Phi from Da Vinci Code
Like other mathematical tales the story of φ begins, in ancient Greece. The Greeks, with their love for symmetry and geometric order, searched for what they felt was the most pleasing rectangle. Believing that the purest and most aesthetically pleasing form of thought was mathematics, they used math to come up with an answer. Strictly speaking, 1.618 is not exactly the golden ratio. The true value is given by the formula
φ = (1 + √5)/2
Like the ancient Hebrews who could never know the true name of God, we will never know the true numerical value of φ coz its irrational. As an irrational number, φ is like that other mathematical constant π, whose infinite decimal expansion begins 3.14159. Mathematicians prefer π to be more important than φ. But as the math major in Langdon’s class puts it up, “Phi is one H of a lot cooler than pi.” π is hot, but φ is cool.
Having found the golden ratio, the Greeks incorporated it into their architecture, ensuring that wherever they went in their cities, their eyes would be met with glorious rectangles. This may be true, but modern historians question the claim. In fact, the entire story about the Greeks and the golden ratio seems to be without foundation. The one thing the world knows for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value. But he seemed more interested in the mathematics than the architecture, for he gave the golden ratio a decidedly unromantic label: extreme and mean ratio. The term divine proportion first appeared with the publication of the three-volume work of that name by the 15th-century mathematician Luca Pacioli. Calling φ “golden” is even more recent: In 1835 it appeared in a book written by the mathematician Martin Ohm (guess who?).
Like most successful thrillers, The Da Vinci Code switches between different plotlines, and so does our φ Code. For the next episode, fast-forward from ancient Greece to Pisa, Italy, in 1202. Young Italian mathematician Leonardo Pisano (Leonardo of Pisa) has just completed a book called Liber Abaci, or Book of Calculation. Although there is no way Leonardo can possibly know it, his book is going to change the course of human civilization by bringing to the West the method of writing numbers and doing arithmetic that we use today. That system, completed in India 500 years earlier, was much more efficient than the Roman numerals in use at the time. The new number system eventually provided the basis for the European emergence of modern science and engineering in the 16th and 17th centuries.
Here comes the interesting thing... One of the many exercises in Liber Abaci is a whimsical problem about rabbits: A man puts a pair of baby rabbits into an enclosed garden. Assuming that each pair of rabbits in the garden bears a new pair every month, which from the second month on itself becomes productive, how many pairs of rabbits will there be in the garden after one year? It is not hard to see that the number of pairs of rabbits in the garden in each month is given by the numbers in the sequence 1, 1, 2, 3, 5, 8, 13, . . . , which is the Fibonacci sequence (Found the answer by myself and then went to find more (answer is 12th month Fibonacci which is 144)). Fibonacci was the name that mathematical historians gave to Leonardo Pisano. It is derived from the Latin filius Bonacci, or “son of Bonacci”—Leonardo being a member of the Bonacci family. (Now there are two Leonardos in our story: Leonardo da Vinci and Leonardo da Pisa, funny isn't it.)
More interesting facts. As people solved the problem in Leonardo’s book, they began to notice that this frivolous sequence of numbers occurred so frequently in nature that not even the bungling French police captain Bezu Fache in The Da Vinci Code could fail to become suspicious. Here are some exhibits suggesting that the Fibonacci sequence is hiding a secret of nature:
Exhibit A: If you count the number of petals in most flowers, you will find that the total is a Fibonacci number. For instance, an iris has 3 petals, a buttercup 5, a delphinium 8, a ragwort 13, an aster 21, a daisy 13, 21, or 34, and Michaelmas daisies 55 or 89 petals.
Exhibit B: If you look at a sunflower, you will see a beautiful pattern of two spirals, one running clockwise and the other counterclockwise. Count those spirals and for most sunflowers you will find that there are 21 or 34 running clockwise and 34 or 55 running counterclockwise, respectively—all Fibonacci numbers. Other flowers exhibit the same phenomenon; the purple coneflower is a good example. Similarly, pinecones often have 5 clockwise spirals and 8 counterclockwise spirals, and the pineapple frequently has 8 clockwise spirals and 13 counterclockwise spirals.
