Sunday, October 16, 2005

Consciousness and Mathematics

What differentiates us from other species (birds, animals, insects, fish, etc.)?

Is it the look and feel, the intelligence, or the ability to adapt the environment to suit our needs rather than adapting to the environment?

The answer lies in one single word. The most dreaded subject we learned in our school.

"Mathematics"
Mathematics is the language of science. Should we say the language of a highly developed consciousness? It is the only Universal language; you can even talk to an Alien civilization using mathematics.

"An equation means nothing to me unless it expresses a thought of God."
- Srinivasa Ramanujan (1887 – 1920)

"Equations are more important to me because politics is for the present, but an equation is something for eternity."
- Albert Einstein (1879 – 1955)

So, our ability to understand simple counting to complex equations makes us different from other species.
As Ramanujan expressed – "God must be a mathematician then." Math is ingrained in our everyday life even though we are unaware of it.

0101101000100000010010010101011001010111001000000100111101001100
0101010001001100001000000100011101010010010011010010000001010110
0101101001011000010100110010000001001001010101100100100001000110
01001111010001110010000001000111010001100100110101010110

The above zeros and ones look like a lovely pattern. However, it could be your email getting transmitted across the globe. The above data is represented using the binary number system (Computers use the binary number system). There is a hidden message in the above zeros, so see if you can decode the message. 

Clues to decode the message
- I used a cipher, which was used in the biblical times.
- The total number of words in the message equals a Fibonacci number


The following table shows two interesting number sequences in mathematics. These Sequences have a lot of impact on our daily life and our surroundings.

Two interesting number sequences

Fibonacci Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597.......
Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61…........


Now, let us look at some of the wonders/puzzles of mathematics. It includes the Fibonacci Number and golden Ratio, the Number ZeroPi, the Imaginary number, and the Infinite number (read more about these 5 Numbers by Simon Singh). Among these, the Fibonacci number is considered part of Nature.

Fibonacci Series

Leonardo da Pisa (aka Leonardo Fibonacci), an Italian mathematician (1175-1250), introduced the Hindu-Arabic number system to Europe, the method we use today with a ten-digit decimal system with the symbol for zero. He discovered a number series where you add the previous two values to get the new value, later known as the Fibonacci series. He wrote a famous book 'Liber Abaci' on how to do arithmetic using the decimal system in 1202.

Here are some interesting facts about the Fibonacci number related to Nature. Let us look closely at flowers; the interesting point is that the number of petals on a flower is often one of the Fibonacci numbers.

1 petal white calla lily flower, 2 petals euphorbia flower are rare, 3 petals trillium flowers are common, Hundred of species of 5 petals columbine flower in wild and cultivated, 8 petals bloodroot flowers are not common but there quite several species, 13 petals black-eyed Susan. The daisy family's outer ring of ray florets also illustrates the Fibonacci sequence. Daisies with 13, 21, 34, 55, or 89 petals are expected. 21 petals Shasta daisy flower. Ordinary field daisies have 34 petals. (
Read more).

Golden Ratio (Phi = 1.618033988749894….)

If you divide any Fibonacci number by the previous number, you will get a ratio of around 1.618… It occurs in Nature at a golden angle of 137.5 degrees (360-360/phi). Take any leaves on a plant; the growth of the leaves on a stem follows a spiral path to the top at an angle of 137.5 degrees. This angle helps the leaves maximize their exposure to sunlight and minimize the shadow it casts on other leaves. Euclid defined the Golden ratio around 300 BC. Luca Pacioli, the 15th-century Italian mathematician, equated the Golden Ratio with the incomprehensibility of God. The most surprising case for the Golden Ratio is its association with Black Holes, a discovery made by Paul Davies of the University of Adelaide in 1989 (read the article '
The Golden Rule').
The golden ratio governs the nautilus shell's (cross-section is shown on the left side) growth pattern of its chambers. The Greeks incorporated the Golden Ratio into their Art and Architecture – many of their building, including the Parthenon, are considered antiquity's most perfect structure.
Prime Number

Every positive number is either a prime number or a by-product of Primes. However, it's been ages since we hunted for the formula to predict the prime numbers, and the complete procedure to predict the prime numbers still eludes us. Following are the effects of finding the formula to predict prime numbers.
- The collapse of current cryptography (Public / Private Key Infrastructure) standards (used in SSL / TLS).

