The Da Vinci Code: Use of Fibonacci Sequences, Golden

Ratio and Cryptography

A Nicolet College Library Literature Lecture Series

presentation by Gary Britton

Emeritus Professor of MathematicsUW-Washington County

April 16, 2009

The Da Vinci Code book

• Author Dan Brown

• Published in March 2003

• 166 weeks on NY Times Best Seller List

• Over 60 million copies sold

• Paperback released March 2006

             

The Da Vinci Code movie

• Released May 2006

• Starred Tom Hanks and Audrey Tautou

• Grossed $77 million opening weekend

• 2nd highest grossing movie of 2006

• Over $750 million world wide

• Available on DVD

Key characters

• Robert Langdon, Professor of Religious Symbology, Harvard University

• Jacques Sauniere, Curator, The Louvre Museum

• Bezu Fache, Captain, Direction Centrale Police Judiciare (DCPJ), approximately the French equivalent of the FBI

• Sophie Neveu, DCPJ cryptographer and Jacques Sauniere’s granddaughter

• Andre Vernet, president of the Paris branch of the Depository Bank of Zurich

• Sir Leigh Teabing, British religious historian living near Versailles, France

Other books by Dan Brown

• Angels and Demons, July 2001• Deception Point, December 2002• Illuminati, November 2003• Digital Fortress, December 2003• The Solomon Key (anticipated title),

Originally planned release in March 2007, now May 2009? History of Freemasonry and the links between it, founding fathers, and birth of the United States?

Angels and Demons movie

• May 15, 2009 release date

• Directed by Ron Howard

• Starring Tom Hanks as Robert Langdon

• Actually a prequel to The Da Vinci Code

Example of commercial products inspired by the novel

Several commercial products related to The Da Vinci Code

• Cryptex security boxes at http://www.cryptex.org/h_gallery.htm.

• The Da Vinci Quest board game, The Movie Game Inc., www.triviainatrunk.com.

• Cracking the Da Vinci Code Day Calendar 2006, Barnes & Nobel, 2005. (material from Cracking the Da Vinci Code by Simon Cox. See excerpts on several slides at the end of the presentation if time.)

Samples from Cracking the Da Vinci Code Day Calendar 2006

• “The Golden Rectangle is one in which the sides are in proportion of the Golden Ratio: in other words, the longer side is 1.618 times longer than the shorter side. Aesthetically pleasing, it is found throughout world art and culture. Leonardo da Vinci’s drawing of the Vitruvian Man has the outlines of a Golden Rectangle based on the head, one on the torso, and another over the legs.” April 14

An example from the commercial marketplace.

A Borders Books web site search on The Da Vinci Code yielded :86 items on 12/27/05,103 items on 4/18/06,137 items on 10/18/06, 172 items on 11/14/06.84 items on 4/16/09.

(at one time included such things as a CD called: The Diet Code: Revolutionary Weight Loss Secrets from Da Vinci and the Golden Ratio by Lanzalotta, Stephen.)

Overview of today’s talk

• Introductory remarks and key characters• Clues from the opening murder scene• Fibonacci sequence• Golden ratio• Anagram messages• Passwords and the Fundamental Principle of Counting• Mirror messages• Cryptology and coded messages• Websites and references• Questions and answers

Clues found at the opening murder scene of the novel:

• Jacques Sauniere had removed his clothes, placed his body like a snow angel, drawn circular arcs with his arms and traced, in blood, a pentacle (see next slide), centered on his navel.

• The following message was written on the floor beside the body.13-3-2-21-1-1-8-5O, Draconian Devil!Oh, lame saint!

• To try to determine if this message has any coded meaning, first consider just the numbers.

• 13-3-2-21-1-1-8-5

• As a start, let’s arrange them numerically.

