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Science communication is important in today's technologically advanced society. A good part of the adult community is not science savvy and lacks the background to make sense of rapidly changing technology. My blog attempts to help by publishing articles of general interest in an easy to read and understand format without using mathematics. You can contact me at ektalks@yahoo.co.uk

Sunday, 9 September 2018

The Rise of Tech Companies - Threat to Privacy and Freedom - Cyber Dystopia

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There are two good things in life; freedom of thought and freedom of action.
                                                       Somerset Maugham (1915)
Slavery is the complete and absolute subjection of one person to the control and disposal of another person, by legalized force.  S.P.Chase (1844)

The last fifty years have seen a sea-change in the way digital technology (DT) has affected practically all aspects of our lives.  Thanks to DT, many things we can do now would have been science fiction even a few decades ago.  Improvements in communications, transport, medicine and other areas have palpably transformed the way we live and interact with each other. 

Technological capabilities -  data processing speeds and miniaturisation of equipment - are growing exponentially and have enabled the tech companies to gain position of influence in the market place.  
    
Notwithstanding the positive effects of DT on everyday life, negative aspects of DT have concerned many people. Loss of privacy and decline in sociability have probably been the most serious so far. In the very near future, every aspect of our lives will be recorded and archived - our digital profile will be available at the click of a button.  The question is - who owns this information?  The owner will have enormous power to control and manipulate people to his/her best advantage.  

In this blog, I wish to discuss the current situation and wonder if we have already passed the point when reclaiming our freedoms - freedom of thought, action, decision making, access to unbiased information - is unlikely.  It might already be too late to save the global human society from the enslavement by a handful of tech giants. It is a scenario that is worth exploring. 
Let me first explain the situation:

***********************
You might find the following article of interest too.. (Feb. 2020)

Tech Companies' Market Dominance:   In the capitalist system, company size is important in giving power and influence on policy makers.  The rise of tech companies over the past 20 years has been phenomenal.  For  example in 1997, Microsoft was the only tech company in the top 10 global companies by capitalisation.   It was third with net worth of $160 billion.
The following two slides show the top 10 companies in the world in 2010 and in 2018. Capitalisation is in Billion US $.



Tech companies have moved up the ladder in the past eight years to completely dominate the markets.  Besides Apple, the other six top companies were founded in the past 20 years - this is phenomenal rate of growth by historic standards.  Additionally, the top tech companies are all based in USA or China. 
  
Dominance of the economy by a sector is considered bad for obvious reasons - the companies in the sector carry unduly large influence on the development of free markets. For example, energy companies dominated the market in the 20th century.  Abusing their dominant position, energy companies had prevented and continue to impede the development of new energy sources - they were able to manipulate the energy production and supply to their advantage. The resulting damage to the environment and global climate is well documented.

The financial strength of big companies allows them effective lobbying and influence on policy decisions at governmental level.  They can buy out smaller promising start-ups to stop any disruptive innovations.  The mission statement of traded  companies is that they operate to maximise the return for their shareholders - welfare of the environment and people at large is secondary.    

But Tech Dominance is Different:  Big companies in traditional sectors like energy, transport, finance, food, etc. exploit their strong position by manipulating the consumer choice and behaviour - mostly through advertising, suppressing competition and lobbying for favourable regulatory regime.  The man on the street is affected directly, but the effects are limited in their scope.  We can choose to travel by car, train or plane depending on the convenience and cost.  Essentially, the market is governed by competition among the companies in a sector - the size of the biggest companies might allow them to bend the rules more favourably.  

The tech company dominance is very different from the traditional scenario.  Until about 1990, the role of digital technology (DT) was functional - it helped traditional industries in areas like word processing, automation, messaging and general computing.  Computer manufacturers (IBM) and software providers (Microsoft) grew rapidly. Then, the personal computers arrived - portable and powerful devices. 

In the past 20 years, the Internet has grown into an interconnected spiderweb of billions of personal, government, educational and commercial computers and devices. The world-wide web (WWW) - the greatest collection of knowledge in human history - is the information sharing part of the Internet. 
The Internet is also used for emails, messages, file transfers etc. and has allowed unprecedented advance in communication among private individuals throughout the world.
  