- The collapse of Internet Secure Communications (Banking and Credit Card transactions)

Nature & Mathematics

After looking at the above 3 Numbers (Fibonacci, Golden Ratio, Prime), it is interesting to know that Nature still hides a lot of information from us or rather gives subtle clues for us to discover the ultimate truth, and mathematics is one of the critical science which helps us to understand the mysteries of the Nature.

So, as a species, our ability to understand Nature is far superior to other species.

What is the next level of our consciousness? Is it nirvana? Or perfection from the scientific point of view? Are there any species that have achieved the next level of consciousness?

Imagine an entire society in a state of nirvana!

A state where science is frozen while the self blends with the Oneness.

"The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore, all progress depends on the unreasonable."
- George Bernard Shaw (1856 - 1950)


Glossary
Prime NumberA Prime Number is a positive integer not divisible without a remainder by any positive integer other than itself and one. Examples. 2, 3, 5, 7, 11, and 13 are Primes.
Mersenne PrimeNamed after the French Monk and mathematician Marin Mersenne, born in 1588. He investigated a particular type of Prime Number 2P-1, where P is an ordinary prime number.
Fibonacci NumberThe Fibonacci number is a series generated by adding the two preceding numbers. Example. 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. Dividing a Fibonacci number by a previous one. For Example., 21/13 or 8/5 results in an answer close to 1.61803, known as a Golden Ratio. 
Golden Ratio1.618033988749894848204586834365638117720……. This unique number represents the Golden ratio, denoted by the Greek letter Phi. The digits of the Golden ratio go on forever without repeating.
PiPi represents the number 3.14159265… which is the ratio of the circumference of a circle to its diameter. An irrational number that cannot be expressed as the ratio of two whole numbers and has a random decimal string of infinite length.



Further Reading
Internet
1. Plus Magazine –
The Life and Numbers of Fibonacci
2. Guardian UK – The Golden Rule
3. Wolfram Research – Fibonacci Number
4. Math Forum – Fibonacci Number and the Golden Ratio
5. American Scientist – Did Mozart use the Golden Section?
6. Web – The Mathematical Magic of the Fibonacci Numbers
7. Web – Fibonacci Number & Golden Ratio in Art, Architecture & Music
8. Web – The Golden Mean in Art (Leonardo Da Vinci)
9. Answers.Com – Fibonacci Numbers
10. Answers.com – Golden Ratio
11. Web – The Golden Ratio
12. BBC – 5 Numbers by Simon Singh
13. Wolfram Research – Prime Number
14. Answers.Com –
Prime Numbers
15. University of Tennessee – The Largest Known Prime
16. Wikipedia – GIMPS – Great Internet Mersenne Prime Search
17. Clay Mathematics Institute – Riemann Hypothesis
18. Answers.com – Pi

Software Products
1. Wolfram Research –
Mathematica 5.2
2. Mersenne.org – Free GIMPS software to search for primes

Books1. Mario Livio –
The Golden Ratio
2. Marcus du Sautoy – The Music of the Primes: Searching to solve the greatest mystery in mathematics
3. Simon Singh –
The Code Book
4. David Wells –
Prime Numbers: The most mysterious figures in Math
5. David Darling –
Equations of Eternity

Tuesday, October 04, 2005

Land of Straight Lines and Beginning of Time

The Land of Straight Lines.

There is a strange planet called Straight Net somewhere in the Andromeda galaxy. So, what is odd or special about it? The land (planet) is built upon objects with straight lines. Which means the planet is a big rectangular land. They play soccer with a cube and rugby with rectangular boxes instead of spherical objects; their body, trees, roads, and animals are built using straight lines.

Now, try to ask straight-lander the following questions. 

1. How do you explain to them about something in a spherical shape (soccer ball, volleyball, etc.)?
2. Can they think about an object which does not have edges?
3. How do you tell them there is a specific shape where you can't specify the object's center (on the surface)?

Here is a conversation with a person from the land of straight lines and a scientist from planet Earth.

Scenario: 

Straightlineman (the person from the straightnet planet) sits on a beautiful couch of flat rosewood. However, he is a little nervous as his new visitor is from another galaxy. What helps their communication is the new intergalactic language translator he bought recently from Salmart stores. His guest didn't find any comfort sitting on the flat wooden sofa either because he knew he would spend some time with the host.

Straightlineman: Hello, you don't seem to belong to this land.
Earthlian: Yes, you are right. I came from a distant place. A place is different than this place.

Straightlineman: Interesting. So, how different is your world? 

Earthlian: Everything looks the same except for one thing. They don't come near me. You are gonna hurt me.

Straightlineman: I don't get it. We are a peace-loving species; of course, there are exceptions. 