2 21 1 1 8 513 3

Fibonacci Sequence

Ratio of successive terms

       

  0    

  1    

0+1= 1 1/1=1.00000000

1+1= 2 2/1=2.00000000

1+2= 3 3/2=1.50000000

2+3= 5 5/3=1.66666667

3+5= 8 8/5=1.60000000

5+8= 13 13/8=1.62500000

8+13= 21 21/13=1.61538462

13+21= 34 34/21=1.61904762

21+34= 55 55/34=1.61764706

34+55= 89 89/55=1.61818182

55+89= 144 144/89=1.61797753

89+144= 233233/144=

1.61805556

144+233= 377377/233=

1.61802575

233+377= 610310/377=

1.61803714

Sweat bee “parents”

A male sweat bee has only a female parent.(i.e. hatches from an unfertilized egg)

Sweat bee “parents”

A female sweat bee has a male and female parent. (i.e. hatches from a fertilized egg)

# of female ancestors in each generation for

a male sweat bee. (read up from M)

• FMFFMFMFFMFFMFMFFMFMF13

• F MF F MF MF F MF F M 8• F M F F M F M F 5• F M F F M 3• F M F 2• F M 1• F 1• M

# of female ancestors in each generation for a female sweat bee. (read up from F)

• FMFFMFMFFMFFMFMFFMFMF13

• F MF F MF MF F MF F M 8• F M F F M F M F 5• F M F F M 3• F M F 2• F M 1• F

Ratio of female to male ancestors in each generation for sweat bees

# F # M Ratio F/M

34 21 1.619

21 13 1.615

13 8 1.625

8 5 1.600

5 3 1.667

3 2 1.500

2 1 2.000

1 1 1.000

Fibonacci sequence appears in other places.

Construct a spiral based on the inscribing quarter-circle arcs in squares of size 1,1,2,3,5,8,13,…

The resulting spiral appears in the shell of the chambered nautilus (mollusk of the So. Pacific Ocean), the placement of leaves on some plants, the hexagonal scales of a pineapple and the seeds in the head of a sunflower.

Chambered nautilus

Seeds in the head of a sunflower

Divine Proportion

Divide a line so the ratio of the line to the longer segment equals the ratio of the longer segment to the shorter. In the line above, C divides AB in such segments. The number is called the Golden Ratio and is 1.618033989…. or approximately 1.62 as in the diagram.

A BC

AB = 603.001 pixels

AC = 373.125 pixels CB = 229.876 pixels

ABAC = 1.62 AC

CB = 1.62

Φ

• The divine proportion is denoted by the Greek letter phi. (though Dan Brown chooses not to use the symbol for any of his frequent references to the number)

• The exact value is (1+√5)/2.• Also called the golden ratio or golden

section.• The ratio of successive numbers in the

Fibonacci sequence approaches Φ.

Φ (continued)

• In the book Langdon recalls his lecture about PHI for his “Symbolism in Art” class, where he references many appearances of the golden ratio in art, architecture, music and in the pentagram. (Recall the pentagram that Sauniere traced on his body. See next slide.)

• In Technical Analysis of stock and commodity prices Φ -1 and (Φ-1)2 are used to determine potential support, resistance and price levels. [Ross, p. 63]

Rectangles with (almost) golden ratio dimensions. (red, …, blue, …)

13-3-2-21-1-1-8-51-1-2-3-5-8-13-21

O, Draconian Devil!Oh, lame saint!

• Assuming the text lines are also anagrams, let’s try to rearrange the letters into a meaningful message for Langdon and Neveu one line at a time.

Anagrams for the audience

Each line below is a separate anagram of something well known to you. Ignore spaces.

ELECT LOGIC LEON

 

AS DEER WHIRL INN

O D R A C O N I A N D E V I L

When decrypting a message we usually delete all spaces and punctuation.

O D R A C O N I A N D E V I L

Partially decoded message on museum floor:

1-1-2-3-5-8-13-21

LEONARDO DA VINCI

OH, LAME SAINT!

O H L A M E S A I N T

Complete decoded message from Jacques Sauniere.

1-1-2-3-5-8-13-21LEONARDO DA VINCI

THE MONA LISA

Decoded anagrams for the audience

ELECT LOGIC LEON

AS DEER WHIRL INN

Nicolet College

Rhinelander Wis

SO DARK THE CON OF MAN

Message written by Sauniere on the glass of the Mona Lisa.

S O D A R K T H E C N O F M A NO

Decoded message written on the glass of the Mona Lisa.

Madonna of the Rocks

Madonna of the Rocks leads to:

• Gold electronic key with 24 Rue Haxo written on it.

• Escape from the Louvre.

• Wild chase through Paris.

