1
Summer 2024
THE INFINITY
Dear Readers,
EDITOR IN CHIEF
KRTIN GOEL
CHIEF OF PRODUCTION
ANANT JAIN
SENIOR EDITOR
Over the past few years, in my capacity as an editorial board
member of the Infinity, I have not only had the privilege of
deepening my own mathematical passion but also honing my
skills for expressing ideas succinctly and impactfully. However,
as we curated articles for this issue,I couldn’t help but notice a
concerning trend that seemed to be common across all forms.
Unlike the time in my junior forms, it is not the inquisitiveness and
genuine passion that fuels article writing, but the desire to secure
a position in the Editorial Board just to add to their list of titles in
school. From witnessing completely AI generated and plagiarised
content to articles lacking authenticity where the writers seemed
to struggle with understanding their very own topics, it was clearly
evident that the essence of curiosity and exploration was missing.
CORRESPONDENTS
Despite striving for excellence, at Infinity, we always assert that
our primary purpose as a mathematical publication is to foster
mathematical thinking within the community rather than preparing
a flawless issue where our very writers fail to comprehend the very
content they are delivering to the rest of the readers. Therefore, as
a final piece of advice, for all juniors reading this, serving as an
editorial board member in any publication is helpful particularly
due to the essential skills you embark and the knowledge you
gain through your experiences, not because it may constitute
a few characters of the Statement of Purpose or the Common
Application that you are going to submit few years down the line.
MASTERS IN CHARGE
Finally, I would like to express my heartfelt gratitude to this issue’s
editorial board for their tireless efforts, commitment and support,
especially in the final days before the release. Had they not
worked the way they did as a team, we would have never been
able to produce an issue which was anything similar to what we
had envisioned from the very beginning.
DESIGN
It has been my utmost pleasure to work with all of you on this
issue and I can proudly say that this has been an experience that
I would not substitute for any other.
ARJUN MITRA
DIVYAANSH SURANA
SENIOR CORRESPONDENTS
ARNAV TIWARI
HARSHIL MAKIN
KAI KUBO
KRISH AGRAWAL
SHRIKRITT JAJOO
ANSHUMAN GUPTA
AVIGHNA DARUKA
HARSH AGARWAL
KRISH GHAI
SHREEANSH BANSAL
MR CHANDAN GHUGHTYAL
MR GYANESH PANDEY
ARYAN AGARWAL
RAJVEER AGARWAL
SPECIAL THANKS
MR ANJAN KUMAR CHAUDHARY
MR MANU MEHROTRA
Warm Regards,
Krtin Goel
Editor-in-Chief
The Infinity
3
Pi Day card Making Competition Winners 2024:
D Form
1st Position: Abhir Kohli 706 T
2nd Position: Samik Garg 668 K
3rd Position: Raghav Agarwal 693 K
C Form
1st: Advay Goel 27 H
2nd: Aryaman Agarwal 201 H
3rd : Shivam Shamanur 37 T
33 Mathematical
THE INFINITY
Marvels in AI Art
CONTENT
AARAV ANAND
35 The
Mathematical
Puzzle of the
Rubik’s Cube
ABHINAV DANGWAL
37 Tessalations:
A timeless
pattern
AADI TULSYAN
07 Cryptology: The
Art of Secrecy
09 A tool to
ARNAV TIWARI
venture into
the unknown
AVIGHNA DARUKA
11 Hodge’s
conjecture
SHAAKTAM
13 The enigma
of unproven
truth
ANSHUMAN GUPTA
31 Transformative
Mathematics in AI
SHRIKRITT JAJOO
39 Infinity
society’s Pi
day report
AADI KHANDELE
41 Unravelling
Connections:
An introduction
to graph theory
MR GYANESH PANDEY
43 What a magic
square is
MR CHANDAN GHUGHTYAL
45 Mastering
report writing
skills
MS ABHA DEB
5
17 Timeline
ARNAV TIWARI
23 Millenium
Prize Problem
ARNAV TIWARI
25 Reimann
Hypothesis
ANSHUMAN GUPTA
27 Darpa Maths
Problem
AVIGHNA DARUKA
29 Unlocking
Dimensions
SHREEANSH BANSAL
SPECIAL
SECTION
THE INFINITY
Introduction
Cryptology, which is known as the science
of secrecy, has blended mathematics,
computer science, and engineering to
facilitate a safer and more secure world –
from securing communication channels
to protecting sensitive information, it
creates messages whose meanings are
hidden to the common eye. According
to Eric Conrad in his book - the ‘Eleventh
Hour CISSP,’ our entire cyber system
is dependent on the field, whether it
be encryption for online chatting or
allowing secure transfer of confidential
communications.
The mathematical foundations of
cryptology use numerous mathematical
algorithms and modular arithmetic
programs, making it pervasive in many
aspects of life, including secrecy
and safety precautions as seen in
cryptography.
Fundamentals
Mathematical techniques shape the
foundation of encryption and decryption
algorithms. However, to grasp the
understanding of these algorithms, a few
basic terms are to be established. Firstly,
the plaintext is the unencrypted form of the
message, while the ciphertext pertains to
the encrypted (algorithmically complex)
form of the message. While a text is
being Encrypted, plaintext is converted
to ciphertext, while decryption converts
ciphertext back to plaintext. These
antagonistic processes ensure that the
encoded message remains confidential
and indecipherable to anyone without the
ability to decipher the algorithm (done
with a key). These keys are generated
using modular mathematics, where
encryption is done using huge prime
numbers, since factoring huge numbers
into prime factors is computationally near
impossible, encryption in this way makes
it computationally difficult for an attacker
to retrieve the key.
Algorithms
One of the earliest, and perhaps simplest,
algorithms that have been used to
transform plaintext into ciphertext, and
vice versa, is the substitution cypher,
where each letter in the plaintext is
replaced with another letter. For the
English alphabet with 26 letters, 26!
possible different substitutions exist,
which translates to a gargantuan 40
3,291,461,126,605,635,584,000,0
00 permutations. However, with the
advent of supercomputers, decoding
these substitutions is easy per se.
“ Transposition cyphers, on
the other hand, are used to
rearrange the order of letters.
This happens without the
need to alter the actual letters
themselves. ”
Cryptology:
The Art of
Secrecy
ARNAV TIWARI
So, cryptographers have developed more
complex algorithms to overcome the problem.
Transposition cyphers, on the other hand, are
used to rearrange the order of letters. This
happens without the need to alter the actual
letters themselves. Rail Fence cypher is a
classic example of transposition cypher, where
plaintext is written diagonally on successive
“rails’ ‘ of an imaginary fence.
Thirdly, Elliptic Curve Cryptography (ECC) is
another branch of public-key cryptography (in
which separate keys are used for encryption
and decryption), where algebraic structures of
elliptic curves are employed over finite fields
of cryptographic purposes. It has a multitude
of applications, from computational resources
to bandwidth and storage. These uses are
applicable in mobile phones, and smart cards–
all of which are part of Internet of Things (IoT)
devices. Moreover, it ensures security at a
higher level, achieved via smaller key sizes
than bigger cryptosystems such as RSA.
Zoom-in to the RSA Algorithm
The RSA algorithm (abbreviation for Rivest,
Shamir, and Adleman–the three founders)
is reliant on the burdening task of factoring
large composite numbers. A more simplified
instance of generating key and encoding
cyphertext is as follows:
For instance, choose two prime numbers. Let’s
say, p and q. Now, compute the product of
these two numbers, N (N = p × q). This number
N would serve as a modulus for both the keys,
public and private. Then, select e, which is
an integer such that 1 < e < φ(N). φ(N) is
Euler’s totient function, a count of the positive
integers from 1 up to N-1, who are prime to N,
relatively. Now, take d, which is the modular
multiplicative inverse of e modulo φ(N). We
take (N, e) as the public key, while (N, d) is used
as the private key. To encrypt the plain text,
convert it into an integer m such that 0 ≤ m ≤
N. All that remains to do is compute the cypher
text c as c ≡ m^e (mod N). However, on the
other hand, compute the plaintext message m
≡ c^d (mod N) to decrypt.
Conclusion
The point where science, technology and
mathematics converge–as shaped by
countless mathematicians and computer
scientists–takes
shape
into
secure
mechanisms to safeguard the internet.