The Internet and IoT result in gigantic amount of data to be created - Big Data.  Big Data is controlled and is available for analysis by the big tech companies.  Essentially, in the near future no aspect of our lives will be private and a complete physical and mental profile of the world citizens will be available to the controllers of Big Data.  Profiling happens just now and one sees that in targeted advertisements, news delivery etc.  Most people use social media and happily provide detailed personal information and do not appear to worry about the loss of privacy.

Smart phones record your location 24 hours of a day, they keep a record of your conversations, messages made to your contacts - not much you do in a day in communicating with others is private any more.  
In not too distant future, mosquito-sized drones will be able to infiltrate any location you are in and can transmit information about all you do.  This is a given in the surveillance society we are already happily marching towards. 

Face-recognition technology is improving rapidly.  In due course, even if you have left the smart phone at home, you can be identified by CCTV mounted everywhere.  You shall not be able to hide. 

Eventually, may be in the next 20 years or so, your thoughts could be read by devices in the vicinity.  Already, it is claimed that your eyes are a good guide and disclose what you might be thinking.  But, the capability to capture electromagnetic signals from your brain activity will be the final link when you lose privacy of thought.  

Enslavement of the Global Population:  Digital technology makes possible enslavement of the global population and the signs of this happening are already visible.  

a.  Manipulation of thought:  Internet has vastly diluted the differences among different cultures.  We are spending more and more hours looking at the Internet and reading/learning the same contents irrespective of where we are.  A child in India does not think much differently anymore than a child in UK.  Sadly, these days parents are too busy to provide wholesome upbringing to their children - but Social Media and Internet are there to help.  Additionally, we are relying on DT to provide us with personal entertainment in the form of music, games etc. Try to spend a day without your smart phone and TV - this is even more acute for children when they are at their most sensitive stage in life. 

Essentially, DT already provides a big chunk of entertainment and cultural education for children and they are very happy to accept material from the Internet in preference to interacting with their peer group.  They are not acquiring the social skills that they would learn in the past by mutual interaction with others in the society.  More seriously, our thoughts and personalities are now open to manipulation by those who control the material on the Internet.

b.  Life in Cyber Space:  Interesting developments are happening in augmented reality (AV) and virtual reality (VR).  Essentially, it will soon be possible for one to experience whatever one wishes - you may be travelling in a space ship, climbing Everest, going on a date with Miss Universe or have best seat to watch your favourite sport. All at the flick of a button - or may be by just thinking about it.  Life in cyber space can be a very happy and pleasant experience.  But powers who control the Internet will use it not only as a marketing tool but also as a reward and punishment strategy.

c.  Cyber Apartheid:  At present 1% of the population owns about 85% of the wealth.  Many of these people are very influential and interestingly they would also own shares in the big tech companies and support how the companies operate.
The rich 1% can also buy cyber expertise.  Essentially, they can afford the most luxurious life style in the real and in cyber space. 

On the other hand, only a tiny fraction of the world population is STEM savvy. Here, we have a situation where most users of Internet and smart phones and other electronic gadgets do not have much money and do not have the know-how to understand the intricacies of what they are using (poor and naive!).  They have also no where to hide in terms of information relating to their physical and thought profiling. Their cultural background and information sphere is being manipulated to a high degree. 
A good fraction of the world population is ready for exploitation and control. Ironically, vast majority of people may still feel happy - they will have good food, good entertainment and good VR friends.  As long as they do what the Internet masters wish them to - and it seems there should be no problem in it - life could be pretty satisfying. They will also not know how the rich live and there would be no reason to feel jealous or unhappy with your lot.  This is similar to how Aldous Huxley envisioned a ruling oligarchy could amuse the masses into submission with a mind numbing drug (Soma) and endless casual sex.   

"... most men and women will grow up to love their servitude and will never dream of revolution" - Aldous Huxley (1949) letter to George Orwell
"None are more hopelessly enslaved than those who falsely believe they are free." - Goethe (1809)

Of course those, who might stray and have independent thoughts, will not be able to hide and fight back to change the system.  They would be treated harshly and eliminated. The reach of surveillance society will be complete.   

Could things work out differently?: I have discussed a scenario on the basis of current state of digital technology and trends.  The current technology can not be un-invented - it is to stay with us and hope to grow exponentially in the foreseeable future. The unknowns mainly relate to the development and impact of artificial intelligence.  It is mooted that within the next 10 to 30 years, AI will reach human-intelligence levels (AGI), also called strong-intelligence. 