Earthlian: You have sharp edges. As a matter of fact, everything in this land is made up of straight lines, which ends up having sharp edges for everything.

Straightlineman: What is so strange about it? Everything in this world is built using straight lines. This whole universe is made using straight lines. Even you are like that.
Earthlian: True. That's because your brain is tuned with the laws of your world. So it rejects things it finds outside that law, and it's projecting my image in a form you can understand easily.

Straightlineman: That's funny. Now you are saying I should not believe my eyes.
Earthlian: You see the mental projection of your brain's interpretation. Your brain interprets much information and filters out information it feels is unnecessary. End of the day, you see what your brain wants you to see. So, even though my image is different, it doesn't look different within the context of your laws.

Straightlineman: OK, OK. No arguments. I need to prove you are right and wrong. However, I would like to understand this new concept rather than try to understand it. (smiles)
Earthlian: Before I explain the new concept, I would like to show you a paragraph (translated to straightnet language by the intergalactic language translator).

"I cdnuolt blveiee taht I cluod aulaclty uesdnatnrd waht I was rdanieg The phaonmneal pweor of the mnid Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is taht the frist and lsat ltteer be in the rghit pclae. The rset can be a taotl mses and you can still raed it wouthit a porbelm.

Tihs is bcuseae the mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe. Amzanig huh? yaeh and I awlyas thought slpeling was necessary."


Earthlian: So, did you notice anything strange?

Straightlineman: Despite the messy spelling, I could read everything correctly on the first attempt. This shows that my mind (brain) interpreted everything correctly when my eyes read it wrong. Now I am getting a feeling where you are going to head. However, I will get uncomfortable with it.

Earthlian: OK, Let us take this step by step. Can you think about a shape that doesn't have any edges?

Straightlineman: Is that a valid question? If you take any shape, let us say a triangle, square, hexagon, pentagon, or any 3-dimensional shape cube, pyramid, anything. Everything has an edge.

Straightlineman: Let us continue this exciting discussion with a cup of coffee. Straightlineman 2
 served two cups of coffee for the stranger and the straightlineman. The cup was made of a cube with the top opened and served in a flat square saucer.

Earthlian: Coffee is pretty good. Let us come back to our topic. Before I answer your query, let me pose another question. Take any of the shapes you mentioned. For example, take a triangle – How many center points (on the surface) will you find?

Straightlineman: Of course, just one central point on the surface, whether it is a triangle, square, hexagon, etc. In the case of 3-dimensional objects, you will find one center point on every side.
Earthlian: So far so good. How about a 3-dimensional object, where you can have infinite center points.

Straightlineman: Infinite number of center points. That's impossible. It is mathematically not possible.
Earthlian: It is possible, however challenging, to explain with the current set of laws.

Straightlineman: By enhancing our laws or with a new set of rules, you can show the mysterious object that may contain infinite center points on the surface.
Earthlian: Yes. Here is a 3-dimensional object made up of triangles.

Straightlineman: Looks interesting. But it still has a lot of surfaces.
Earthlian: Let us add more triangles to this object by reducing the size of the triangles and see what happens?

Straightlineman: It is getting weird. 

Earthlian: Let us add more and more triangles to this object.

Straightlineman struggling to find his breath, he stares at the object and sees a transformation in the visitor. The visitor looks completely different. The visitor has acquired an image that is beyond his imagination.

Epilogue 

Thanks to the magic of triangles ending up as a sphere, in the end, it took great effort to illustrate the concept of the sphere to a person who lived in the Land of Straight Lines.

In the same thought process, is it possible to think about a concept without beginning for Time!

If there is no beginning for Time, then what is the relevance of the concept of past and future?

Are we in a matrix (created by our mind) that imposes the constraint of Time moving in a direction where the past is behind us while the future is in front of us and we live in the present? "
There is no reason to suppose that the world had a beginning. The idea that things must have a beginning is really due to the poverty of our imagination." - Bertrand Russell.


Further Reading

Internet

1. Answers.com - Triangle, Sphere, Geometry
2. Geometry Thru Art -
http://mathforum.org/~sarah/shapiro/triangle.diagonals.html
3. Edwin A. Abbott (1838-1926)
Flatland – A romance of many Dimensions
4. PBS – Time Travel
5. How Stuff Works – How Time Travel will work?
6. Stanford University – Time Travel and Modern Physics
7. Caltech University – Time Travel in Flatland
8. Wikipedia – Grandfather Paradox
9. Absolute Astronomy – Causality (Physics)

* Image courtesy of Intergraph Computer Systems