Pure Ingenuity. The world has become
interconnected and as a result, there is a greater
need for more complex forms of cryptography
that will be able to overcome the challenges
posed by quantum computers and advanced
technology. From ancient substitution cyphers
to modern-day asymmetric encryption, the
evolution of cryptology extends to underscore
the importance of preserving confidentiality
and privacy.
7
THE INFINITY
Before the advent of digital navigation
and satellite systems ,humans and global
navigators employed clever and accurate
mathematical tools to navigate precisely
through their surroundings. From triangulation
to celestial navigation, mathematics in
navigation was prominent and deemed
accurate while simultaneously inexpensive,
effective, and accessible. Bartholomew Diaz
for example, used a trusty tool called an
‘Astrolabe’ to discover the cape of good hope,
something that has proved pivotal for the
furtherance of our society as a whole. These
tools and mathematical navigation calculations
spearheaded human advancement into the
unknown world helping them to reach places
which we certainly could not have had reached
in centuries to come.
To understand how modern-day navigation
works, the principle of trilateration must be
explained at length to leave minimal room
for doubt. Trilateration is when the distance
between three satellites is used to determine
the geological location of a point. Imagine
you are standing someplace on Earth. With
only one satellite, you’d only be able to tell
where you were in a circle with a much greater
radius, making positioning not very reliable.
However, given a signal from a second satellite,
you may narrow down to where the two rings
surrounding that satellite intersect. Finally, a
third satellite condenses everything down
to a single point, which is your location. The
distances are calculated by measuring how
long it takes the signal to travel from the satellite
to the receiver. This is possible because the
speed of light is known.
This is a very simple explanation of how
communication satellites are used to display
your Global Positioning Systems Location
extremely accurately and effectively as this
process utilises only a fraction of seconds to
conclude your location regardless of the relief
features around you or any other factor.
Fig 1.1
Triangulation, on the other hand, is a much more
manual process which is used to determine the
distance between two points. Sailors rely on
celestial bodies or distant objects visible on the
horizon. A navigator can compute the vessel’s
latitude and longitude by calculating the angles
between these known positions and the ship’s
location, using trigonometric principles. After
calculating the angle between these known
reference points, the distance between a third
reference point can be calculated using either
the sin function or whatever is appropriate for
the particular situation.
“ Mathematics has long been
deemed the queen of science,
but its usage in our day-to-day
lives has often been overlooked
and given a bat of the eye.”
A tool to
venture into the
unknown
AVIGHNA DARUKA
Fig 1.2
But apart from the principles of triangle
geometry, mathematical tools such as a
‘theodolite,’ ‘astrolabe’ and an ‘atlas’ have
also been employed to help determine the
position of an object. Let us understand how
these tools operate and why they were so
pivotal in determining the effectiveness and
efficiency of these other principles.
First, the theodolite is a tool employed in
triangulation by navigators to calculate the
angle between two points. It measures the
vertical and horizontal rotation of a moving
magnifying telescope while stabilising
its viewport using a ‘bubble leveller.’ It is
extremely accurate while measuring the
angle between two known points, but
errors can occur frequently, as at that time
the theodolite rotation was not always
perpendicular to the reference plane, leading
to exacerbated errors and uncertainty in
distance calculations.
As mentioned before, Bartholomew Diaz
used the astrolabe to navigate the oceans.
In explanation, the astrolabe has a disc with
markings depicting the positions of stars
and constellations. It also includes a rotating
ring that displays the various times of day
and night. By looking through the astrolabe’s
sight and aligning it with a star, a navigator
may use the markings and ring to estimate
the star’s position in the sky as well as the
time of day. This uses the same principle of
the heavenly bodies to discover the position
of an unknown object but it is slightly more
mechanical in its workings.
Mathematics has long been deemed the
queen of science, but its usage in our dayto-day lives has often been overlooked and
given a bat of the eye. Due to the rise of the
digital world, everything seems simple and
not overly complicated, but if the details of
the situation are understood, we can simply
conclude how mathematics at its core,
controls our paradigm.
9
THE INFINITY
Mathematics, as we know, is a landscape
of ideologies and beliefs. What defines
mathematics is the work done by the erudite
members of its diaspora available to those like
us who seek to learn, in the form of theorems,
laws, observations and conjectures, each
different from one another. Among these,
the Hodge Conjecture looms large, with its
elegance and profundity enticing modern
mathematicians for the past century.
The Hodge Conjecture has captivated
generations of mathematicians, defying easy
categorization and offering glimpses into the
deeper structures of mathematics.
First formulated by W.D Hodge in the 20th
Century, the conjecture suggests that there’s a
significant relationship between the algebraic
and topological properties of certain varieties
in algebra, which are defined by spatial
polynomial equations.
Algebra and geometry are two separate fields
altogether but what unites them are complex
algebraic varieties. The ground these work
on involves polynomial equations being the
genesis of rich algebraic structures. Topology
explores the global properties of spaces, avidly
focusing on the notions of continuity and
deformation.
Firstly, we must understand what cohomology
is all about to truly have a perspective of
what the conjecture is. Imagine you have
a shape resembling that of a doughnut,
and you want a way of measuring holes
in that shape. If you have a doughnut, you
can stretch and bend it, but you can’t turn it
into a ball without ripping it because of the
hole in the middle. Cohomology provides a
distinguishing framework for studying the
geometric and topological properties of such
spaces. Cohomology groups capture essential
geometric information by associating algebraic
objects, referred to as ‘cohomology classes’, to
subspaces of a given manifold. These classes
encode essential features such as holes,
handles, and higher-dimensional structures,
enabling mathematicians to discern the
underlying shape of a space.
When we talk about the Hodge Conjecture, we
must refer to a specific type of cohomology
classes called ‘Hodge Classes,’ which
admit a particularly elegant representation,
but lay its framework within the basis of a
technique strengthened by differentiation.
These differential forms, which capture
infinitesimal geometric properties, serve as
the bridge between algebraic and topological
perspectives. As per Hodge’s observations,
every Hodge class can be represented by a
linear combination of algebraic cycles, thereby
establishing a deep connection between
the algebraic and topological structures of
complex algebraic varieties.
Essentially, making the Hodge Conjecture
links abstract mathematical concepts and
concrete geometric intuition. Its implications
make sense throughout conventional
arithmetic, offering insights into diverse fields
from number theory to theoretical physics.
“ The Hodge Conjecture has
captivated generations of
mathematicians, defying easy
categorization and offering
tantalising glimpses...”
Hodge’s
Conjecture
SHAAKTAM
Now as unique and special as it may seem,
what benefit does Hodge’s Conjecture have
within the status quo? Firstly, it sheds light
on questions of moduli spaces, deformation
theory, and the birational geometry of
higher-dimensional manifolds. Moreover,
the conjecture intersects with arithmetic
geometry, providing crucial tools for studying
the arithmetic properties of algebraic
varieties over number fields.
In physics, the Hodge Conjecture plays a
pivotal role in string theory—a framework for
defining quantum mechanics and general
relativity. Complex algebraic calculations,
known as Calabi-Yau manifolds, serve as
the geometric backdrop for string theory,
providing a conceptual idea as to how the
vibrational modes of strings propagate.
The Hodge Conjecture offers profound
insights into the geometry of Calabi-Yau
manifolds, illuminating the intricate fabric
between geometry, topology, and quantum
field theory. However, it does not limit itself
to purely mathematics or physics. Even in
computational biology, the Hodge Conjecture
has its role to play in helping us understand
the geometric properties of biomolecular
structures. Techniques derived from algebraic
geometry, such as algebraic statistics and
computational topology, are employed to
analyse the shapes and configurations of
biological molecules. The used techniques
from algebraic geometry and calculus are
employed to model the dynamics of asset
prices and examine the risk profiles of
investment portfolios. Financial analysts can
develop more sophisticated risk models and
hedging strategies, mitigating the impact
of market fluctuations and enhancing the
stability of financial systems, all with the help
of Hodge’s conjecture.
Despite decades of intense scrutiny, the
Hodge Conjecture remains as one of the most
elusive unsolved problems in mathematics.
Its resolution promises to unlock new vistas of
understanding, unveiling deeper connections
between algebraic geometry, topology, and
theoretical physics. As mathematicians
continue to grapple with the enigma of the
Hodge Conjecture, they remain guided by
a shared sense of wonder and curiosity,
perennially mindful of the transformative
power of mathematical inquiry.