Many experts have argued that at some stage, AI will be able to reprogram and improve itself.  A cycle of recursive self-improvement can be very rapid, it will also be unstoppable.  The intelligence explosion will lead to the emergence of super artificial intelligence (SAI) - a technological singularity where SAI may be able to invent/discover anything.  SAI will be superior to AGI what human intelligence is to microbes!

SAI might decide not to support the continued existence of humans and their institutions. Even if we can program, from an early stage, a friendly version of AGI leading to SAI, it is a moot point as to how long before it will reprogram itself.  If you were a million times more intelligent and resourceful - would you wish to preserve Boris J. and Donald T. of this world? 
  
Final Word: This work has come out of some crazy thoughts and I have written it because I do feel that we are sleep walking into some sort of cyber dystopia.  The scenario I have painted might be completely off the point but thinking logically always seems to push me back to it.  
Love to hear from you - ektalks@yahoo.co.uk

Update:  November 2019:  A vivid example of how digital technology is being used in China to control the islamic population in Xinjiang province (Reference:  Nature).  'Mass Internment and Arrest by Algorithm'.  The article in Nature states:

...the AI-powered policing platform purports to predict crimes based on computer generated findings alone.  ...  The system is able to amass vast amounts of intimate personal data through warrantless manual searches, facial-recognition cameras..'.
Through the collection of large-scale data run through AI and machine learning, the algorithm can ... predict ahead of time, places of 'crime' and possible perpetrators. 
With batteries of facial-recognition cameras on street corners, endless checkpoints and web of informants, the 'programme' generates a sense of omniscient, omnipresent state that can peer into the most intimate aspects of daily life.

The example above is sure to be repeated throughout the world.  Governments love control - using big data and suitable algorithms just makes so much sense! 

The above example is using opressive methods of Orwell's 1984.  I think, Huxley's Brave New World had a much better way of controlling people - at least everybody stays happy.

Cyber Dystopia has almost arrived...Good luck.

Saturday, 11 August 2018

A Simple Estimate of Global Mean Sea Level Rise due to Increase in Global Temperatures


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During the 20th century, mean sea-level rose by about 10 cm and is expected to rise at a much faster rate in future. Rising sea-level increases the intensity of severe stroms and  tides causing serious flooding.  Cities and infrastructure near coastlines are vulnerable to damage. Rising sea-level also increases the risk of coastal erosion and shoreline retreat. 

Climate change is given as the reason for the accelerated rise in global mean sea-level and under current scenarios, we should expect a further rise of 60 to 100 cm by the year 2100.  For a lay person, it is difficult to relate a warming climate to a sea-level rise, and how the estimates are made might appear confusing.  

In this blog, I shall calculate sea-level rise due to melting of ice on land and thermal expansion of water. Both estimates may be reliably made using school level arithmatic.  I shall use  numbers rounded to two decimal places to make the calculations look tidy.

Thermal Expansion of water:  Water expands with rising temperature. For each degree centigrade rise in temperature, the volume of water increases by a factor equal to 2.14 x 10-4.   The average sea surface temperature is 17oand during the 20th century the surface water temperature  increased by ~ 0.5oC

The interesting part of this calculation is to appreciate that the water temperature in the sea drops steadily with depth and reaches 4oC
 at about 700 m.  At 4oC, water has the highest density and sea water below 700 m does not change much in temperature - hence does not expand in volume and does not contribute to sea-level rise. 


When the sea warms, it is mainly the layers up to 700 m depth that warm and expand.  We can calculate the increase v in the volume V of water  when the temperature changes by dT.  Since the surface area A of the sea may be assumed constant, the increase h for dT = 0.5oC
 is  
                
                  v = 2.14 x 10-4 x 0.5 x V 
or               h A = 2.14 x 10-4 x 0.5 x 700 A
or               h = 0.075 metres = 7.5 cm in 100 years 

Over the past 20 years, the global warming rate has accelerated and it is estmated (IPCC 2013 Report) that in the next 100 years the sea level will rise by at least 20 cm due to thermal expansion of water (IPCC Senario RCP4.5).

