### Da Vinci, Mathematics and Chess

This post is based on the premise that Leonardo Da Vinci, along with many other artists, used a particular number ratio (the Golden Ratio or Golden Mean) in his work - even extending to the ratio between the dimensions of the surrounding picture frame. On this occasion we are even confronted with its use in the drawings of some chess pieces. If you wish to skip the math and get to the chess - please just go to the next heading.

The joy of numbers is there for all of us, mathematicians or not. I remember a school trip to a science fair in 1967 when I first saw an electronic calculator. It was a magnificent piece of kit the size of a briefcase, glowing red numerals at least a couple of inches high. It seemed as if the numbers stretched to infinity, maybe 20 or 30 digits. I soon made an excuse to leave the rest of the party and asked the attendant to let me try it out.

The obvious thing to do was find the Holy Grail - otherwise known as the Golden Ratio. Those of you in the know will appreciate the use of the term Holy Grail - in that it can never be found. We begin with the Fibonacci sequence which begins with two numbers, say 1 and 1. The next number in the sequence is found by adding the two previous ones, thus, 1 + 1 = 2. So we have:

1, 1, 2

Did I say the two previous numbers? Well, the last two numbers in the above are 1 and 3. So we add them together to find the next number, thus 1 + 2 = 3 which gives us:

1, 1, 2, 3 (we have added the two previous numbers together, 2 and 1)

Then 2 + 3 and so on, thus:

1, 1, 2, 3, 5, 8, 13, 21 ...

It is an interesting exercise to divide each pair of numbers in the sequence, starting with 1 and 1, thus:

1 / 1 = 1

2 / 1 = 2

3 / 2 = 1.5

5 / 3 = 1.3333333 ...

8 / 5 = 1.6

13 / 8 = 1.625

21 / 13 = 1,6153846153846153846153846153846 ...

This then was the Holy Grail that I pursued all those years ago. These results would seem to be zoning in on some final value that is in the region of just over 1.6 - or are they?! If you want to know the value to a million decimal places you can go here - in my case the teacher dragged me back to the rest of the student group before I could get that far!

Let me just point you at two videos dealing with Fibonacci Numbers, the Golden Ratio and how these mathematical entities are an intrinsic part of our universe. The first one is visually entrancing while the second one contains a little more detail about the way the sequence is built up.

Perhaps because the Golden Ratio would seem to be bound up with our universe in some fundamental way, it is completely independent of the initial starting numbers - you can start off with any two numbers you wish - and you will still end up with the Ratio. If you want to get moving nice and quickly then it makes sense to start with two numbers which are already close eg 1 and 1.618, thus:

1, 1.618, 2.618, 4.236, 6.854 ...

Dividing these last two digits on a calculator gives us the value 1,6180358829084 ... which is already accurate to 5 significant figures ie 1,6180. The main thing is that you can still get there with any two starting numbers you like - it just takes a little longer is all. And this post took a little longer to get to the chess too - but we are there now.

Apparently there is some possibility that our friend Leonardo was the world's first chess-puzzle illustrator! The story is that a long forgotten leather bound book, containing some chess problems, was discovered in 2006 - and was illustrated by Leonardo (shown below). The treatise, "De Ludo Schaccorum" (Latin for "Of the Game of Chess") includes more than 100 chess problems - I quote here from USA TODAY.

Here is a quote from an article in the Guardian from a Milanese researcher, Franco Rocco. - "the proportion of the pieces, and especially the pawns, coincides with the Golden Mean [an arithmetical ratio of approximately 1: 1.618], which fascinated both Leonardo and his friend Pacioli."

In case you are wondering, yes, there is apparently a lot of evidence that Leonardo at least knew the moves of the game, thus to quote from the same article) "Leonardo, who drew a portrait of the marchioness, is known to have understood, if not played, chess. He used a technical term from the game in one of his many manuscripts". This is well accepted even if the above attribution to the leather bound book is not.

If you are interested in the actual position shown on the right hand page in the image below, there is no better place to go than an article by Raymond Keene in which he analyses the position in great detail and theorises about the historical background. However, the actual solution is given by James O'Fee.

In the position shown we see that there is an implausible looking white pawn on d1- here is what Keene has to say about it:

"It should also be observed that there is an illegal white pawn on d1 in the original. I have replaced this with a white knight, which makes no difference to the solution. It is possible that the original Renaissance copyist put down a pawn instead of a knight — or after the massive rule changes around 1475 there may have been some local disagreement about the laws of the game which might perhaps have permitted a pawn to move backwards. Somehow, though, I doubt this and favour the solution that implies a miscopied piece".

The joy of numbers is there for all of us, mathematicians or not. I remember a school trip to a science fair in 1967 when I first saw an electronic calculator. It was a magnificent piece of kit the size of a briefcase, glowing red numerals at least a couple of inches high. It seemed as if the numbers stretched to infinity, maybe 20 or 30 digits. I soon made an excuse to leave the rest of the party and asked the attendant to let me try it out.