11
THE INFINITY
Mathematics
has
always
bewildered
mathematicians and seasoned experts with
curious phenomena such as unprovable
theories. These theories provoke and shake the
foundation of mathematics and question the
deep hole existing at the subject’s core. Many
prominent examples of such baffling theorems
have implications for the world of mathematics.
Firstly, the Twin Prime Conjecture. It is a puzzle
challenging the very core of arithmetic. The
theory proposes that there are infinite pairs
of prime numbers that are precisely two units
apart, like 5 and 7, 17 and 19, and more. This
hypothesis has intrigued mathematicians for
centuries and is one of the oldest unsolved
problems arising in the discipline of arithmetic.
Even after extensive research on the theory
it could not be proved right for numbers till
infinity. The significance of the Twin prime
conjecture extends far beyond what most of
you think. Despite a mere logical repetition of
twin prime numbers, great preachers like Euclid
himself couldn’t find justice to this problem.
Mathematicians have approached this
conjecture using techniques such as number
theory and algebra, but no prominent progress
has been visible. This conjecture emphasizes
the complexities of prime number behaviour.
While the hypothesis currently defies formal
proof, mathematicians see it as a testimony to
the wonders of number theory and continue to
pave the way for a concrete explanation.
Next, dealing with infinity has always been a
concept that defies conventional mathematical
understanding. Simply put, Infinity represents
a boundless quantity, one that cannot be
expressed in terms of numbers. In the late 18th
century many mathematicians put forward the
idea of how ‘some infinities would be larger
than other infinities’. This statement shook
the mathematical world and was coined The
Greater Infinity. Some phenomenal work has
been produced on this topic and one of the
most respected works has been given by
Georg Cantor in the 19th Century. He titled his
work as the ‘Diagonalization Proof’. With his
work, he proved half of the problem that infinity
indeed cannot be defined and expressed in
terms of numbers. This was the foundation
stone for the concept of infinity.
His work said that there are infinite numbers
between the numbers 0 and 1. To prove this he
first listed down as many numbers he could,
lying between 0 and 1 which consisted of as
many digits as he could imagine. Then came
the turning point, to create a new number he
took the first digit from the first number, the
second digit from the second number and so
on until he did this till the last number, now
he added one to all of the digits to create a
new number. This ensured no repetition as
the number was different from all others in at
least one way. This method could be continued
forever and never stop. This in turn proved
the existence of infinity but what it could not
prove was how one infinity could be greater
than another infinity. To this day this question
remains a profound mystery with no answer
question.
Finally, Gödel’s Incompleteness Theorems
are subject to forever altering our perspective
of the foundations of mathematics. Kurt
Gödel introduced these theorems, going
completely against the flow of the field and
shattering the dream of a complete and
consistent algebraic system for mathematics,
revealing limitations within their reasoning.
“ The incompleteness theorem
asserts the existence of true
statements that are impossible
to prove. These statements
can exist in any mathematical
system capable enough to...”
The enigma
of unproven
truth
ANSHUMAN GUPTA
The incompleteness theorem asserts
the existence of true statements that are
impossible to prove. These statements can
exist in any mathematical system capable
enough to express the basic properties of
numbers. Gödel’s proof of this system was
constructing a number system of his own
known as the Gödel numbering. This system
used the paradox of self-reference to devise
a statement that essentially established its
own unprovability in the system. So just by
exploiting the encoding Gödel got to the fact
that true statements themselves sometimes
cannot be proven right.
Using Gödel’s theorem itself proves how it is
not possible to prove the proof of some true
statements in Mathematics. The existence of
these unprovable theories makes us question
the integrity and logical reasoning behind
mathematics. Are there truths that lie beyond
proof? Can we ever hope to fully comprehend
the complexities of mathematics? Questions
like these continue to be the driving force
for the exploration of knowledge. As said by
the great mathematician David Hilbert, “We
must know, we will know”, and with this, the
journey towards unwrapping the mysteries of
mathematics continues.
13
Special Section
THE INFINITY
The journey of mathematics through time is a
testament to humanity’s relentless pursuit of
understanding the universe’s intricacies. From
ancient civilizations like the Babylonians and
Egyptians, who developed rudimentary arithmetic and geometry for practical purposes
like land measurement and commerce, to the
groundbreaking discoveries of figures like Pythagoras, Euclid, and Archimedes in classical
antiquity, mathematics evolved as a tool for unlocking the mysteries of the cosmos. The Middle Ages saw the preservation and expansion
of mathematical knowledge by scholars like Fibonacci and al-Khwarizmi, paving the way for
the Renaissance, where visionaries like Descartes and Newton revolutionized mathematics with their contributions to algebra, calculus, and physics. The 19th and 20th centuries
witnessed the rise of abstract mathematical
theories, such as set theory, group theory,
and topology, reshaping our understanding of
the fundamental structures underlying reality.
Today, mathematics continues its journey, intertwining with technology and driving innovations in fields as diverse as cryptography,
artificial intelligence, and theoretical physics,
reaffirming its timeless significance in shaping
the course of human progress.
15
THE INFINITY
Invention of Calculus:
The Timeline
1637
1675
Pierre de Fermat developed
methods to find maxima, minima,
and tangents, known as ‘adequality’.
Meanwhile,
Rene
Descartes
introduced analytical geometry
which developed studies behind
the employment of the coordinate
system in data analysis.
Gottfried
Le
differential ca
the notation o
his published
He introduces
equations (ODE
1668
Isaac Barrow published his works
on tangent lines and areas under
curves, further clarifying the
fundamentals of calculus.
17
Antiquity
Ancient Greeks, such as Eudoxus and Archimedes, laid the groundwork
for integral calculus, while ancient Indians developed early concepts,
as seen in works like “Sulba Sutras”, containing geometric methods for
finding areas.
1704
eibniz
developed
alculus, introducing
of calculus through
findings in 1684.
ordinary differential
Es) as well.
Formalization
Early 18th
Century
1687
Isaac
Newton
publishes
“Philosophiæ Naturalis Principia
Mathematica,” where he presents
his method of fluxions (differential
calculus), introducing the laws of
motion and universal gravitation.
Cementing the birth of calculus.
Isaac Newton published “Opticks,”
containing the method of fluxions
where he discusses differential
calculus in depth
THE INFINITY
1734
Leonard Euler publishes “Introductio in
analysin infinitorum,” which serves as
a comprehensive textbook to calculus,
Euler’s contribution to the development
of differential calculus provided powerful
tools for solving practical problems across
scientific disciplines, simultaneously
revolutionising mathematical theory.
1715
Brook Taylor publishes “Methodus
Incrementorum Directa et Inversa,”
introducing the Taylor series.
Something that proved to be
pivotal in the approximation of a
function.
Expansion
Refineme
Late 18th
Century
19
1786
Pierre-Simon Laplace publishes
“Théorie
Analytique
des
Probabilités,” applying calculus to
probability theory.
n and
ent
1768
Joseph-Louis Lagrange introduces
the calculus of variations.
THE INFINITY
1874
Georg Cantor introduces set
theory, which became crucial in the
development of modern calculus
and analysis.
Further
Development
Computa
and Mod
Advance
20th Cen
Present
19th
Century
1854
Bernhard Riemann introduces
Riemann integration, a more
general form of integration than the
Riemann sum.
21
1950s
ational
dern
ements
The advent of digital computers
revolutionises
the
numerical
solution of differential equations.
ntury-
1920s
John von Neumann and others
develop numerical methods for
solving differential equations,
leading to the field of numerical
analysis.
1980sPresent
Advances in computational
mathematics and software
lead to the widespread
use of numerical methods
for
solving
differential
equations.
THE INFINITY
MILLENIUM
PRIZE
PROBLEM
THE FIRST PROBLEM IS THE
BIRCH AND SWINNERTON-DYER
CONJUCTURE
AN ARTICLE BY HARSH AGARWAL
THE POINCARE CO
WHICH IS THE ON
TO BE SOLVED
M
ONJUCTURE
NLY PROBLEM
Millennium Prize Problems along with the
Simon Problem and the Smale Problem
are prestigious and celebrated elements of
the mathematical community. Since their
inception in 2000 by the Clay Institute of
Mathematics, Millennium Prize Problems
reside in the immeasurable expanse of
boundless mathematical exploration. These
problems are regarded as a testament to the
ongoing pioneering in the field of mathematics
and persevering human spirit to develop a
sound apprehension of our understanding of
the universe.