• Paris branch of Depository Bank of Zurich. (see Google search web site on next slide)

• Sauniere’s deposit box.

Fundamental Principle of Counting

• Vernet says “… every key is electronically paired with a ten-digit account number that functions as a password.” (p. 184)

• Sophie calculates the cryptographic odds to be ten billion possible choices.

• The Fundamental Principle of Counting is a procedure for counting total possible outcomes in multistage processes. The number of ways that the entire process can be done is the product of the number of ways each stage can be done.

• Each digit of the password has 10 possibilities so the password has (10)(10)(10)…(10)=1010 possibilities.

Predicting the password

• Notice that Sauniere’s message, 13-3-2-21-1-1-8-5, has 10 digits.

• Sophie enters 1332211185 and realizes she only has one chance to get it right.

• She says “It’s too random. … he would have chosen a number that he could easily remember, … something that appeared random but was not.”

• So she enters “1123581321”.

Success! The password worked.

• Rosewood box obtained from the vault at Depository Bank of Zurich.

• Cryptex is found inside the box.• Langdon and Neveu escape bank and go to

Sir Leigh Teabing’s estate.• Langdon discovers a coded message

written underneath an inlaid rose in the top of the box. It leads to opening the cryptex.

Rosewood box

Description of cryptex

• A cylinder approximately the dimensions of a tennis ball can.

• Five lettered dials that rotate like a bicycle lock. (though on p. 198 it mentions 6)

• The inside is hollow to contain a message.• It opens at the end when all the dials are

properly set. • 265 = 11,881,376 possible settings.• May look similar to the following picture.

Description of cryptex

• A cylinder similar to a tennis ball can.• Five lettered dials that rotate like a bicycle lock.• The inside is hollow to contain a message.• Opens at the end when all the dials are properly set.

• 265 = 11,881,376 possible settings.

Coded verse written in the lid of the rosewood box:

Remembering her Grandfather’s teachings, Sophie determines the code.

an ancient word of wisdom frees this scrolland helps us keep her scatter’d family wholea headstone praised by templars is the key

and atbash will reveal the truth to thee

Sauniere had written this iambic pentameter verse using a mirrored script, the same as Leonardo Da Vinci used in his notebook, now known as Codex Leicester and owned by Bill Gates.

Cipher disk

• Two concentric disks with the alphabet on one disk and symbols or a mixed, reversed or standard alphabet on the other.

• Invented in Italy before 1470.• First large scale use in the U.S. during Civil War.• During, and for several years after, World War I, the U.S.

Army issued the disks to units that needed a cipher that could be carried and used easily.

• Leave the setting on the same predetermined letter through the message or agree in advance to alter it after every letter of the message or daily or weekly.

Caesar-cipher

• Invented by Julius Caesar.

• Shifts the alphabet forward three letters. (variations of this use a shift of 1 or 2 letters forward or backward).

• You can decipher the message by just writing the alphabet in columns down from the message. (add as many letters in each column as the maximum shift to be considered for the Caesar-cipher.)

Example of Caesar-cipher with a shift of 3 letters forward.

a b c d e f g h i j k l m n o p q r s t u v w x y z

x y z a b c d e f g h i j k l m n o p q r s t u v w

That is, 'a' in the message becomes 'x' in the encryption.

Sample encrypted message.

k f z l i b q z l i i b d b

Encrypted message.

k f z l i b q z l i i b d b

One way to break the code, write the alphabet downwards

from each letter of the message. Do as many times as needed.

k f z l i b q z l i i b d b

l g a m j c r a m j j c e c

m h b n k d s b n k k d f d

n i c o l e t c o l l e g e

Caesar Box code writing

• This is a transposition cipher invented by Julius Caesar.

• Consider the encrypted message “wiouannnsgcthtoy”

• Notice there are 16=42 letters so write them in the format of a 4 X 4 box.

wiouannnsgcthtoy

Caesar Box (continued)

Now read the box column by column.W I O UA N N NS G C TH T O Y

So the deciphered message is “Washington County”.

The Atbash cipher

• Dates to 500 BC

• A monoalphabetic substitution cipher (i.e. each plain letter is replaced by a single cipher letter from the same alphabet). 

• Rotational substitution scheme, the first for the last, second for next to last, etc.

• Based on the 22 letter Hebrew alphabet.