Exhibit C: Take a close look at the way leaves are located on the stems of trees and plants. The leaves are located on a spiral path that winds around the stem. Starting at one leaf, count how many complete turns of the spiral it takes before you find a second leaf directly above the first. Let p be that number. Also count the number of leaves you encounter (excluding the first one itself). That gives you another number, q. The quotient p/q is called the divergence of the plant. Here’s the surprising part. If you calculate the divergence for different species of plants, you find that both the numerator and the denominator are usually Fibonacci numbers. In particular, 1/2, 1/3, 2/5, 3/8, 5/13, and 8/21 are all common divergence ratios. For instance, common grasses have a divergence of 1/2, sedges have 1/3, many fruit trees (including the apple) have a divergence of 2/5, plantains have 3/8, and leeks come in at 5/13. Clearly, there is something going on here.
Now, as in any good mystery novel, it’s time to bring the two threads together, to show how the ancient Greeks’ golden ratio and the 13th-century Fibonacci sequence are connected. Take the Fibonacci sequence and divide each number into the one that follows it. Here is what you get:
1/1 = 1; 2/1 = 2; 3/2 = 1.5; 5/3 = 1.666 . . . ; 8/5 = 1.6; 13/8 = 1.625; 21/13 = 1.615 . . . ; 34/21 = 1.619 . . . ; 55/34 = 1.6176 . . . ; 89/55 = 1.6181 . . .
Wait a minute. 1, 1.6, 1.61, 1.618. This is starting to look like the golden ratio. This is no accident. Mathematicians have proved conclusively that, like two shy lovers in a Hollywood romance, the Fibonacci ratios get slowly closer to φ and finally kiss at infinity (great way to say reaches at infinity). Now that our two stories have merged, it’s time to unravel the mystery. What is so special about the Fibonacci sequence and the golden ratio?
A GOLDEN RATIO QUIZ
There are so many false claims made about the golden ratio, and so many surprising truths, that it can be hard to separate fact from fiction. Here are some of the most common statements you will find in the literature, besides the ones mentioned in the article. See how many you can correctly guess are true or false. (True means known for sure to be true; false means there is insufficient evidence to justify the statement.)
1. The Egyptian pyramids were constructed using the golden ratio.
2. Some Egyptian tombs were constructed using the golden ratio.
3. Some stone tablets show the Babylonians knew about the golden ratio.
4. The cubists based much of their work on the golden ratio.
5. The famous French architect Le Corbusier advocated and used the golden ratio in architecture.
6. The Secretariat building at the United Nations headquarters in New York comprises a stack of three golden rectangles.
7. Some Gregorian chants are based on the golden ratio.
8. Mozart used the golden ratio in some of his music.
9. Béla Bartók used the golden ratio in some of his music.
10. When a falcon attacks its prey, it swoops in along a path that is mathematically related to the golden ratio.
11. The poet Virgil based the meter of his poem The Aeneid on the golden ratio.
12. Some 12th-century Sanskrit poems have a meter based on the Fibonacci sequence (and hence are related to the golden ratio).
13. The golden ratio occurs in certain crystal structures.
14. There is a Fibonacci number with exactly 666 digits.
15. If you square any Fibonacci number, the answer will differ by at most 1 from the product of the two adjacent Fibonacci numbers.
1. F 2. F 3. F 4. F 5. T 6. F 7. F 8. F 9. F 10. T 11. F 12. T 13. T 14. T 15. T
First of all, you need to separate the truly significant facts from the accidental or the spurious. Whether or not the ancient Greeks felt that the golden ratio was the most perfect proportion for a rectangle, many modern humans do not. Numerous tests have failed to show that most observers prefer any one rectangle, and preferences are easily influenced by other factors.
Another spurious appearance of the golden ratio is one that appears in The Da Vinci Code. In his Harvard lecture, Langdon says: “Measure the distance from the tip of your head to the floor. Then divide that by the distance from your belly button to the floor. Guess what number you get.” His answer is φ. Why is this spurious? First of all, you won’t get exactly the number φ. You never can; remember, f is irrational. But in the case of measuring the human body, there is a lot of variation. True, the answers will always be fairly close to 1.6. But there’s nothing special about 1.6. Why not say the answer is 1.603? Or 1.698? More to the point, there’s no reason to divide the human body by the navel. If you spend a half hour or so taking measurements of various parts of the body and tabulating the results, you will find any number of pairs of figures whose ratio is close to 1.6—or 1.2. Or 1.8.