The Clay Institute of Mathematics put forward
7 of the most challenging problems existing
in the mathematical world that are either
incompletely solved or unsolved. To make it
interesting, the institution will award a generous
prize of 1 million dollars for the complete proof
or solution to any one of the problems. These
problems delve into an array of concepts with
practically endless scope. The problems range
from number theory to algebraic geometry
and do transcend theoretical physics and
computer science where their heart lies in the
application of fundamental mathematics.
The first problem is the Birch and SwinnertonDyer Conjecture which profoundly entangled
us in a possibility in number theory, particularly
dealing with the study of elliptical curves.
It suggests that the rational points on an
elliptical curve are related to the behaviour
of its function at certain points, highlighting
the algebraic and analytic properties of such
curves to be defined over a range of certain
rational numbers.This conjecture has profound
implications on our understanding of the
distribution of prime numbers and probably
can help us to identify a trend to predict them.
The second problem is the Hodge Conjecture.
This problem delves into the intersection of
algebraic geometry and topology, it essentially
discusses the theory of intersection of
algebraic geometry. The intersection of
algebraic geometry and differential geometry
can teach us a lot about their structural
aspects.
In this list, the third problem is the NavierStokes Existence and Smoothness. The
Navier-Stokes equation presides over the
motion of viscous fluids, the challenge is to
establish not only smoothness but also a
solution to this problem in three dimensions,
this task is still unaccomplished by the
mathematical community and eagerly awaits
the results of the efforts made recently by
Harvard University.
Succeeding this herculean problem is another
challenge, the Poincare Conjecture which is the
only problem to be solved, by Grigori Perelman
in 2003. This conjecture states that any 3-D
manifold which is simply connected is actually
homomorphic (two graphs can be deemed as
homomorphic if they can be obtained from the
same graph by dividing the edges with more
vertices) to a 3-D sphere.
The fifth problem at hand is the Riemann
Hypothesis also known as one of the toughest
and unsolvable problems known to humanity.
This hypothesis is deeply concerned with
the distribution of prime numbers and the
zeros of the Riemann Zeta function. This
hypothesis, if proven, can not only prove the
prime number theorem but also the symmetry
in their distribution much similar to the Birch
Swinnerton-Dyer Conjecture which also
delves into the intrinsic symmetry of existing
prime numbers.
The second last problem is the Yang-Mills
Existence and Mass gap which talks about the
existence of quantum Yang-Mills theory and
the mass gap. Within the realm of Quantum
Field Theory, the Yang-Mills Theory shows how
quarks and gluons interact with each other in a
set of mathematical equations which provide
us with a basic framework to understand this
intricate mechanism. This problem, if solved,
can give valuable insights into the relatively
unexplored world of quantum mechanics.
The last problem, the P vs NP Problem, along
with the Riemann Hypothesis, has occupied
the title of nearly unsolvable. This problem
discusses that P problems can be efficiently
solved in polynomial time while NP problems
can be verified in polynomial time. Now, if it
could be proven that the solution to a problem
can be verified as efficiently as solving it when
time is a polynomial function. Essentially, what
complicates this problem is that cryptography
exists only in the dichotomy of these two
types of problems. As when we enter our
passwords, they can be verified efficiently
in a fraction of time while if the process of
identifying the password within a certain time
is theoretically proven then very well programs
and algorithms can be designed to execute
the same. Therefore, all our passwords could
be calculated as efficiently as they could have
been verified thus making cryptography only
an aspect of the past. Moreover, computer
analysts and scientists see certain possibilities
of partly solving this problem thus changing
the definition of digital security we had till now.
23
THE INFINITY
The Riemann Hypothesis is one of the seven
Millennium Problems, having been first put
forth by Bernhard Riemann in 1859. This
hypothesis is about the intricate distribution of
prime numbers, a problem which has eluded
the mathematical community for over two
centuries.
The hypothesis puts forward the idea of
a specific pattern or line where certain
mathematical points related to prime
numbers exist. If this hypothesis is true, it
would help mathematicians understand how
prime numbers are distributed, which has
implications for many areas of mathematics
and cryptography. A key part of the Riemann
Hypothesis is the Riemann Zeta function,
denoted by ζ(s), initially introduced by Leonard
Euler and further refined by Riemann. This
complex-valued function is defined for
complex numbers with a real part greater
than 1 and is expressed as a sum of reciprocal
powers of natural numbers:
ζ (s) = 1 +12s+ 13s + 14s + 15s …
However the boundaries of a zeta function can
be broken by extending its existing domain.
This can be done by taking imaginary numbers
into consideration and plotting the function
on the complex plane. This process is known
as analyzing and extending a function. This
allows mathematicians to explore its behaviour
in the complex plane, where the input variable
(s) can take on complex values as well.
The zeroes of the zeta function, the values of
(s) for which the function equals out to be zero,
are crucial to the hypothesis. These zeroes can
be categorised as trivial or non-trivial. Trivial
zeroes are those which occur at negative even
integers whereas non-trivial zeroes are found
elsewhere in the complex plane. The nontrivial zeroes and their distribution is what we
explore in the article further.
Multiple studies indicate that the non-trivial
zeroes of the zeta function appear to cluster
around a certain critical line on the complex
plane. Over here the real number part of the
variable(s) equals 1 ⁄ 2. This critical line, often
referred to as the Riemann zeta function is
present on the plane as [Re(s) = 1/2], plays a
pivotal role in the formulation of the Riemann
Hypothesis.
Formally stated by Euler and then reiterated by
Riemann in his research paper, the Riemann
Hypothesis posits that all the non trivial zeros
of the Riemann Zeta functions lie precisely on
the critical line of Re(s)=½ . In simpler words, it
hypotheses the fact that there are no non trivial
zeros present outside the critical line. This
hypothesis has found a momentous place in
the road to proving or disproving the theorem.
The connection between the zeta function
and prime numbers lies in the Prime Number
Theorem, which provides an asymptotic
formula for the distribution of prime numbers.
The Riemann zeta function is linked to this
theorem through its relationship with the
Möbius function and the Euler product formula.
The hypothesis predicts a precise pattern in
the distribution of prime numbers, which may
help us predict their occurrence .
The significance of the Riemann Hypothesis
extends beyond number theory. Its resolution
would not only deepen our understanding
of prime number distribution but also lead to
advancements in cryptography, coding theory,
and broader mathematical concepts.
Despite countless efforts by the greatest
of minds, for over a century and a half, the
Riemann Hypothesis still remains unproven.
Its complexities put forward some of the
most intimidating challenges which require a
different level of knowledge and understanding
in the field. However, ongoing research
continues to unravel its mysteries, fueled
by the collective efforts of mathematicians
worldwide.
To end my effort on explaining this unexplained
mathematical phenomenon, the Riemann
Hypothesis still stands tall as one of the
biggest closed bridges connecting number
theory, complex analysis, and prime numbers.
Embodying an answer to the problem promises
to resolve and deepen our understanding
of the prime number distribution and
illuminate undefined territories in the field of
mathematics. Mathematicians still persist to
untangle the complexities of the theorem.
RE
H
AN ARTICLE BY A
THE REIMANN
HYPOTHESIS
REMAINS UNPROVEN
25
EIMANN
HYPOTHESIS
ANSHUMAN GUPTA
THE HYPOTHESIS PUTS
FORWARD THE IDEA OF A
SPECIFIC PATTERN OR LINE
THE INFINITY
DARPA’S
MATH
QUEST
AN ARTICLE BY AVIGHNA DARUKA
IN THE REALM OF NETWORK
SCIENCE, DARPA CHALLENGES
MATHEMATICIANS
THE INTERACTION BETWEEN
MATHEMATICS AND BIOLOGY IS
PARTICULARY EXCITING
The Defense Advanced Research Projects
Agency (DARPA) has given 23 math
challenges which have been discussed
below. According to DARPA, solving any of
these difficulties would impact “dramatically
revolutionising mathematics and thereby
strengthening the scientific and technological
capabilities” of the US Department of
Defense. These problems are especially
challenging due to their vague nature, no one
has a specified direction on how to approach
a specific issue and hence they still need to
be solved.
Firstly, DARPA’s mathematics centered goal
is focused on developing a functioning model
of the brain. Achieving this, DARPA aims to
transform our understanding of neuroscience
and also open the door for significant
advances in artificial intelligence.