Clues from the verse

• Ancient word of wisdom frees the scroll

• Headstone praised by Templars

• Atbash will reveal the truth

Langdon finally realizes the stone head is that of Baphomet, so apply the Atbash cipher to “Baphomet”.

“Baphomet” and Hebrew alphabet

A B G D H V Z Ch T Y K

Th Sh R Q Tz P O S N M L

• Vowel sounds not written in Hebrew spelling.• With vowels in lower case and with Hebrew letters it becomes

BaPVoMeTh, which has five letters as desired.• Using the cipher table we get Sh-V-P-Y-A.• “Sh” can be pronounced as “S”, and “P” as “F”.• “SVFYA” becomes “Sofya” or “Sophia” when sounded out.• “Sophia” means wisdom in Greek.• But in ancient Greek is spelled “sofia”.

Contents of the cryptex

Inside the cryptex is a smaller cryptex which also has five dials that need to be set using a five letter code. This smaller cryptex is wrapped inside a protective lambskin vellum with another iambic pentameter verse written on it. The verse, in English, read:

In London lies a knight a Pope interred.His labor’s fruit a Holy wrath incurred.

You seek the orb that ought be on his tomb.It speaks of Rosy flesh and seeded womb.

Puzzles for the audience.• What is the five letter word that is needed to

open the second cryptex?In London lies a knight a Pope interred.His labor’s fruit a Holy wrath incurred.You seek the orb that ought be on his

tomb.It speaks of Rosy flesh and seeded womb.• What character’s name is an anagram for the

(possible) biblical reference “Oh! Supine Eve”?

Web sites related to Dan Brown and The Da Vinci Code.

• www.danbrown.com, is the author’s own web site. A must read for those interested in pursuing some of the background on the novel and related information. Click on “Common Questions” to read Dan Brown’s own thoughts on some of the questions raised by, and about, the novel. You can even sign up for a Dan Brown newsletter if you explore the site deep enough.

• www.johnlangdon.net, is the web site of John Langdon, close friend of Dan Brown’s father, creator of the ambigrams in Angels & Demons, and possibly the inspiration for the name Robert Langdon in The Da Vinci Code.

Web sites (continued)

• www.symbols.com, is an excellent site for studying symbols mentioned in The Da Vinci Code.

• http://kasmana.people.cofc.edu/MATHFICT/default.html, provides many references to works of fiction that use mathematics. For The Da Vinci Code it points out the mathematical shortcomings in the explanations and descriptions in the book, including some of the material I have shared with you. The cryptography section is quite thorough.

• www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html, is a good web site for the Fibonacci sequence.

Web sites (continued)

• www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html, provides good information and additional links about PHI, the Golden Ratio.

• www.murky.org/cryptography/index.shtml, contains a primer on cryptology and ciphers.

• www.wordsmith.org/anagram/advanced.html, has an anagram builder.

• www.anagramgenius.com, has software for purchase or a free trial basis that creates anagrams.

• www.wordles.com/getmycruypto.asp, includes a cryptogram builder.

Bibliography

Brown, Dan, The Da Vinci Code, Doubleday, 2003. Brown, Dan, The Da Vinci Code, Special Illustrated

Edition, Doubleday, 2004.Burnstein, Dan, Editor, Secrets of the Code, CDS Books,

2004.Lewand, Robert Edward, Cryptological Mathematics, The

Mathematical Association of America, 2000.National Security Agency, Codes, Ciphers, and Puzzles

Activity Book. Ross, Debra Anne, Master Math: Geometry, Thomson,

2005.Sharma-Jensen, Geeta, et.al., Milwaukee Journal Sentinel,

April 5, 2006, pp. 1E & 8E.

Questions, answers, comments?

Answers to puzzles for the audience.• What is the five letter word that is needed to

open the second cryptex?In London lies a knight a Pope interred.His labor’s fruit a Holy wrath incurred.

You seek the orb that ought be on his tomb.It speaks of Rosy flesh and seeded womb.

• Hints:– Alexander Pope (A. Pope)– Presided at Sir Isaac Newton’s internment– “orb”, “rosy flesh”, “seed”

• APPLE

Answers to puzzles for the audience.• What character’s name is an anagram for the

(possible) biblical reference “Oh! Supine Eve”?