Some artists have flirted with φ, but as with the Greeks and their architecture, you have to be careful to distinguish fact from fiction. The oft-repeated claims (which appear again in The Da Vinci Code) that Leonardo da Vinci believed the golden ratio is the ratio of the height of a perfect human face to its width and that he used φ in his drawing Vitruvian Man, which has a prominent role in the early part of Brown’s novel, seem to be without foundation. So, too, are the equally common claims that Sandro Botticelli used φ to proportion Venus in his famous painting The Birth of Venus and that Georges Seurat based his painting The Side Show on φ. Painters who definitely did make use of φ include the 20th-century artists Louis-Paul-Henri Sérusier, Juan Gris, Gino Severini, and Salvador Dalí; but all four seem to have been experimenting with φ for its own sake rather than for some intrinsic aesthetic reason.
The data on flowers and plants, however—Exhibits A, B, and C—are another matter. Such frequent appearances of Fibonacci numbers in nature cannot be an accident. Although many of those observations were made a hundred years or more ago, it was only in the 1990s that mathematicians and scientists were finally able to figure out what is going on. It’s a question of nature being efficient.
Coming to engineering point of view - to achieve maximum efficiency, flower heads and plant leaves grow in a spiral fashion governed by the golden ratio. Since φ is an irrational number and the number of petals, spirals, or stamens in any plant or flower has to be a whole number, nature “rounds off” to the nearest whole number. Because of the kissing property, this will be a Fibonacci number.
That still leaves two basic questions. Why spirals? And why spirals that follow the golden ratio? In the case of leaves, each new leaf is added so that it least obscures the leaves already below and is least obscured by any future leaves above it. Hence the leaves spiral around the stem. For seeds in the seed head of a plant, nature wants to pack in as many as possible. The way to do this is to add new seeds in a spiral fashion.
As early as the 18th century, mathematicians suspected that a single angle of rotation can make all this happen in the most efficient way: the golden ratio (measured in the number of turns per leaf, and so on). However, it took a long time to put together all the pieces of the puzzle. The final step came just over a decade ago, in 1993, with some experimental work by two French scientists, Stéphane Douady and Yves Couder. Today, then, we have a good scientific explanation of why φ plays such a crucial role in plant growth: It’s the ratio that gives the optimal solution to growth equations. The mathematical explanation is that of all irrational numbers, φ is, in a very precise, technical sense, the furthest from being representable as a fraction.
And so the story ends. Or does it? In addition to bringing the plot to a satisfactory conclusion, a good novel should leave you wondering about some aspect of life. For The Da Vinci Code, the lingering questions left behind are how much of traditional church teachings are fact, and why people are so willing to accept things that are probably not true. Our φ Code ends with two similar questions. The literature on the golden ratio is full of claims that have little or no basis in fact and in some cases are known to be false. Why is it that these myths live on? And why are we so eager to believe everything we are told?
In his book Elements, Euclid shows how to divide a straight line running between points A and B into two by a point P so that the ratio of the longer segment (AP) to the shorter one (PB) is exactly the same as the ratio of the entire line (AB) to the longer segment (AP).
A P B
AB = AP
It doesn’t matter if the actual length of the line AB is one foot, one meter, or one shoelace length. It’s the ratios that count. So let’s say the length of PB is 1 to simplify our discussion.
With PB = 1, the length (x) of AP is what we now call the golden ratio. To calculate its value, we have to do a bit of algebra. The length of AB will be x + 1. This means we can rewrite the above geometric identity as the equation
x + 1 = x
This can be rearranged by cross multiplying to give 1 (x + 1) = (x) x, which becomes x + 1 = x2. We can then rearrange this to give the quadratic equation x2 – x – 1 = 0.
If you think back to your high school algebra class, quadratic equations have two solutions, and there is a formula to give you those solutions. When you apply this formula to the above equation, you get the two answers:
x = 1 + √5 and x = 1 - √5
Using a calculator to three decimal places, the answers are 1.618 and –0.618, respectively. The golden ratio, φ, is the first of these two solutions—the positive number.
You start to suspect there’s more to φ than meets the eye when you ask what happened to the negative solution to the quadratic equation, –0.618, which also goes on forever as a decimal. Apart from the minus sign, it looks the same as the first solution (φ) but with the initial 1 missing. But that turns out to be a false lead. Calculate a few more decimals and you will see that the two numbers are not the same. But if you dig a bit deeper still, you will find a surprising identity. The negative solution is equal to 1 – 1/φ. Hmmm. That doesn’t usually happen with quadratic equations.