Next, DARPA encourages mathematicians to
create tools that accurately model behavior
in large-scale dispersed networks. From
communication systems to social networks,
the capacity to forecast and comprehend
complex network dynamics is critical
for solving new difficulties in the digital
era. By adding this as DARPA’s second
problem, DARPA ensures that the future is
mathematically modelled.
The third DARPA mathematical problem
invites mathematicians to create approaches
that capture persistence in chaotic contexts.
In a world full of uncertainty and fluctuation,
such methodologies hold the key to gaining
deeper insights into changing systems. An
example would be understanding the Euler’s
pendulum, a chaotic system of operations
which can only be modelled by employing
mathematical algorithms.
DARPA’s fourth problem is a departure from
classical fluid dynamics, and seeks for new
ways to address the intricacies of foams,
gels, and liquid crystals—a challenge that
might transform areas ranging from materials
research to medicine delivery.
The intersection of mathematics and biology
presents another fertile ground for exploration.
In this proxy, DARPA fifth problem plans to use
quantum and statistical methods to predict
pathogen evolution, potentially altering our
approach to combating infectious diseases.
DARPA’s challenges span a wide range of
mathematical areas, each providing unique
opportunities for investigation and discovery.
As data collection grows, in its following
problems, DARPA challenges mathematicians
to extract the most information from
minimal sensing—a task that needs creative
approaches based on entropy maximization
methods.
The relationship between mathematics
and biology is very fascinating, with topics
ranging from defining new concepts of
distance to increasing symmetries and
action laws in biology. DARPA’s 10th problem
highlights
mathematics’
fundamental
interconnectedness with the natural world,
emphasizing its role as a universal language
that underpins all scientific investigation.
The search for the underlying laws of life
is DARPA’s computational biological goal,
a challenge that transcends academic
boundaries and emphasizes mathematics’
role in our understanding of the natural
world. The fifthteenth DARPA problem
recognizes that addressing this challenge
would require using collective insights
learned from prior challenges, emphasizing
the interconnectedness of mathematical
frontiers.
As mathematicians embark on the journey
to tackle DARPA’s 23 challenges, they are
poised to unravel profound mysteries and
pave the way for unprecedented scientific and
technological advancements. In the crucible
of these mathematical conundrums lies the
promise of a future where the boundaries of
human knowledge are continually pushed,
propelling humanity towards new frontiers of
discovery and innovation.
27
THE INFINITY
Once, when William Rowan Hamilton was
walking by the Royal Canal in Dublin, he
suddenly experienced a moment of clarity.
This changed the way how mathematics is
seen worldwide. In the Brougham Bridge,
he wrote the formula for quaternions, which
is a mathematical concept used to extend
complex numbers. This is where the journey
of higher dimensional numbers starts, ranging
from real numbers to the realms of quaternions
and octonions.
Higher dimensional numbers begin with
the square root of negative values. They are
known as a product of the imaginary unit ‘i’
which represents the root of the negative unit.
The complex numbers comprise a real and
an imaginary part (a+bi). These were used
to provide solutions to cubic equations that
lacked a real solution. Their acceptance and
use were less during the initial stages, as the
idea of imaginary numbers contradicted the
principles of real numbers. As time passed
by, the use of complex numbers in simplifying
equations was more frequent. Then slowly they
gained a foothold in mathematical practice.
Today complex numbers are a necessity
in fields such as electrical engineering and
quantum mechanics.
The journey of Quaternions and their
introduction is marked and remembered by
Hamilton’s realisation that three-dimensional
space can be algebraically encoded.
Quaternions extend the complex numbers to
four dimensions, expressed as ‘a+bi+cj+dk,’
where a, b, c and d are real numbers whereas
i, j, and k represent the three imaginary
components. Opposite to real and complex
numbers, quaternion multiplication is noncommutative which means that the order in
which quaternions are multiplied affects the
outcome. Quaternions dramatically simplify the
computation of three-dimensional rotations,
making them valuable in fields like computer
graphics and aerospace engineering. In these
fields specifically, they allow for the smooth
rotation of spacecraft and have also been
instrumental in building up the algorithms that
keep virtual reality environments stable and
oriented.
Octonions push the boundary even further. It
extends the dimensions to eight, with numbers
of the form ‘a+bi+cj+dk+ el+fm+gn+hp.’
It was discovered by John T . Graves and
independently by Arthur Cayley, octonions
are even more abstract than quaternions. The
reason for this is that they are non-associative
(the associative property ((xy)z =x(yz)) does
not hold). Even though they have an abstract
nature, they play a crucial role in the quest for a
unified theory, particularly in string theory and
quantum gravity.
Furthermore, quaternions also simplified the
visualisation of the orientation and motion
in fields of engineering like computer vision.
Octonions do impact theoretical physics,
especially in particle physics and string theory.
Moreover the extra 8th dimension predicted by
the properties of octonions opened new doors
for research and mathematical advancements.
Higher dimensionality further accentuates
the beauty and complexity of mathematical
development. It gives us a tough challenge
in our perceptions of space and time,
providing powerful tools for understanding
and manipulating the physical world. As we
go deeper into the universe, the significance
of these numbers only becomes more
pronounced which proves that mathematics is
the language in which the universe is written.
U
DIM
AN ARTICLE BY S
THE JOURNEY OF
QUATERNIONS AND
THEIR INTRODUCTION
29
UNLOCKING
MENSIONS
SHREEANSH BANSAL
THE WHOLE CONVERSATION
OF HIGHER DIMENSIONAL
NUMBERS BEGINS
THE INFINITY
Using ChatGPT or a similar generative AI
such as Gemini to solve problems or complete
assignments has become commonplace over the
past 2 years. However, people don’t appreciate,
how, at the core of ChatGPT, the mathematical
wonder known as a Large Language Model (LLM)
helps it function.
Imagine a library teeming with billions of books.
The LLM is akin to a “super librarian” who has
read every book in this library and can remember
every word. It doesn’t rely on databases.
Instead, it predicts word sequences based on
its vast databank. It “thinks” in predictions. This
revolutionizing approach marks a shift toward
creativity and cognition, a step away from merely
regurgitating information.
The process is fine-tuned by manipulating billions
of parameters. Like interconnected nodes in a
graph. Analyzed for weighted connections every
time a response is formulated. Revolutionary.
A shift toward creativity and cognition. Not just
spitting out information.
LLMs fundamentally rely on the “transformer
architecture.” Picture a master juggler. Juggling
billions of parameters simultaneously. Catching
and throwing data with precision to generate
relevant responses. Judging which parameters
matter for the user’s input.
Fundamentally, Large Language Models rely on
what’s termed the “transformer architecture”. In
other words, think of the transformer as a master
juggler. It juggles several billion parameters at the
same time.
It heavily uses linear algebra. An efficient way to
handle the massive number of parameters. Linear
algebra enables scaling, rotating, and transforming
input data in multiple dimensions. So the model
can dynamically adjust word importance and
relationships. Fundamental for “learning” from
data. By adjusting its internal word and sentence
representations over time, it improves response
accuracy.
Gradient descent and other optimization
techniques make these adjustments possible.
Iteratively refining parameters to minimize
prediction error. Linear algebra is the bedrock.
Its use of linear algebra reveals something
interesting. A key concept: probability. Picture
every word sequence as roads in a vast network.
Each road has a sign showing the likelihood of
reaching the next word. The model calculates
these probabilities. Choose the most promising
word. Like choosing the best road based on traffic
signs. It’s a sprawling highway of word possibilities.
The model is your navigation app. Computing
probabilities. Assessing each route’s likelihood of
reaching the next word.
Another crucial aspect is attention mechanisms.
Imagine a flashlight illuminating parts of previous
words to decide what to say next. The brighter
the light shines on a word, the more important it
is considered by the LLM in its current decisionmaking process. This helps it choose the wording
of its response to match the user’s intentions. This
31
is enabled through the use of matrices. Through
matrices, you can calculate the most probable
output based on weights in a shorter time frame
compared to brute force. By using matrices in
conjunction with linear algebra the model can
quickly resolve the query from possibilities in
multidimensional space to a workable output,
this is also the reason many alternative outputs
are available for a single query; an equation
in multiple dimensions always has various
solutions.
“ This revolutionising approach
marks a shift toward creativity
and cognition, a step away
from merely regurgitating
information.”