Melting of Ice on Land:  At present, glaciers cover 10% of the land area and store about 75% of world's fresh water (of all the water on Earth, only about 2.5% is fresh water).  Twenty thousand years ago, at the peak of the last ice age, glaciers covered ~32% of the total land area!  
Ice is also present in the sea in the Arctic Ocean; but the  melting of ocean ice, as in the Arctic Ocean, does not change the sea-level.  It is only the land ice that can add additional water to the seas and cause its level to rise.
Ice on land is present in inland glaciers and ice sheets - the Greenland Ice Sheet (GIS) and Antarctic Ice Sheet (AIS). So far, the inland glaciers have contributed the largest amount of  melt-water; IPCC estimate (5th asssessment report 2013, chapter 13) that melting inland glaciers will contribute 12 ± 6 cm sea-level rise by the year 2100. 
  

The Greenland Ice Sheet, GIS covers 80% of Greenland surface and has an area equal to 1.7 million km2, its thickness on average is 2 km.  GIS is losing ice quite rapidly; during the decade 2002 to 2011, the average ice loss was 215 billion tons per year!

Antarctic Ice Sheet, AIS is much bigger.  It covers 98% of Antarctic continent and has an area 14 million km2, and is up to 4 km thick at some places.  It contains 26 million billion tons of ice.  While most parts of the ice sheet are considered reasonably stable, over the past few years, loss of ice from West AIS has been increasing and is currently about 80 billion tons per year.



Before we calculate the rise in sea level due to melting of the ice-sheets, let us note that the Earth has a surface area of 510 million km2. Of this about 71% is ocean - 360 million kmand the land is 150 million km2.



GIS is losing 215 billion tons or 215 x 1012 kg of ice per year (1 ton is equal to 103 kg).  The density of ice is 917 kg m-3Therefore, volume of water produced per year =  215 x 1012/917 or 2.35 x 1011m3.  
This water will raise the level of sea by 2.35 x1011m3360 x 1012m2  = 0.653 mm per year
In 100 years GIS will increase the sea level by 6.5 cm.
However, it is expected that the rate of ice loss will increase significantly due to continued warming of the planet resulting in a much greater rise of global mean sea level due to GIS ice loss. 

The amount of ice lost per year by the Antarctic Ice Sheet is about 80 billion tons per year and will contribute only 2 cm to the rise of sea level over the next 100 years. This might be a gross underestimate of what could happen if our planet continues to heat up.  AIS has large areas of ice plates (shelves) projecting over the sea with their base in contact with the sea water (the grounding line). Warmer sea water is already melting the underside of these plates and adding extra water to the oceans. See slide.




The grounding line also moves inland increasing  the probability of thinned ice shelves breaking and falling in the ocean to raise sea levels. IPCC 2013 report estimates that, by the year 2100,  this (Ice Sheet Rapid Dynamics) can add 10 cm to the sea level rise. 

But what happens if all of the GIS and AIS melt over the next few hundred years.  How much the sea level rise be?

Considering that GIS is on average 2 km thick and covers an area 80% of 1.71 million km2; the volume of water that the ice in GIS will generate is 2.5 million km3; The resulting sea level rise will be 7 metres.  At 250 billion tons of ice lost per year, it will take GIS 10000 years to melt completely; however, the melting rates are expected to increase and many scientists think that GIS might melt away within the next 1000 years. The 7 meter sea-level rise will be catastrophic for our civilisation. 

See Also: https://www.bbc.co.uk/news/science-environment-48337629 
              May 21, 2019 report
See also https://www.nationalgeographic.com/environment/global-warming/sea-level-rise/
                           February 2019
Interesting Read:  https://edition.cnn.com/interactive/2022/04/world/climate-sea-level-rise-iceland-marshall-islands-cmd-intl/ 



Friday, 3 August 2018

Myths and Hypes about the Ubiquity of the Amazing Golden Ratio; Its Relation to Fibonacci Numbers; Logarithmic Spiral, Phyllotaxis and the Pentagram

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'In nature, Golden Ratio (Phi) and Fibonacci Numbers (FN)  are common, probably reflecting the practicalities of life.  In our human world, we might see them where they don't exist, but where they do, we find them pleasing.  Whether we are attracted to them by the mystic of mathematics or the aesthetics they produce is uncertain.'    ... Tim Entwise in Blueprint for Living  

Watch a pleasant 2.5 minute presentation on the Golden Ratio here but do not believe everything that is said in the video. 
Indeed, there is a lot of hype about Phi and FN -  the golden ratio, also known as the golden section or golden mean, is claimed even to connect humans to God!!  (It is claimed that in ɸ, He has crossed nothingness (0) by unity (1) to obtain the symbol for the golden ratio. But note that the symbol ɸ was adopted only recently for the golden ratio!).  It is stated that 'The Golden Section, or Phi, found throughout nature, also applies in undertanding the relationship of God to Creation'.


