'm not Langdon and don't profess to be. So Langdon really ought to have known that Dürer could have got “his” square out of a book he would certainly have known about.

Whether Dürer did get his square this way is of course another question, one for serious historical research.