The obvious thing to do was find the Holy Grail - otherwise known as the Golden Ratio. Those of you in the know will appreciate the use of the term Holy Grail - in that it can never be found

1, 1, 2

Did I say the two previous numbers? Well, the last two numbers in the above are 1 and 3. So we add them together to find the next number, thus 1 + 2 = 3 which gives us:

1, 1, 2, 3 (we have added the two previous numbers together, 2 and 1)

Then 2 + 3 and so on, thus:

1, 1, 2, 3, 5, 8, 13, 21 ...

It is an interesting exercise to divide each pair of numbers in the sequence, starting with 1 and 1, thus:

1 / 1 = 1

2 / 1 = 2

3 / 2 = 1.5

5 / 3 = 1.3333333 ...

8 / 5 = 1.6

13 / 8 = 1.625

21 / 13 = 1,6153846153846153846153846153846 ...

This then was the Holy Grail that I pursued all those years ago. These results would seem to be zoning in on some final value that is in the region of just over 1.6 - or are they?! If you want to know the value to a million decimal places you can go here - in my case the teacher dragged me back to the rest of the student group before I could get that far!

Let me just point you at two videos dealing with Fibonacci Numbers, the Golden Ratio and how these mathematical entities are an intrinsic part of our universe. The first one is visually entrancing while the second one contains a little more detail about the way the sequence is built up.

Perhaps because the Golden Ratio would seem to be bound up with our universe in some fundamental way, it is completely independent of the initial starting numbers - you can start off with any two numbers you wish - and you will still end up with the Ratio. If you want to get moving nice and quickly then it makes sense to start with two numbers which are already close eg 1 and 1.618, thus:

1, 1.618, 2.618, 4.236, 6.854 ...

Dividing these last two digits on a calculator gives us the value 1,6180358829084 ... which is already accurate to 5 significant figures ie 1,6180. The main thing is that you can still get there with any two starting numbers you like - it just takes a little longer is all. And this post took a little longer to get to the chess too - but we are there now.

**Leonardo da Vinci and chess**Apparently there is some possibility that our friend Leonardo was the world's first chess-puzzle illustrator! The story is that a long forgotten leather bound book, containing some chess problems, was discovered in 2006 - and was illustrated by Leonardo (shown below). The treatise, "De Ludo Schaccorum" (Latin for "Of the Game of Chess") includes more than 100 chess problems - I quote here from USA TODAY.

Here is a quote from an article in the Guardian from a Milanese researcher, Franco Rocco. - "the proportion of the pieces, and especially the pawns, coincides with the Golden Mean [an arithmetical ratio of approximately 1: 1.618], which fascinated both Leonardo and his friend Pacioli."

In case you are wondering, yes, there is apparently a lot of evidence that Leonardo at least knew the moves of the game, thus to quote from the same article) "Leonardo, who drew a portrait of the marchioness, is known to have understood, if not played, chess. He used a technical term from the game in one of his many manuscripts". This is well accepted even if the above attribution to the leather bound book is not.

If you are interested in the actual position shown on the right hand page in the image below, there is no better place to go than an article

In the position shown we see that there is an implausible looking white pawn on d1

"It should also be observed that there is an illegal white pawn on d1 in the original. I have replaced this with a white knight, which makes no difference to the solution. It is possible that the original Renaissance copyist put down a pawn instead of a knight — or after the massive rule changes around 1475 there may have been some local disagreement about the laws of the game which might perhaps have permitted a pawn to move backwards. Somehow, though, I doubt this and favour the solution that implies a miscopied piece".

White to move and checkmate in 9.Can you find it? |

The original image is from the Abaco museum site and the modern version is from Susan Polgar's blog. The former site has a video in which you actually see the book itself (look for the link "Guarda il video").

"It was like a Holy Grail of chess," said Serenella Ferrari Benedetti, cultural coordinator of the foundation that manages the Coronini estate. "We knew it existed but nobody had ever seen it". "It was like a Holy Grail of chess," said Serenella Ferrari Benedetti, cultural coordinator of the foundation that manages the Coronini estate. "We knew it existed but nobody had ever seen it."

The illustrations of the red and black chess pieces were themselves a puzzle. The slender, abstract design was so unusual that Ferrari Benedetti asked Rocco to study the drawings.

After a year of research, Rocco concluded that Pacioli enlisted Leonardo's help to draw the pieces.

Rocco, in a telephone interview with The Associated Press, noted that the two men had earlier collaborated in Milan when Leonardo helped illustrate a treatise on proportion while also painting The Last Supper.

Rocco said the futuristic style of the chess pieces is in sharp contrast with the way other pieces were represented at the time.

"Every piece also was proportionally related to each of its parts and to the other pieces, a trademark of Leonardo's art, he said. In addition, some pieces directly recall other works by Leonardo, including a queen similar to a fountain drawn in one of the artist's manuscripts".

More from the Fondazione Palazzo Coronini Cronberg library collections where the book is kept can be found here.

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