Many attempts have been made to bring order to the situation regarding hypes and myths about Phi and FN.  Please click here and here.  

Phi is an amazing number with some unique mathematical properties and this is what we shall be looking at in this blog - the aim is to give some examples invoving Phi and FN that have a wow factor - that is what recreational maths is all about.
First, we define the golden ratio and Fibonacci Numbers: The golden ratio is denoted by the greek letter capital Phi ɸ and its inverse by lower case phi φ.  
(Click on a slide to see full page image.  Press ESC to return to Text) 

The following slide is from

Fibonacci Numbers are intimately related to the Golden Ratio and are claimed to occur widely in nature.  I define them in the following slide - a more general and detailed discussion is available in my blog.


First, I would like to show how closely, Phi that is purely geometrical in origin is related to Fibonacci Numbers which are formed from numbers following a simple mathematical prescription.   

The second term in the expression for Fn becomes progressively smaller as n increases, and to a very good approximation Fn increases as ɸ to the power n for n greater than about 10.  
Also, notice that the second term is negative when n is even and positive for odd values of n. For small values of n, Fn oscillates about the values calculated for ɸ to the power n.  This is explained in the next slide:

Puzzle 1:  Climbing Steps:  The puzzle may be stated as follows:

You would like to climb six steps.  You can either climb one step at a time (s1) or two steps at a time (s2).  How many differnt ways can you climb the steps?

One way to solve the problem is to work sequencially 
First Step: - only one way --  1s1; N1 = 1
Second Step:  2s1 or 1s2.  N2 = 2
Third Step: 3s1; 1s1+1s2; 1s2+1s1.  N3 = 3
Fourth Step: 4s1; 2s1+1s2; 1s2+ 2s1; 1s1+1s2+1s1; 2s2. N4 = 5
Fifth Step: 5s1; 3s1+1s2; 2s1+1s2+1s1; 1s1+2s2; 1s1+1s2+2s1; 2s2+1s1; 1s2+1s1+1s2; 1s2+3s1.  N = 8  (6th Fibonacci number) 
Sixth Step:  6s1; 4s1+1s2; 3s1+1s2+1s1; 2s1+2s2; 2s1+1s2+2s1; 1s1+2s2+1s1; 1s1+1s2+3s1; 1s1+1s2+1s1+1s2; 3s2; 2s2+2s1; 1s2+4s1; 1s2+2s1+1s2; 1s2+1s1+1s2+1s1.  N = 13  (7th Fibinacci Number)

Notice the sequence of Fibonacci numbers 1, 2, 3, 5, 8, 13 appears here.  It is now straightforward to calculate the number of different ways you can climb 10 steps.  It is FN at n = 11 --- this is 144.  

Puzzle 2:  Seating Arrangement: 

At a school function with lots of children (C) and adults (A), the seating arrangement requires that an adult (A) must not sit next to another adult.  If there are N chairs then how many different ways they may be seated?

According to the puzzle,  combinations AA are not allowed.  Let us start with one chair

1 chair:   C or A.  N1 = 2
2 chairs: CC; CA; AC. N2 = 3
3 chairs: CCC; CCA; CAC; ACC; ACA. N3 = 5
4 chairs: CCCC; CCCA; CCAC; CACC, CACA; ACCC, ACCA; ACAC.  N4 = 8
5 chairs: CCCCC; CCCCA; CCCAC; CCACC; CCACA; CACCC; CACCA; CACAC; ACCCC; ACCCA; ACCAC; ACACC, ACACA.  N5 = 13
and so on...

Again the sequence of Fibonacci numbers 2, 3, 5, 8, 13,...appears. The number of different ways the visitors may be seated increases by ɸ = 1.618033 each time an extra chair is added.  