Transformative
Mathematics
in AI
SHRIKRITT JAJOO
In summary, ChatGPT’s “thinking” combines
understanding language patterns with complex
probability calculations and optimization
through billions of input/output parameters. All
are powered by the transformer architecture. The
journey from a simple database to pioneering
mathematics? Fascinating. A testament to
the synergy of mathematical ingenuity and
technological advancement.
THE INFINITY
AI architecture is a technological advancement
that leverages the capabilities of mathematics
to create realistic images not bound to human
imagination. This digital space is characterised
by image replications which are fascinating
because of the mathematical principles that
combine with computer programming to
produce attractive scenes.
The AI-based probability and statistics
functions direct how long each rendering
part should take. For instance, probabilistic
methods are used in Generative Adversarial
Networks (GANs) and various Autoencoders
(VAEs) for approximating real-world images
where art imitates life. GANs form a complex
interplay between a generator pipeline
producing images from some plausible
distribution and a discriminator network
delineating genuine against synthetic ones.
GANs achieve this remarkable feat of creating
novel imagery through a complex interplay of
mathematical functions. The resulting outputs
occupy a fascinating space between the
pragmatism of real-world representation and
the evocative utilisation of artistic expression,
effectively demonstrating the potential for
mathematically-driven creativity within the
field of artificial intelligence.
AI generators use Linear Algebra as a
necessary instrument. Using techniques such
as Convolutional Neural Networks (CNNs)
make it easy to manipulate pixel values to
heighten details of the proposed images.
Through matrix multiplication and validation
which are basic algebraic operations, CNNs
can decode complex patterns in data to
enhance the conditions of image replication.
By finding such distinctive structures in the
data, CNNs produce visual artefacts that are
unparalleled in their realism, thus illustrating
the complexity of visual space.
Optimization algorithms act as enhancers for
refining pictures into optimal states. In this
way, AI models seek to minimise pre-specified
loss functions through methodologies such
as gradient descent. This iterative process
is comparable with an adversarial contest
in which at each stage the generator and
discriminator try to outsmart one another. The
interaction between these two forces yields
compelling artistic expressions that transcend
mere verbal description.
Bridging the information gap is achieved by
comparing backdrops in Information Theory.
This effort involves, for most autoencoders
(VAEs), a search for an underlying space
where the image is represented not as pixels
but as a constellation of data points. With
VAEs, the pixels’ hidden abilities are unearthed
by separating images and conserving
their fundamental traits. As the bedrock of
contemporary AI, Deep Learning enables
generative models to unfold the details of
picture synthesis. Darknet engines project the
originality of artificial data by interpreting the
cryptic codes stowed away in pixels to render
“
b
g
l
g
“ To be inspired and informed
by nature’s splendour in art,
geometric symmetry springs to
life as an influential font that
guides the way.”
imaginative images. With awe-inspiring
networks and intriguing mechanisms, these
architectural masterpieces are pushing
image technology beyond known limits
into uncharted realms, enabling unheard-of
strides in creativity and invention.
To be inspired and informed by nature’s
splendour in art, geometric symmetry springs
to life as an influential font that guides the
way. Created by fractal algorithms are living
shapes, forms, and textures possessing
verisimilitude and practical properties.
These algorithms, characterised by their
iterative and self-similar nature, generate
intricate and visually stunning patterns that
defy conventional artistic boundaries. This
convergence of mathematical principles and
artistic expression transcends established
aesthetic norms, revealing a unique form of
beauty rooted in multidimensionality.
The
strong
connection
between
mathematics and imagination rules the
world of AI image production. At the very
heart of image creation, the probabilistic
character of GANs or the fractal’s geometric
grace, both computational techniques form
a unity with mathematical principles. Such
mathematical marvels would surely guide us
towards unseen horizons in the fields of art
and technology as we explore further into the
enigmatic realm of AI generation.
Mathematical
Marvels in
AI Art
AARAV ANAND
33
THE INFINITY
Remember the first time you picked up a
Rubik’s cube? The first attempt inevitably
ends in a total colour mess. Despite being an
amazing party trick, have you ever thought
about it? Are there any other ways to solve it
other than the traditional method of learning all
the algorithms and steps? The creator of the
Rubik’s Cube, architect Erno Rubik, built the
popular 3D puzzle in his mother’s apartment.
Originally known as the Magic Cube or Rubik’s
Kocka in Hungarian, the initial purpose of
creating this 3-D puzzle by Erno was to teach
his university students about the architecture
of 3-dimensional figures. It was painted in
six different colours. Due to factors like every
square being painted invariably differently
and its ability to turn upwards, downwards,
clockwise, and anti-clockwise and split
into many different smaller cubes, It has a
whooping forty-three quintillion permutations,
which means that if every person on the planet
Earth turned a Rubik’s cube randomly, only one
cube would return to its original state.
It took Erno himself a month to understand
and comprehend his invention. Which was
released for the public to enjoy in the 1980s.
Ever since it was released, there have been
numerous attempts to solve the cube using
numerous distinct mathematical equations
and algorithms. In 1981, only one year after it
had been made public, British mathematician
Morwen Thistlewaite established the evidence
that using a branch of algebra that deals with
symmetry and transformation, known as the
‘group theory’, it was always possible to solve
a Rubik’s cube in fifty-two moves or less.
Rather than using the traditional method, we
can use another alternative method named
after Morwen Thistlewaite, the founder of
the method. The ‘Thistlethwaite algorithm’
involves dividing the cube into smaller subgroups, each involving a peculiar symmetry.
‘Thistlethwaite algorithm’ involves the usage
of group theory, cognizant of the cube’s
symmetry that can be transformed into
multiple configurations using scores of moves,
redeeming itself as the most reliable method.
By analysing the symmetry of the cube,
mathematical speculators have concluded a
solution that involves breaking the cube down
into smaller versions with more manageable
parts. After closely keeping all these facts in
mind, a professor at Binghamton University,
Jessica Fridrich, came up with a method
popularly known as CFOP (Cross-F2L-OLL
– PLL). Sometimes also called the Fridrich
method, the invariably used approach to a
speed cube. Harnessing the principles of
group theory, The method involves solving
the cube layer by layer, starting with the
cross, then solving the first two layers (F2L),
followed by orienting the last layer (OLL),
and finally permuting the last layer (PLL). The
CFOP method requires memorising several
algorithms for each step, but with enough
practice, it can be a very efficient way to solve
the cube. Many speedcubers use the CFOP
method, which is often used among Rubik’s
cube solvers.
The reason why mathematicians find the
Rubik’s cube so bewildering is because,
despite being so colourful, like children’s toys,
it can challenge one’s mind in ways not even
expected. Even a simple movement to the left
and then a rotation on the top layer makes a
lot of difference. Even putting it back in the
same order so you can see how it works,
the kind of displacement that happens, and
which pieces change when you scramble
it. As mentioned before, it is an amazing
example of group theory, and it can be used
to understand and manipulate the object in
certain ways.
“despite being so colourful, like
children’s toys, it can challenge
one’s mind in ways not even
expected.”
As we delve into advanced research with
regards to the mathematics behind the
Rubik’s Cube, we discover more about the
secrets and solutions to it and uncover a
journey that narrates the mathematical
prowess humans possess. From algorithms
to theories, we gain a deeper understanding
of a Rubik’s cube’s complex structure and its
symmetry. So that day, when Erno Rubik was
trying to make an object to help his class learn
better, he, by mistake, invented something
that would continue to remain an object of
mathematical beauty for years to come.
The Mathematical
Puzzle of the
Rubik’s Cube
ABHINAV DANGWAL
35
THE INFINITY
Patterns are perhaps the most common of
all mathematical concepts that recur in the
world around us, from the phases of the moon
to the music we listen to. Patterns are mere
repetitions of a sequence to create an order,
and as mentioned before, they are present all
around us, making them a phenomenon that is
both easy to understand and intriguing to learn
about.
Tessellations, also known as tilings, are an
amalgamation of the beauty of art and the
rigidity of mathematics. They are a type of
pattern formed by connecting shapes while
making sure there are no gaps or spaces left
in between. Tessellations are notable for being
quite a sight for the eye when considering the
simplicity of the sub-structures that create
them. From the tiles that cover our floor to
the tightly packed hexagons in a beehive,
tessellations are widely found and liked in
today’s world due to their mesmerising yet
simple nature
The core of any pattern lies in its mathematical
foundations, and tessellations are not an
exception. Shapes used in tessellations
must possess a special property known
as “tolerability.” Any shape harbouring this
property can be replicated and reused multiple
times to create an infinite pattern and a flat
plane or flat surface. Apart from this rule, there
are two primary types of tessellations: ‘regular
tessellations’ and ‘semi-regular tessellations.’
