Research: The Endless Frontier
____________________________________________
Rajendra K. Bera
Chief Mentor, Acadinnet Education Services India Pvt. Ltd., Bangalore
Lecture delivered at the Indian Institute of Information Technology, Bangalore
08 April 2017
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Opening Remarks
It was with immense pleasure that I accepted the invitation to give a talk
on the occasion of this Open House. I thank Professor Sadagopan,
Professor Srinath Srinivasa, and Professor Shiva Kumar Malapaka for the
great honor. I have fond memories of teaching Quantum Computing, and
Intellectual Property Rights at this Institute during the five years (20062010) I spent here. Some of the brightest students attending those
courses are now my cherished friends. This lecture is dedicated to them.
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Research is addictive
What is it about research that people like Galileo, Newton, Einstein, Maxwell,
Planck, and hundreds of others seek, not wealth but fame, by discovering some
nugget of fundamental knowledge about Nature? Their discoveries have
changed the face of human civilization via development of new technologies.
I too am a researcher, bitten by the same bug and addicted to it. By the time I
come to thanking you for patiently hearing me, I hope I would have conveyed
to you some of what I have learnt and why I want to learn more.
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A time when “Truth” was almost grasped
In 1894, the American physicist Albert Michelson said, “The more important fundamental laws and facts of physical science have all been discovered, and these are now
so firmly established that the possibility of their ever being supplanted in consequence
of new discoveries is exceedingly remote . . . Our future discoveries must be looked for
in the sixth place of decimals.”
In 1895, Lord Kelvin (William Thomson, 1824-1907) had confidently said, “heavierthan-air flying machines are impossible”.
By 1905, the known “Truths” about the world were precisely and concisely stated in
mathematical terms, as the next slide shows.
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State of classical physics, 1905
_________________________________________

E 
0
B  0
B
E  
t
c 2  B 
Ref. Feynman’s Lectures on Physics, p. 941.
d
mv
(p)  F, where p 
dt
1 v2 / c2
j
0

E
t
F  G
m1m2
er
2
r
F  q (E  v  B )
Note the remarkable brevity with which all of classical physics is encoded here. The
tomes of physics text books and other literature you sweated through as a student is all
condensed here in mathematical language. If you are sure of your mathematics, you can
independently derive all you have learnt and more from the above equations. Such is
the expressive power of mathematics.
Since the Industrial Revolution (1760-1840) human civilization has thrived on these
equations. Surely, physicists had discovered some “Truths”.
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Cataclysmic changes in physics
Between 1900 and my birth in 1945, the world of physics underwent a cataclysmic
change. By then it had dawned on scientists that “Truth”, whatever one may mean by it,
was not knowable. The main upstarts were Gödel, Turing, Einstein, Planck, and the
Wright brothers.
Kurt Gödel
Alan Turing
Albert Einstein
Max Planck
Orville & Wilbur Wright
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The changes included …
In 1900, Planck, sowed the seed of quantum mechanics.
In 1903, the Wright brothers designed and flew a heavier-than-air, powered aircraft.
In 1905, Einstein showed space and time are entwined and relativity theory was born.
In 1916, Einstein showed that matter warps space and time.
In 1931, Gödel showed that mathematics has limitations.
In 1936, Turing showed that even computations have limitations.
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Know your competitor
But the most remarkable change in technology was in computing. Per U.S. dollar, you
can get about 1018 times more computations done today than in 1900!
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What does a computer do?
A linear sequence, S, of binary bits in a digital computer’s memory (including
all the registers) represents an integer, n. In every clock cycle, some bits flip
and the new S turns into a new integer. That’s it.
A computer changes its state from one integer to another where each integer
has encoded within it a prescription for generating the next integer! And it is
completely mechanizable. When I understood this, I remembered Leopold
Kronecker’s (1823-1891) famous words, “God made the integers, all the rest is
the work of man.”
This binary string can be encoded in many ways and interpreted in many more
ways to carry and convey messages. In fact, a binary string can be viewed as
being in a state of superposed interpretations in a non-quantum way.
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Superposed interpretation!
Here is an example!
How many of you saw a pretty
girl?
So, if the universe is a computer,
there are innumerable ways of
interpreting it.
The encoded integer stands between
you and your employability. You
must know how to interpret, not just
compute.
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We communicate through language
Thoughts get communicated through language and emotion (body language). Benjamin
Lee Whorf (1897–1941) said, “Language shapes the way we think, and determines
what we can think about.” And Ludwig Wittgenstein (1889 –1951) said, “The limits of
my language mean the limits of my world.” Thus, there is a need to heed Albert
Einstein (1879-1955) who said, “A man should look for what is, and not for what he
thinks should be.”