An Example from Biology:  Let us look at the family tree of bees. The situation may be stated as follows:
In a beehive, there is one female queen who lays all the eggs.  
If an egg is fertilized by a male bee, then the egg hatches into a female bee.  
But if the egg is not fertilized then it hatches into a male bee (a drone).  
Worker female bees do not lay eggs.

Essentially, a drone has one parent while a female bee has two parents. We wish to map out a family tree for bees.  I have prepared the next slide to show this:
Notice that at generation 8, the number of males, females and also anscestors increase by 1.6154 already.  The increase per generation will be equal to the Golden Ratio (=1.618) for n ≧ 10. 

Interestingly, FN and ɸ appear in the most unlikely places - before I discuss more examples, let us expand on the scope of the golden ratio that has  been defined for a line (one-dimension) so far.  Extend to two or three dimensions and we encounter some fascinating observations.

Golden Rectangle:   The sides of a golden rectangle are in the ratio ɸ = 1.618033.  

Claims have been made that the proportions (aspect ratio) of a golden rectangle are aesthetically most pleasing, and this is reflected in architectural designs; art; paintings; aspect ratio of books, cards and many other objects.  I think it is fair to say that a proprtion around ɸ might be preferred by many but the popular range for aspect ratio b/a of objects is more like ɸ ± 20%.  Similarly, claims about proprtion of human body parts etc. do not stand up to scrutiny.  
However, an aspect ratio based around the golden ratio may have some truth. I shall stick my neck out here and say that when we look at the view in front of us, we see a larger horizontal span but the size of our vertical view is truncated by the ground we are standing on.  This trains our mind to function best when we have a view that is about 1.5 times wider than it is higher - and it is possible that a preference of this kind of aspect ratio might be hard-wired in our aesthetics. 

Fibonacci Spiral:  Also referred to as a Golden Spiral and a source of untold confusion in popular science articles.

If you google 'spirals in nature and design', you will find a large collection of articles about discovering spiral patterns in nature and art with some very nice pictures too.  Mostly it is the logarithmic spiral that one observes but in many publications it is claimed that what we are seeing is the golden spiral with every 90 degrees turn, the size of the spiral increases by the golden ratio (1.618033...).  This generally does not stand up to scrutiny - for instance, a popular example is the size of the nautilus shell.  The shell does grow as a logarithmic spiral but the growth per ninety degree turn is not the golden ratio (1.618033...) but has been measured to range from 1.33 ± 20%  to 1.7.  

I find the logarithmic spiral fascinating and shall discuss it in more detail here.  First thing to note is that Fibonacci spiral has constant 'local' curvature (quarter circles in successive squares) while a logarithmic spiral (golden spiral is a special case of it) has a continuously varying curvature. 
The next slides list some interesting properties and examples of logarithmic spirals:








Did you know that a peregrine falcon, while hunting,  swoops at speeds up to 220 miles per hour (~360 km/hour) - fastest speed of any animal in the world!! 


Golden Angle:  If we divide the circumference of a circle in two parts according to Euclid's prescription (effectively, change the straight line to make the perimeter of a circle) then the smaller of the two angles is called the Golden Angle.




The question is:  For a new primodium to start,  the plant must know where the least crowded spot on the meristem is?  The best location seems to be situated at an angle equal to the golden angle from the last primordium - but how does the plant know this? This is where it gets technical - A full description of phyllotaxis is not yet established and lot of questions remain.   I shall give a  very brief summary. 
Molecular-genetic experiments indicate that active transport of the plant hormone auxin is the key process regulating phyllotaxis.  Auxin is a plant hormone produced in the stem tip that promotes cell elongation.  Organ primordia produce an inhibitory field - depleted local auxin levels - that prevents organ initiation within a certain proximity.
Mechanical stresses from cell deformation at the site of primordia are also hypothesized to play an important role.  
Locating new primordia at exact separation of 137.5 degrees is not understood although some attempts have been made on the basis of packing seeds in a limited space.  

Regular Pentagon and The Petagram:  I discuss these as a pentagram has some unique mathematical properties and the sides and angles of a pentagram are intimately connected to the golden ratio.  Because of its symmetry, a pentagram has been attractive to mathmaticians, designers etc.