My lucky pen
Things do change in life. My first beta parker I had was a silver top one. It had been with me like my alter ego. I started using it the day I left to the US (to be precise on August 8th 05). It got emptied (first time in my life) on May 5th 06. During this 8 month span I had battered it to the fullest extent. In 3 months it lost its silvery color on the top. Its pocket holder pin became loose coz of my continuous pulling of it when I was thinking. Most of my americans identified my pen instantly coz it was the dirtiest looking in the whole department. There was even an occassion in which Morgan asked me to lend this pen and my answer was NO. Let me give u some other pen. Ran to the office (3 floors high) and got one from Penny (office assisstant) and gave it to Morgan. He gave a smirk...
The best part of it is that I had lost it atleast 5 times in the 8 months. Once while studying for the math exam I had dropped it on the floor. All my freinds searched it for nearly 30 mins and dint find it. I was really upset. I said that I shall give $5 for anyone finding it. After 2 days Kirthi gave it me saying that it was in her bag. Whether I gave the money or not is another issue. The last time I lost it was when I was playing soccer in Cooperstown. In that garguantun (loved this word) field there was no hope of finding it. Searched while playing itself for nearly 2 hours. But I never succeeded in finding it. In fact Morgan was pleased in seeing it disappear into thin air. Bless my eye sight after the game when everyone started to leave, it was right there where I am not supposed to search, near the goal post shining in that dusk light.
I was curious my whole 2nd sem and the end of the 1st sem that 'How long will this pen write?' I've been using it for 8 months. Can u believe it 8 months of course work and it dint get dried. I dont know whether u call it coincidence or its intimate contact with me it got over on the last day of my spring semester that too at the end of my Transport exam. I had finished the exam and was going to submit it. I noticed that I havent written my name. I opened my pen cap, took it in my hand as usual. The letters I wrote were 'Hariharasudha', n wasnt visible coz there was no ink. It was instantaneous. It had fetched me new ideas, best grades in my lifetime, great qulities of determination and much more. The way I am going to respect the pen which loved me is by keeping it as my most renowned possession. I had kept it safe in my case. I have seen, heard and read storied in which great people have heartly possessions, for me this pen is the first in my list.
Currently am using the gold top second pen which I bought. I am also making a note of its weight every week to analyse how long it will take to empty it and whats the original weight of ink in the refill. U may call me as a fool, a cracked researcher, or a sentimental idiot, but as per me That Pen Has Been My Best Friend For These Hard Eight Months Of My Life. I wanted to post an image of that pen, but did nt want to do so coz in the future that pen will cost millions when I am famous.
Hoping that the second pen shall do me equal good as the old one. I reverre my first Pen.. I have heard of Penpal, but a pen being the pal it starts from me.....
Amidst all the incomprehensible resuts that I have been getting, there appeared a silverline in my research work - Prediction to the core.... I've been toiling all day long in my 'silence of the dead' Ujjala colored lab for the past 4 months and finally had a chance to test my research. I've had this training on this instrument called the atomic force microscopy (AFM) during my course work itself. But I had problems with it when I had my amateur training. I used it its fullest extent that it sparked on one instance and went dead for the rest of the past 2 months. Finally a ray of hope arrived from my guides mail stating 'Hi Hari, I've contacted Molecular Imprinting and they are sending an expert to train you. PS he is bringing the new heads'
Training as expected was defenitely from the eyes of an expert, but to say the least there were these new students who also desperately wanted to learn the technique. As expected the expert started from basics. Day one was a utter waste of time and energy from my side. Managed to gain some extra space at the end of the days session by coeding George the AFM Janitor to allow me use the machine at 11:00 pm. Thats right its 11 pm and not am. Had my samples ready. Scanned 3 of them till 1:00am. Saved the crappy images in my USB and went home.
The next day (infact I've slept at 2am so its the same day) went a lil late (by 10 mins) and my guide started to leave voice mails in my cell... And what did I miss, Nothing!!... This was worse than yesterday. Two of the new students sat in front of the machine like watch dogs and did experiments on blank samples.
I persuaded them to try my samples on the new setup. Guess what?? The machine crashed instantly.. The expert after 2 hours of his sweaty analysis found out that the guy sitting in front like the watch dog accidentally pulled the power cord for one of the electronics in use. Time was 12 noon. By american standards that's lunch time. But for me its an opportunity to test my sample. Kept it in and scanned. New and inconclusive results. But the images were artistic. If Picasso lived he would have bowed to my art for sure.