Regular tessellations are formed by repeating
any particular kind of ordinary polygon in such
a way that all their dimensions remain equal to
prevent gaps to act as obstructions between
the figures. Triangles, squares, and hexagons
are known to be the only shapes to form
perfect regular tessellations considering their
distinctive and prevalent shapes.
On the other hand, semi-regular tessellations
are formed by creating a mix of two or more
different kinds of shapes, usually regular
polygons. This gives the tessellations a more
diverse perspective. Understanding geometric
properties like angles and side lengths is
extremely important for creating tessellations.
Symmetry also plays a huge role in perfecting
such visually pleasing structures. Reflectional,
rotational, and glide symmetries help create
repeating patterns that fill the plane effortlessly.
The Alhambra Palace in Granada, Spain, is
evidence of the stunning effects achieved
through the use of symmetrical tessellations.
Mathematicians have developed formulas to
determine which shapes can be tessellated
and how they can be arranged. These formulas
unlock the possibilities for creating unique and
visually stunning patterns.
Euler’s polyhedron formula relates the number
of vertices (V), faces (F), and edges (E) in
a uniform polyhedron (a 3D tessellation). It
can also be applied to planar tessellations by
setting the genus (a measure of “holes” in the
shape) to 1. The formula is ‘V + F - E = 2 - 2g.’
For a simple, flat tessellation with no holes (g =
37
“ The core of any pattern
lies in its mathematical
foundations, and tessellations
are not an exception.”
Tessellations:
A Timeless
Pattern
AADI TULSYAN
1), it is ‘V + F + E = 0.’
Furthermore, the Tiling Theory delves into the
question of which shapes can tessellate the
plane and under what specific conditions. It
explores semi-regular tilings, where multiple
shapes with specific properties come
together to create a pattern. For example,
in 2023, a discovery was made by Dr. Craig
Kaplan in which a single, oddly shaped tile
could cover a flat surface infinitely, but never
in a repeating pattern. Unlike traditional tilings
where a small pattern is repeated endlessly,
this “aperiodic monotile” could create a
unique pattern every time a new tile is added.
Tessellations are also a widespread feature
in various works of art, especially mosaics
and graphic design. Their interlocked and
packed figure gives them a solid, 3-D look,
despite being 2-D in nature. One of the most
famous examples of tessellations used are
in the works of Maurits Cornelis Escher.
Escher was a Dutch graphic artist whose
works were based on mathematical objects
and their structures. Moreover, Escher
even communicated with mathematicians,
continuing his ardent research on tessellation.
With everything present in the modern
world having a makeover in the digital realm,
tessellations are no exception to this rule.
The repetitive nature of tessellations makes
them interesting from a coding perspective.
Algorithms can be designed to generate
tessellation patterns based on specific rules
and shapes. This has applications in areas like
computer graphics and image compression,
similarly as to what helped Dr. Kaplan in his
research.
Tessellations are more than just mere
mesmerising patterns. They are a bridge
between the artistic and the mathematical,
offering a glimpse into the elegant language
of shapes and space.
THE INFINITY
The Infinity Society, the mathematics society
of our school, orchestrated a vibrant and
intellectually stimulating Pi-Day celebration on
March 14th, 2024. Led by our “Infinity Society”
boy-in-charge, the event was a testament
to the power of mathematical curiosity and
creativity within our school community.
The festivities commenced with an art
competition, inviting students from the 8th
and 9th grades to unleash their artistic talents.
Participants were tasked with creating posters,
brochures, gift cards, or any form of artwork
inspired by the theme of Pi. Such was the
enthusiasm that the competition extended
over two days, from March 4th to March 5th,
due to the overwhelming response. The quality
of entries was exceptional, showcasing the
depth of creativity and passion among our
students.
Acknowledging the outstanding efforts of
the participants, the top three entries from
each class were honored with awards by
our esteemed Headmaster. This recognition
not only celebrated individual creativity but
also fostered a culture of appreciation for
mathematical exploration and expression.
Preparations for the grand celebration were in
full swing a day before Pi-Day. The school’s
gazebo became a canvas for mathematical
expression as it was adorned with various
pulses, symbolizing the mathematical
significance of the occasion. The collaborative
efforts of the Infinity Society, the art department,
and the school art secretary, Aditya Saraf, were
instrumental in bringing this vision to life. Their
dedication and teamwork exemplified the spirit
of unity and collaboration that defined the
event.
On the auspicious day of Pi-Day, the Infinity
Society took center stage to enlighten the
school community about the significance of
this mathematical constant. In an impassioned
address, the “Infinity Society” boy-in-charge
emphasized the beauty of mathematics and
encouraged students to embrace it without
fear. This message resonated deeply as
students shared self-written poems about Pi
and fascinating mathematical facts, further
enriching the celebration with their creativity
and intellect.
As the day progressed, the entire school
community gathered to indulge in a culinary
delight – special carrot cakes served during
lunch. This gastronomic treat served as a sweet
reminder of the joyous occasion, fostering a
sense of camaraderie among students and
teachers alike.
However, the celebration did not end there.
True to the spirit of giving back, the pulses used
for decoration were later donated to the needy,
39
embodying the ethos of compassion and
generosity instilled in our school community.
“ On the auspicious day of PiDay, the Infinity Society took
center stage to enlighten the
school community...”
In conclusion, the Pi-Day celebration
organized by the Infinity Society was
a resounding success, thanks to the
unwavering dedication and collaborative
efforts of all involved. It not only showcased
the creative talents of our students but also
reinforced the importance of mathematics
as a source of inspiration and wonder. As
we reflect on this memorable event, let us
continue to nurture our love for mathematics
and strive to make a positive impact on the
world around us.
Signed,
Aadi Narayan Kandele
Boy-in-Charge, Infinity Society
Infinity Society’s
Pi-Day
Celebration Report
AADI KANDELE
THE INFINITY
In the intricate web of interconnected data,
from social networks to molecular structures,
lies a fundamental abstraction known as graph
theory. Graph theory serves as a powerful tool
for understanding and analysing relationships
among objects, entities, or elements. Originally
conceived in the 18th century by the Swiss
mathematician Leonhard Euler, graph theory
has since burgeoned into a vast field with
applications spanning computer science,
biology, sociology, and beyond.
Graph theory, as a mathematical discipline,
was born in the 18th century through the
pioneering work of Leonhard Euler, a Swiss
mathematician. The genesis of graph theory
can be traced back to a famous problem posed
by the citizens of Königsberg, a city in Prussia.
Königsberg was characterized by its division
by the Pregel River, which formed two large
islands connected to each other and the
mainland by seven bridges. Residents of the
city pondered whether it was feasible to devise
a route that would allow them to traverse each
bridge exactly once, returning to their starting
point. Despite numerous attempts, a solution
remained elusive.
Fig 1.4
In 1735, Euler, then residing in St. Petersburg,
Russia, approached the problem with a
fresh perspective. He thought of the bridges
and landmasses as vertices and edges,
respectively, thereby creating what is now
known as a graph. Euler astutely observed that
the specific configuration of the bridges was
inconsequential; what truly mattered was the
underlying pattern of connections between the
landmasses.
Euler’s crucial insight was that a viable path
traversing each bridge exactly once would
only be possible if each landmass (vertex)
had an even number of bridges (edges)
connected to it, with the exception of possibly
two landmasses having odd degrees. This
revelation laid the groundwork for what are
now recognized as Eulerian paths and cycles.
By reducing the conundrum to its abstract
graph representation, Euler conclusively
demonstrated that no such path existed in the
case of Königsberg’s bridges. This seminal
work not only resolved a practical puzzle but
also inaugurated the field of graph theory.
Euler’s seminal paper, “Solutio problematis
ad geometriam situs pertinentis” (“Solution
to the problem of the Seven Bridges of
41
Königsberg”), published in 1736, is hailed as
the cornerstone of graph theory.
Euler’s groundbreaking contributions to
graph theory established it as an independent
branch of mathematics, laying the foundation
for subsequent developments by generations
of mathematicians and scientists. Today,
graph theory stands as a vibrant field with
diverse applications across mathematics,
computer science, engineering, and various
other disciplines.