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To express Nature, you need mathematics
“Philosophy [i.e. physics] is written in this grand book, the universe, which stands
continually open to our gaze, but it cannot be understood unless one first learns to
comprehend the language and interpret the characters in which it is written. It is written
in the language of mathematics, … without which it is humanly impossible to
understand a single word of it.” (Galileo, 1623)
“To those who do not know mathematics it is difficult to get across a real feeling as to
the beauty, the deepest beauty, of nature. … If you want to learn about nature, to
appreciate nature, it is necessary to understand the language that she speaks in.”
(Richard Feynman, 1965)
Our formal scientific knowledge may well be constrained by the expressive power of
mathematics as a language, and computing power constrained by known technology.
But, for the moment, the power of mathematics appears awesome.
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The wonder of mathematics
Here is how modern physicists have come to admire mathematics.
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences." (Eugene
Wigner, 1960)
“Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning. … [I]t is impossible to explain honestly the beauties of
the laws of nature in a way that people can feel, without their having some deep
understanding of mathematics.” (Richard Feynman, 1965)
“Our reality isn’t just described by mathematics – it is mathematics … Not just aspects
of it, but all of it, including you.” In other words, “our external physical reality is a
mathematical structure.” (Max Tegmark, 2014)
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Scientific knowledge is unbounded
While I have a lot to share about my fascination with science, on this occasion I shall
restrict myself to one topic—the potent power of iteration in exploring Nature.
By 1934, Karl Popper (and many other scientists, e.g., Einstein) had concluded that
“The game of science is, in principle, without end. He who decides one day that
scientific statements do not call for any further test, and that they can be regarded as
finally verified, retires from the game.”
The game is without end and it is a game of iteration.
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The power of iteration
Popper postulated that humans gather knowledge by trial and error, i.e., by making
conjectures and refutations. Broadly, we guess laws of Nature as follows:
(1) Hypothesize a causal relationship among things that raise our curiosity, and then try
to demolish (refute) the hypothesis, say, by deep analysis or seeking new information
that may demolish it.
(2) Continue the demolition effort till such time the hypothesis shows defects. If the
hypothesis is good enough, finding a defect may take a long time (even centuries), so a
lot of patience and tenacity spanning generations may be required.
(3) When a defect appears (experience shows, one day it will), modify the hypothesis
or make a new one (this usually requires talent, being in the right intellectual
environment, serendipity, or just plain luck) and go to step (2). The loop ends when one
gives up voluntarily or involuntarily.
I will now turn to some fascinating examples of iterative processes studied in mathematics and some of the meanings I attach to them. You may find other meanings too.
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Iterated function systems (The Chaos Game)
_________________________________________
Using iterations, Barnsley came up with something dramatic. Consider the
iterative scheme below, where in each iteration, a, b, c, d, e, f are selected randomly
from one of the four possible rows (1-4) in the table with some probability p.
 xn   a b   xn1   e 
   
   ,
 
 yn   c d   yn1   f 
a
b
c
d
e
f
p
1
+0.000000
+0.000000
+0.000000
+0.172033
+0.496139
-0.090510
+0.010000
2
+0.075906
+0.312285
-0.257105
+0.204233
+0.494173
+0.132616
+0.075000
3
+0.821130
-0.028405
+0.029799
+0.845280
+0.087877
+0.175709
+0.840000
4
-0.023936
-0.356062
-0.323405
+0.074403
+0.470356
+0.259738
+0.075000
Note: p is the probability of selecting a given function a, b, c, d, e, f.
If you omit the first few iterations, and plot xn, yn, what do you expect to see?
Its called the Barnsley Fern. It beats Euclidean geometry hollow! You can
also mimic Euclidean geometry by this method.
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The chaos game (contd)
_________________________________________
Here is an example, called the Sierpinski triangle,
created by its related iterated function system by
Barnsley.
Animation source: "Sierpinski chaos animated" by Zoharby - Own work. Licensed under Public domain via Wikimedia Commons http://commons.wikimedia.org/wiki/File:Sierpinski_chaos_animated.gif#mediaviewer/File:Sierpinski_chaos_animated.gif
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The predator-prey game
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Consider now the iterative function, called the logistic map (studied by various
people before Barnsley),
xn+1 = r xn (1 – xn), r > 0.