O H S U P I N E E V E

S O P H I E N E U V E

End of presentation.

Selected entries from Cracking the Da Vinci Code Day Calendar 2006

• “Sauniere’s mirror-image poem on the rosewood box provides the key to its own meaning: ‘Atbash will reveal the truth to thee.’ The Atbash Cipher, which dates from around 500 BCE, uses the letters of the Hebrew alphabet in a simple substitution system; each letter is replaced by another an equal distance from the opposite end of the alphabet, i.e. the first letter for the last, the second for the second to last, and so on.” January 3

• The many meanings of the pentagram are at the heart of The Da Vinci Code. In Christian tradition, the pentagram was once used to represent the five wounds, or stigmata, of Christ. To the Pythagoreans, the five points represented the five classical elements: fire, earth, air, water and idea, or divine thing. The Pythagoreans also saw within the pentagram the mathematical perfection of the Golden Ration of 1.618….” January 28

• “Leonardo’s drawing, Vitruvian Man, was originally an illustration for a book on the works of the architect Vitruvius. It was accompanied by a translation of Vitruvius’s theory about the measurements of the human body: ‘that 4 fingers make 1 palm, and 4 palms make 1 foot, 6 palms make 1 cubit, 4 cubits make a man’s height,’ and so on.” February 6

• “The strange properties of the Golden Ratio - which occurs in the Fibonacci sequence – meant historically that it was seen as divine in its composition and infinite in its meaning. The ancient Greeks believed that understanding the Golden Ratio would help them get closer to God: God is ‘in’ the number. Many things in the natural and man-made worlds conform to its proportions, from the human face to widescreen televisions.” February 25

• Langdon finds a message of male and female harmony in Leonardo Da Vinci’s Vitruvian Man, his drawing of a spreadeagled nude male contained within a circle. This is unlikely. The composition is based wholly on Vitruvius’s dimensions of the human body, the emphasis being on rationalization of the geometry, by means of small whole numbers, to build the composition.” March 8

• On this date in 1727, Sir Isaac Newton died in what is now the London Borough of Kensington and Chelsea. He was buried in Westminster Abbey, the first scientist to be accorded this honor. His study and understanding of light, his invention of the reflecting telescope, and his revelation in Principia of the mathematical ordering of the universe are all represented on his monument.” March 20

• “The Golden Rectangle is one in which the sides are in proportion of the Golden Ratio: in other words, the longer side is 1.618 times longer than the shorter side. Aesthetically pleasing, it is found throughout world art and culture. Leonardo da Vinci’s drawing of the Vitruvian Man has the outlines of a Golden Rectangle based on the head, one on the torso, and another over the legs.” April 14

• “The mortally wounded Louvre curator Jacques Sauniere scribbles the puzzling reference on the floor of the Grand Gallery to a ‘Draconian devil’. This briefly brings to Langdon’s mind the Athenian legislator Draco, remembered for his harsh law code of 621 BCE, which imposed the death penalty for relatively trivial crimes. In this contest, the word ‘draconian’ describes harsh or repressive legal measures.” April 20

• The flame-haired thirty-two-year-old Agent Sophie Neveu studied cryptography at the University of London’s royal Holloway College, on the city's western outskirts. Cryptography, the science and study of analyzing, writing or deciphering codes and ciphers, forms part of a relatively new Masters degree in Information Security, of which Royal Holloway’s was the first course of its kind in the world, and is reputed to be the finest.” April 21

• Langdon’s experience of epigraphical ciphers by the English writer, philosopher and scientist Francis Bacon (1561-1626) helps him crack Sauniere’s code. In Book VI of his Proficience and Advancement of Learning Divine and Humane, Bacon expressed his preference for ciphers whose ‘vertues’ include that they ‘bee without suspition’ – in other words, for codes that do not appear to be enciphered messages.” May 1

• Leonardo’s mirror script, featured in Sauniere’s cryptex riddle, may have been a ‘code’ or merely an idiosyncratic, left-hander’s approach to writing. Either way, he would not have wanted to broadcast his most mind-blowingly prescient observation that the driving creative force is not God, but Nature, for which ‘necessity is the mistress and teacher’. Some three hundred years later, Darwin’s theory of evolution was still considered heresy.” May 11