“ Euler’s groundbreaking
contributions to graph theory
established it as an independent
branch of mathematics...”
Unravelling
Connections: An
Introduction to
Graph Theory
MR GYANESH PANDEY
THE INFINITY
I usually start my lessons in my classroom
with an engaging warm-up or introductory
activity that stimulates my students’ interest
in mathematics. I recently asked my D form to
use the numbers 1 through 9 to form a magic
square. While some of them found it difficult,
others finished it in a matter of minutes. I
saw several students get interested in magic
squares as they worked. They inquired as to
what a magic square is. Who thought of it?
Why do we investigate it? And the classic, will
it be on the May Test?
A magic square is a square arrangement of
numbers where the sum of the numbers in each
row, column, and diagonal is the same. This
sum is known as the magic constant. Because
of their mathematical elegance and symbolic
meaning, magic squares have fascinated
people for centuries. Magic squares have an
interesting origin. According to a Chinese tale
dating back to around 2600 B.C., the first 3x3
magic square was discovered by Emperor Yu
on the shell of a turtle. This discovery, known as
the Lo Shu square, is believed to bring stability
and prosperity.
In ancient China, during the reign of Emperor
Yu, there was a belief in the Kingdom that
offering sacrifices, particularly grains, to
the river God, known as the river Lou, could
prevent floods. Emperor Yu himself once
went to offer sacrifices to the river. During this
event, an unusual occurrence took place: a
turtle appeared from the river, having a unique
pattern on its back. The emperor and those
around him were astonished to see this pattern,
interpreting it as a divine sign. They carefully
noted down the pattern and believed it was
auspicious. Since that momentous event, the
pattern has been regarded as a symbol of
good fortune and prosperity. As it is thought
to bestow prosperity and stability upon the
occupants, it has become common in China to
include this fortunate design at the foundation
of all new construction.
Over time, the magic square spread from Asia
to Africa, Europe, and other places. It became
an important part of different cultures and was
used in many ways, such as for beauty and fun.
Despite once being considered magical, magic
squares are now mostly valued as challenging
math puzzles. Still, their transition from mythical
beginnings to mathematical challenges show
how people have always been fascinated by
patterns, symmetry, and the mysteries of the
world.
How to make a 3 by 3 magic square? By
taking any value of x the magic square can be
formed as follows. By letting x as 10, a magic
square is constructed in which the magic
constant is 30. Thus, making magic squares
Fig 1.5
43
is a fascinating activity, irrespective of its
historical, cultural, and aesthetic importance.
To learn mathematics, it is essential to have
a logical mindset. There are various methods
and various types of magic squares; studying
Fig 1.8
from right to left and six parallel lines slanting from left to right. Adjust the numbers
that are left out of the grids, such as 16 and 22, moving to the 5th grid from left to
right, and 4 and 10 moving from right to left. Similarly, 1, 2, 6, 24, 20, and 25 will shift
to the blank grids. With these adjustments, a magic square with a magic constant
of 65 is constructed.
“ A magic square is a square
arrangement of numbers where
the sum of the numbers in each
row, column...”
Fig 1.6
these patterns and arrangements will surely
enable learners to develop an interest in the
subject. The starter activities, like playing
with numbers, stimulate the development of
mathematical thinking.
To construct a magic square of 4 by 4 we can
take any 16 consecutive numbers and arrange
the in a 4 by 4 grid. Then by rearranging the
certain numbers in a particular order will
give you a magic square. By interchanging 2
and 15, 3 and 14, 5 and 12, and finally 8 and
9, the above arrangement will give a 4 by 4
magic square. The same can be done by
interchanging the corner numbers diagonally
and the numbers of the inner 2 by 2 grids also
diagonally, giving a 4 by 4 magic square with
a magic constant of 34. When each number
of a magic square is multiplied or divided by
the same number, or when the same number
is added to or subtracted from each entry, the
magic constant also changes the same way.
Even with the date of birth of a person, we
Fig 1.7
What a Magic
Square is
MR CHANDAN GHUGHTYAL
can create a magic square of 4 by 4 order. Let
‘D’ be the date, ‘M’ be the month, Y is the first
two digits, and Z is the last two digits of the
year. The date of birth of Krtin is November
20, 2006. For him, D is 20, M is 11, Y is 20, and
Z is 6. By following the set arrangement in the
grid below, the birthday magic square of Krtin
can be constructed. The magic number for
Kritin is 57.
In addition to being fun, magic squares
encourage mathematical curiosity and
problem-solving abilities by displaying
number patterns and symmetry. They
represent harmony and order in the cosmos
and are present in many cultural practices
around the world. Magic square research
demonstrates the connection between
mathematics and human expression,
stimulating creativity in literature and design.
Magic squares improve critical thinking,
creativity, and a respect for differences in
culture.
To create a 5 by 5 magic square, start by
arranging numbers from 1 to 25 in square
order. Then, draw six parallel lines slanting
THE INFINITY
In today’s digital landscape, the ability to
communicate effectively through writing is
paramount. Whether it’s a tweet, a blog post, or
a comprehensive report, the way information
is presented can make or break its impact.
This article explores the importance of honing
report writing skills, specifically tailored for
social media and website use, drawing parallels
from the world of mathematics.
The Inverted Pyramid Style: Building a
Strong Foundation
In the realm of report writing, one technique
stands out as a cornerstone of effective
communication: the inverted pyramid style.
Just as in mathematics where foundational
concepts precede more complex ones, the
inverted pyramid presents information in
descending order of importance. Imagine
writing a synopsis for a mathematical
experiment – you’d start with the key
hypothesis and results before delving into the
methodology and finer details. This structure
is ideal for social media and website reports,
catering to the fleeting attention spans of
today’s audience.
Fig 1.9
Crafting Compelling Headlines:
Gateway to Engagement
The
The headline serves as the entry point to your
report, akin to the title of a mathematical paper.
Just as a well-crafted title encapsulates the
essence of a study, a catchy headline captures
the essence of your content. Take, for instance,
the headline “Discovery Unveiled: Experts
Shed Light on Unexplained Phenomenon” –
it’s brief, informative, and entices the reader to
delve deeper.
The Art of Introduction: Setting the Stage
Now, let’s dissect the content of your
report, drawing parallels to the process of
summarizing a mathematical experiment. The
beginning acts as the introduction, setting the
stage for what follows. In a math synopsis, this
is where you’d outline the problem at hand
and its significance. Similarly, in your report,
a compelling lead introduces the incident or
event, capturing the reader’s attention from the
outset.
45
Unfolding the Narrative: Building Block by
Block
As you progress through the report, think
of it as unfolding a mathematical proof.
Each paragraph builds upon the previous
one, leading the reader through a logical
sequence of events or ideas. In the context
of a mathematical experiment, this translates
to presenting the methodology, results,
and analysis in a coherent manner. Just
as mathematical proofs require clarity and
precision, so too should your report convey
information concisely and accurately.
Breathing Life into Your Writing: The
Power of Speech
“ In the realm of report
writing, one technique stands
out as a cornerstone of effective
communication...”
Mastering
Report
Writing Skills
MS ABHA DEB
To breathe life into your report, incorporate
direct and indirect speech, much like including
quotes from researchers or participants in
a mathematical study. Direct speech lends
authenticity, while indirect speech provides
context and interpretation. For example,
quoting a mathematician’s excitement upon
discovering a solution adds a human element
to your report, making it more relatable to
readers.
Power of SEO: Enhancing Visibility
In the digital realm, visibility is key, and
this is where the power of SEO, keywords,
and hashtags comes into play. Just as
mathematical concepts are indexed and
categorized for easy reference, optimizing
your content with relevant keywords and
hashtags ensures its discoverable by your
target audience. Think of keywords as the
variables in a mathematical equation –
strategically chosen to enhance searchability
and engagement.
Mastering the Art of Report Writing
In conclusion, mastering report writing skills
is essential for effective communication in
today’s digital age. By employing techniques
such as the inverted pyramid style, crafting
compelling headlines, and incorporating
direct and indirect speech, you can create
content that captivates and resonates
with your audience. Much like presenting
a mathematical proof, clear, concise, and
compelling writing ensures your message is
understood, appreciated, and shared across
digital platforms.
THE INFINITY