It may be viewed as representing the antagonistic dynamics between two classes of
players, say, predator and prey, where in the total population (predators + preys), xn is
the fraction of predators, and hence (1– xn) the fraction of preys at iteration n; 0  x0 
1 is the fraction of predators at the start of play (i.e., players of both sides must be
present). The lone parameter r determines the rate at which the dynamics is driven.
Interestingly, depending on the r value, the result may quite well be settled, dithering
or in a totally confused (chaotic) state.
Inter alia, the map can also be used to understand how a mind struggles to decide
between two alternatives, e.g., when reason and emotion take opposing stands in
voting for a politician or disciplining your son.
The map also depicts the dynamics of a laser under certain circumstances.
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The predator-prey game (contd)
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To capture the long-term dynamics of the map (and not how it gets there), i.e., the
destiny of x, we proceed as follows. Given, xn+1 = r xn (1 – xn), r > 0,
 Choose a value of r, starting with r = 0.
 Choose a seed value x0 at random between 0 and 1.
 Calculate the orbit of x, i.e., the sequence xn for n = 0,1, 2, 
 Ignore the first 100 or so iterates (to exclude the transient part of the orbit) and plot the orbit
subsequent to that (this will represent the long-term behaviour of the orbit).
 Increment r by a small amount and go to step 1 if r  4, else stop.
The plot you get is …
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The predator-prey game (contd)
_________________________________________
… called the bifurcation diagram because of the presence of innumerable 2-pronged
pitchforks in the diagram.
The diagram caught mathematicians by great surprise because of its unexpected rich
complexity. We shall only describe a few of its features.
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The predator-prey game (contd)
_________________________________________
There is no difficulty in intuitively understanding the map from r = 0 to r = 3 when
the first bifurcation occurs and oscillations manifest between a low and a high value
of x, about the x value at r = 3. So, if x represents the poor (rich), then for r > 3, one
not only sees substantial swings in the population of the poor (rich), but also the
swings’ range becomes wider and complex; at r = 4, society is in a topsy-turvy
(chaotic) state. Interspersed in between, there are regions (white spaces) of relatively
less complex swings (e.g., the 3-point attractor which, mathematicians have
discovered, ominously predicts that there will be chaos somewhere in the system’s
dynamics, i.e., at one or more values of r). Such complex behavior comes not from
where the system started (i.e., x0) but the rate r at which the system is driven.
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The predator-prey game (contd)
_________________________________________
Beyond r = 3, this is not only non-intuitive but also unnerving in an economy that is
gathering momentum (i.e., increasing r to produce goods and unavoidable waste, in a
world of finite recyclable resources, to accelerate socio-economic progress). It shows
that very rapid poverty removal programs can throw society into chaos as the poor
cannot become productive overnight. Cycles of prosperity and poverty will inevitably
appear.
A high-speed economy is essentially a difficult beast to manage, grow, and control.
Beyond r > 3, the economy is not only increasingly unpredictable, but increasingly
vulnerable to wide swings in wealth distribution within the economy. It is rather
obvious that rapid advances in multiple technology areas contribute to increasing r.
My guess is that the global economy is moving at r > 3.5.
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The Mandelbrot set
_________________________________________
Mandelbrot created an amazing and the most complex mathematical object known to
man. We now call it the Mandelbrot set, M.
A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales.
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The Mandelbrot set (contd)
_________________________________________
Quite astoundingly, the Mandelbrot set, when magnified enough, is seen to contain
rough copies of itself, tiny bug-like objects (molecules) floating off from the main
body, but no matter how great the magnification, none of these molecules exactly
match any other (see Fig.).
The boundary of M is where a Mandelbrot set computer program spends most of its
time deciding if a point belongs there or not. The simplicity of the iterative formula
and the complexity of the Mandelbrot set leaves one wondering how such a simple
formula can produce a shape of great organic beauty and infinite subtle variation.
Figure by Ishaan Gulrajani, A zoom sequence of the Mandelbrot set showing quasi-self similarity, 01 October 2011,
https://commons.wikimedia.org/wiki/File:Blue_Mandelbrot_Zoom.jpg (Placed in public domain)
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You can visit the site to interactively explore the Mandelbrot set
by zooming various parts of it. Try it!
Image source: The Mandelbrot Set, Malin Christersson ◊◊◊ 2015-04-02
http://www.malinc.se/m/Mandelbrot.php
under an Attribution-NonCommercial-ShareAlike CC license
The Mandelbrot set (contd)
_________________________________________
Beautiful and amazingly aesthetic computer-generated plots can be created by colorcoding points in the escape set depending on how quickly they diverge to infinity.
