Friday, 21 September 2018

Perimeter and Area of Regular Polygons - From Triangles to Circles - The Iso-Perimetric Theorem

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The Iso-Perimetric Theorem (IP Theorem; iso - same perimeter) concerns with the relation between the perimeter and the area of two dimensional (2-D) surfaces - plane figures.  For a historic account click here. A rigorous proof of IP Theorem is complex (see also) - I shall present here proofs that only need school level mathematics. 

IP Theorem may be stated as follows:

a. Among all 2-D shapes with the same perimeter, a circle has the largest area; or equivalently - 
Among all 2-D shapes with the same area, a circle has the shortest perimeter.

A short 2 minute video on youtube has a demonstration of the IP theorem.

For polygons (2-D shapes with n sides; n ≥ 3), IP Theorem may be stated as:

b.  For an n-sided polygon (fixed n; n-gon) with the same perimeter, the regular polygon (all sides of the same length) has the largest area.
c.  Among all regular polygons (any n) with the same perimeter, the one with the largest number of sides (largest n) has the largest area.
A circle may be considered a regular polygon approaching infinite number of sides and has the largest area for a given perimeter (points b and c). Statement c is thus equivalent to statement a of the theorem. 

A corollary of the IP Theorem may be stated as

d.  For a regular polygon for arbitrary n, the ratio of its area to its perimeter is equal to half the radius of the inner circle (half its apothem).

About Polygons:  The following two slides list the properties of regular polygons.  They relate the area and perimeter of n-sided polygons to their basic parameters like side length, circumradius and the apothem.  Centre of a regular poygon is the point that is equidistant from all vertices of the polygon - it is the centre of the circumcircle.

(Click on a slide to see full page view, Escape to return to text)
Slide 1

Slide 2
Eq.7 gives us the ratio of the area A to the perimeter of a regular polygon.  For a given parameter P, the maximum value the ratio (A/P) may have, is r/2 and that is for a circle for which n tends to infinity and cos (𝜋/n) = 1. For all other values of n, area A of the polygon is less than Pr/2. I show this in the next slide


Slide 3

We can make an interesting observation, from the graph on this slide:  The ratio of the area and the perimeter of a regular polygon is equal to half the length of the inner circle radius (the apothem). This result is independent of the number of sides in the polygon. Length of the apothem y does increase with n according to eqns. 1 and 2; namely y = r cos(𝜋/n).


Slide 4

We also notice that the circle has the largest area for a given perimeter (A/P approaches 0.5 r as n tends to infinity).
In the above, we have proved statements a, c and d of the isoperimetric theorem. Next let us look at statement b.

Regular n-gons have the largest area for a given perimeter:  To understand this, we use a heuristic approach and look at triangles formed on a chord in a circle with the vertices of the triangles on the circumference. See the following two slides:


Slide 5

Slide 6

 The Equilateral Triangle:  We shall use algebra to show that 
for a triangle (n = 3), for a given perimeter an equilateral triangle has the largest area.  
I shall use the well known Heron's formula for area of a triangle.

Slide 7
Slide 8

The Square:  First we look at the general four-sided polygon (a quadrilateral) and show that the rectangle (all internal angles equal to 90 degrees) has the largest area for a given perimeter.  Then we use the 
 AM-GM theorem (the arithmetic mean of a set of numbers is always greater than or equal to their geometric mean) to show that of all rectangles, a square (a regular 4-gon; all sides are equal and angles = 90 degrees) has the largest area.
Here, I have followed the analysis by Martin Joseffson in Geometricorum 13, 2013, 17-21

Slide 9

Slide 10








An interesting historical remark is that the formula K = (a+c)/2 x (b+d)/2 was used by the ancient Egyptians to calculate the area of a quadrilateral, but it’s only a good approximation if the angles of the quadrilateral are close to being right angles. In all quadrilaterals but rectangles the formula gives an overestimate of the area, which the tax collectors probably didn’t mind.

AM-GM Theorem: I now show that the largest area for a rectangle is for a square when all four sides are equal. 
Slide 11


Notes:  
1.  I have considered convex polygons in the discussion.  For completeness, I present a slide to explain the difference between a convex and a concave polygon.

Slide 12

2.  I find the case when y = 2 very interesting and have prepared a slide to show the numbers for some polygons. Please refer to slides 1 and 2 for notations.
Slide 13
In a circle, the inner and circumradius are always the same.  

3. I have drawn all the slides in this blog.  You can use them but acknowledge the source of this blog as 
http://ektalks.blogspot.com/2018/09/perimeter-and-area-of-regular-polygons.html

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:

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