Final Note:  I have really enjoyed working on this blog article.  The Golden Ratio is an amazing number and I only wish that people do not hype its relevance to nature and humans too much.  It is a pure geometrical delight and its close relation to Fibonacci Numbers is a great surprise to me - that is why mathematics is so much fun - great feeling of wonder and very satisfying when you have completed a proof.
I am not finished with Phyllotaxis and hope to return to it sometime in future - first I need to find a friendly botanist in the University with some spare time!
By the way, you can look here to see how they even find Golden Ratio and Fibonacci Numbers in the stock market - almost as fantastic as our theology friends.

PS:  The golden ratio springs up in unexpected places - a popular maths puzzle is   6ܑⁿ + 4ⁿ = 9ⁿ.  The value of n is actually related to the inverse of the golden ratio. Amazing!  

Pass the web link to your friends if you have enjoyed reading it.

Friday, 6 July 2018

Derivation of nth Term of a Lucas Sequence - Fibonacci Numbers and the Golden Ratio

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I present a full derivation of the general term Gn of a Lucas sequence, and discuss the cases of the Fibonacci and Lucas numbers.  The Golden Ratio Φ (Greek letter Phi) enters in the discussion naturally and is an integral part of the derivation. 

The unique properties of Fibonacci Numbers and the Golden Ratio have captivated scientists, artists, architects and others over many centuries.  They are abundantly found in nature - particularly in the arrangement of petals and branches in plants.  There are many places where their properties are described but it is difficult to find a derivation of the general term without going to mathematical journals where the subject is treated formally and also much background in mathematics is assumed.  
The presentation here is suitable for someone with a knowledge of school level mathematics.      

I define the Lucas sequence as follows:   A Lucas Sequence is a series of numbers where the nth term is formed by adding the two terms immediately preceding it. 
The series starts from n = 0 and is represented by the recurrence relation:

                            Gn+1 = Gn + Gn-1    for n ≥ 1          ...  eq. 1 

The first two terms, G0 and G1, uniquely determine the rest of the elements of the series.  For example, the choice  
G0 = 0 and G1 = 1, gives the Fibonacci numbers 0,1,1,2,3,5,8,13,21, ...    while 
G0 = 2 and G1 = 1, gives the Lucas numbers 2,1,3,4,7,11,18,29, ...
  
In order to find the nth term, we define a generating function G(x) as a power series in x whose coefficients are the elements of the Lucas sequence.

[Click on a slide to see full page image, press Esc to return to text]


Equation 6 defines the generating function of the Lucas sequence with the first two terms equal to G0 and G1.  
The next step is to expand G(x) in a power series in x.  This is achieved in the following slides:  





Fibonacci and Lucas numbers:  These number series are formed for particular choices of the first two terms 
G0 = 0 and G1 = 1, gives the Fibonacci numbers
 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,  ...

     
G0 = 2 and G1 = 1, gives the Lucas numbers
 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322,  ...

The full expression for the nth term is given in equation 13 and with the particular choices allows us to write: 
Equations 14 and 15 are expressions for general F and L series while equation 16 gives the approximation for large values of n.
We notice the very close similarity of the Fibonacci and Lucas series and their intimate relationship with the Golden ratio.

Extensions:  The recurrence relation in eq. 1 that we had used is
           Gn+1 = Gn + Gn-1    for n ≥ 1          ...  eq. 1

Many different sequences may be constructed by modifying eq. 1.  A general form might be as follows:

           Gn+1 = P Gn - Q Gn-1    for n ≥ 1          ...  eq. 17 

It is also possible to have a term equal to the sum of, say, preceding three terms (the tribonacci series) etc. 

One of the more intriguing extensions is the Random Fibonacci Sequence where P and Q in eq.17 are allowed to take values of +1 or -1 in a  random fashion with probability 0.5. Divakar Vishwanath found that for large n, the sequence increases as (1.13198824...)n with a probability of 1.- a completely counterintutive result.  The number 1.13198824 has been named as the Vishwanath number.  

Final Word:  In this blog I have set out the background to the development of Fibonacci numbers and establish their close relationship to the golden ratio.  I feel the derivation of the general term has a great heuristic value as the method detailed here for the calculation of the generating function for a given recurrence relation may be applied to other cases as well.  I was guided by the book 'Generatingfunctionology' by Herbert S Wilf.

In my next blog, I wish to explore the wonderful worlds of Fibonacci numbers and the Golden Ratio.  These two characters prop up in completely different situations generating surprise and a delightful feeling.  
Congratulations if you are reading this sentence - well done.