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The Mandelbrot set (contd)
_________________________________________
Images Courtesy Michael Hease
llhttps://michaelheasell.com/blog/2014/10/14/functional-fractals/
Some more pretty pictures. Exploring the Mandelbrot set is compute intensive, so you
need very powerful computers.
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The Mandelbrot set (contd)
_________________________________________
To me the Mandelbrot set represents, inter alia, the working of the mind. Its infinitely
many variations embedded within itself seems to say that once the mind latches on to
an idea and begins to deeply explore it, it does so by investigating its many variations,
often in a random fashion (i.e., choosing c randomly) but does not abandon the core
idea (the iterated function). On the other hand, if a mind randomly discovers a few of
the dispersed similar looking sets, it begins a search for the mother set, M, itself.
Is it then surprising that researchers often solve unsolved problems through random
exploration based on a hunch (the iterated function), and if they are persistent enough,
a solution finally emerges if the hunch is right? Conjectures & refutations at play.
Image source: Clipart Library,
http://clipart-library.com/clipart/6cp5abXqi.htm
Image source: Pixabay,
https://pixabay.com/en/smiley-happy-face-smilelucky-559124/
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The predator-prey game (contd)
_________________________________________
We now know that a relationship exists between the
Mandelbrot set M and the logistic map. The intersection of M with the real axis is only in the interval
[-2, 1/4]. The parameters along this interval can be
put in one-to-one correspondence with those of the
real logistic map and is given by
Take some time to ponder over it after this lecture as to how mathematics links ideas.
Figure by Georg-Johann Lay, Diagram showing the connexion between Verhulst dynamic and Mandelbrot set, 07 April 2008,
https://commons.wikimedia.org/wiki/File:Verhulst-Mandelbrot-Bifurcation.jpg (Placed in public domain)
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Benoit Mandelbrot
Benoit Mandelbrot (1983) observed, “Clouds are not spheres,
mountains are not cones, coastlines are not circles, bark is not
smooth, nor does lightning travel in a straight line.”1
Before Mandelbrot, Nature was generally regarded as noisy Euclidean geometry; e.g., a
mountain is a roughened cone. Paul Cezanne instructed his young painters:
“Everything in Nature can be viewed in terms of cones, cylinders, and spheres.” Surely,
we beg to differ today.
1 Mandelbrot,
Benoît B. (1983). The Fractal Geometry of Nature. San Francisco: W.H. Freeman, p. 1.
Mandelbrot photo: http://www.math.yale.edu/mandelbrot/Templates/downloads/Mandelbrot.jpg
Cezanne quote: http://classes.yale.edu/fractals/
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The seductive power of mathematics
Mandelbrot had also observed:
Being a language, mathematics may be used not only to inform
but also, among other things, to seduce.
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Mandelbrot extended
_________________________________________
Here is a 3D version of the Mandelbrot set
created by using distance estimate data to
create a height map.
“[Distance estimate] is calculated by taking
the running derivative of the function
[rather than escape time]. It produces small
values for points very close to the Mandelbrot set, and larger values for points further
away. Distance estimate values are floating
point values,  Escape time plots tend to
be much more “spiky”, ”
Here are some more mathematically “seductive” pictures.
Figure and quote from Duncan C., Golden Mandelbrot landscape, 12 December 2009, at http://www.fractalforums.com/images-showcase-(ratemy-fractal)/golden-mandelbrot-landscape/
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Mandelbrot extended (contd)
_________________________________________
Olbaid-ST, http://olbaid-st.deviantart.com/art/Mandelbrot-84-Timeless-wisdom-417708138 (L)
Olbaid-ST, http://olbaid-st.deviantart.com/#/art/Mandelbrot-61-Something-to-keep-in-mind-359602977?hf=1 (M)
Pharmagician, http://olbaid-st.deviantart.com/art/Tumbleweed-211094173 (R),
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Mandelbrot extended (contd)
_________________________________________
All of these can be turned
into 3D objects using 3D
printing. Now THINK!
Kattvinge, http://kattvinge.deviantart.com/art/December-Orbs-2-params-342924324 (L)
Kattvinge, http://kattvinge.deviantart.com/art/Ran-21c-params-318244165 (M)
Kattvinge, http://kattvinge.deviantart.com/art/Lilith-29-params-293859692 (R),
Aren’t we all 3D objects
created by some iterated
function system called the
Laws of Nature?
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So it is this iterative exploration of the
world that truly fascinates me!
With this I thank you for your attention.
Thank you!
________________________________________________________________
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