• “Sophie and Robert Langdon puzzle over Sir Isaac Newton’s tomb in Westminster Abbey for the solution to the final clue. The monument was completed in 1731 by Michael Rysbrack to the design of William Kent. Mathematical and optical instruments and some of his most influential books, including the Principia, surround the reclining figure of Newton. On top of the globe sits the figure of Astronomy. John Maynard Keynes said of Newton after his death, ‘He regarded the universe as a cryptogram set by the Almighty.’” June 2

• “The Da Vinci Code introduces to readers many complex words from the field of code-breaking: pictogram, codex, symbology, iconology, cryptex and polyalphabetic substitution cipher, to name but a few. Not all are inventions or adaptations by Dan Brown.” June 25

• “Jacques Sauniere’s use of mirror writing in the first riddle pays homage to Leonardo da Vinci, whose virtually illegible scrawl travels from right to left across the pages of his numerous private notebooks. In these, his observations, accompanied by sketches and drawings, touch on science, philosophy, art, architecture, engineering, astronomy and anatomy.” July 6

• “Langdon recalls his bewildered first impression of the Codex Leicester, a compilation of loose-leaf notes written in Leonardo’s mirror-image version of Renaissance Italian between 1506 and 1510. It contains his observations on subjects ranging from astronomy to the properties of rocks, water, fossils and air. It is now owned by Bill Gates, co-founder of Microsoft and world’s richest man, and is on display at the Seattle Art Museum.” July 10

• “The downfall of the Knights Templar was partly brought about by charges of worshiping the mysterious idol, Baphomet, whose name is the key to opening the first cryptex in Dan Brown’s novel. By converting the name Baphomet into its Hebrew spelling (‘B PV M Th’), and then applying the Atbash Cipher to these letters, Sir Leigh Teabing reveals the word sophia, the Greek word for ‘wisdom’.” August 10

• “Sauniere arranged his dying body into the image of Leonardo’s iconic drawing, Vitruvian Man. Vitruvius was a Roman engineer, writer, and architect of the late first-century BCE and early first century CE. His one extant book, De Architectura, contains ten huge encyclopedic chapters on human proportions. His rediscovery in the Renaissance fueled the growth of classicism. Vitruvian Man is Sophie Neveu’s favorite Leonardo work.” September 7

• “Knowledge of the Atbash Cipher enables Sir Leigh to open Sauniere’s first cryptex. The name Atbash comes from the first two letters of the Hebrew alphabet (aleph and beth, a and b in English) and their equivalents in cipher (tac and shin, for t and s). Hebrew scribes translating the books of the Old Testament applied the cipher to place names like Sheshach, which biblical scholars later revealed as Babel.” October 10

• “Jacques Sauniere’s scrambled Fibonacci sequence provides the ten-digit PIN for the Priory keystone at the Depository Bank of Zurich, as well as an opportunity for some interesting musings on phi, the Golden Ratio. Leonardo Fibonacci (c.1170-c.1250) was born in Pisa, Italy, and educated in North Africa. Returning to Pisa around 1200, his mathematical genius earned him a place at the court of the Holy Roman Emperor, Frederick II.” October 18

• “Sir Leigh Teabing was not the first scholar to apply the Atbash Cipher to the word ‘Baphomet’ to reveal sophia. This hidden meaning was, in fact, first brought to light by the Dead Sea Scrolls expert Dr. Hugh J. Schonfield, author of The Passover Plot, who felt that the Knights Templar must have known about the cipher through their dealings in the Holy Land.” October 29

• “The device used to protect sensitive information in Dan Brown’s novel is called a cryptex, said to have been constructed from Leonardo’s blueprint designs. Unfortunately this isn’t true. The cryptex is Dan Brown’s invention, and a flawed one at that – the vinegar stored inside would not be acidic enough to destroy papyrus.” November 25

• “Arguably the father of modern physics, Isaac Newton was born on this date in 1642 (by the Julian calendar). His Principia described universal gravitation, the laws of motion and formed the bedrock of classical mechanics. In The Da Vinci Code, the line from the second cryptex, ‘In London lies a knight a Pope interred” leads Robert Langdon and Sophie Neveu to Newton’s tomb in Westminster Abbey, London.” December 25