1pm new mode of AFM and not my kind of stuff. More for the harcore electronics ppl. Man I wonder how dry can their subjects be. The student was talking greek and latin even to the expert. He was very slithery. Everythin he did was at uncontrollable pace and dropped every damn thing that he had in his hand. Then the expert said something which really attracted me he said "The fastest way to do a thing is to do it slowly"..... How philosophical of him... By 3:30 I finally got a chance to do my very own expert supervised experiment. Did great guns in settin up the machine and making the tip to resonate properly. But the result I got was not that we've been expecting.
To make this blog short - "When I had the time there was no machine, When I had the machine I had no expert, When I had the expert I had no samples, When I had samples I had no proper operator, When I had an operator I got inconclusive results"
The one thing I learnt from this experience is 'Come on man, this is PhD, if u get results as expected at the right time then y do u want to do this for 5 yrs' Patience is a bitte plant but it has a sweet fruit...
Alas I am relived that I had learnt something out of nothing in my life....
How bad can driving be?
May 2nd 06. I had planned to take my road test so that I can get my driving license. My plan was to take the license and go to Indiana for my cousins graduation walk. The day was not usual. Had my exams before and after that day. I had to get up early in the morning. It was pouring heavily and had to call up the kentucky county office to confirm if theres gonna be a road test in this weather. They doubted it and asked me to call after half an hour. Got the approval to go for it from them. Went to Abhay's room so that he can accompany me to the county clerk office. We went to Vivians house and got her keys (car). The one thing she stressed "If anything happens to my car, call me and say it immediately"
Took her car, and went for practice before the road test. Was sailing around downtown. Abhay was happy seeing my driving. It was around 20 minutes before my scheduled appointment and so we decided to go for parralell parking practice. We went into this street where there were 3 cars to my right. Two were together (An SUV and an old jeep) and one standing alone (Wolskvagon, forgot the spelling, in short it was a bug). I was going towards the SUV so that I can do parallel parking practice behind it. Mid way I noticed in my rear view mirror a toyota accord waiting. So I gave way to them by moving behind the SUV. The car drove past me. I shifted to the rear gear and moved back. Then shifted to forward gear and drove in front of the SUV so that I can do my practice. Abhay said that the instructor would ask u to also move back smoothly. So he asked me to do parrallel parking behind the bug about 25 feet away from me.
So I shifted to rear gear and went smooth. Went near the bug, and suddenly noted a red car in my rear view mirror. It was standing in the stop signed intersection. So decided to go forward and park so that they can move. I shifted the gear and moved, but Abhay said that no problem u finish what u were tryin to do. So shifted to rear and went smoothly. Then coz of the impatience I had, I brushed the bug for a second or two. Thats it. First things from my mouth "F*&@k, What the hell was I doing" I got down stood near the bug, Abhay parked the car and came to me and we kept a note on the bug, giving my contact details. We checked the damage and found out that "The bug hardly had a scratch (not even 2 cms). Whereas Vivian's car had some paint removed and it was surely gonna b noted by her. After 10 mins of confidence boost up finally decided to go for the test.
Parked the car and went in. I entered the room where the instructor was. He called me and asked me may i help u. I said " I am here for my scheduled road test"
Instructor: " ur permit plz"
me: here sir
Instructor: what is this?
me: the permit....
Instructor: y r u havin this?
me: coz this was what that was given to me on my written test day
Instructor: this is not the permit
Instructor: this is the result of ur passing after ur written test.
Instructor: u r supposed to give this to the lady sittin in the next room. she shall then take ur photo and give ur permit....
me: then y dint u inform me this earlier on the day of my written test?
Instructor: we thought u knew it...
me: plz inform this to other international students from now on
me: no more dialogues... the face said it all....
then went took my permit and went to vivians house. gave her a call on the way and said her abt the accident. she was defenitely pissed off. she was in her lab. had to keep her car in her house and then went to univ and gave her keys at her lab.... she was fine by the time i reached her....
the rest of the day wasnt that great either...
the one thing which went positive on that day was that everyone loved this episode of mine and that everyone had a very good boisterous laugh... one kid even asked me to drop him. i said u can come in bicycle.. they laughed more.....
Everyone has a bad day...
The FUN PART IS THAT I USED THIS PIECE OF PAPER AS MY PERMIT DURING MY SPRING BREAK. DROVE AROUND 50 MILES. IF A COP HAD STOPPED US I WOULD B IN JAIL...