Let’s play the following “chaos game”. Consider a regular triangle and take some initial point in it. Now pick one of the three vertices at random, connect it with the current point, and mark this line’s midpoint as the new point. If you repeat this process many times, what will the resulting shape look like? You might expect to find the points simply shattered everywhere, without much structure, but surprisingly, the points will in fact create a Sierpinski fractal.

There are lots of variations on this theme, some of which can be found on this Wikipedia page.

Animation made in Mathematica.

n roots of unity

When one is dealing with complex numbers, it is many a times useful to think of them as transformations. The problem at hand is to find the n roots of unity. i.e

As is common knowledge z = 1 is always a solution.

Multiplication as a transformation

Multiplication in the complex plane is mere rotation and scaling. i.e

Now what does finding the n roots of unity mean?

If you start at 1 and perform n equal rotations( because multiplication is nothing but rotation + scaling ), you should again end up at 1.

We just need to find the complex numbers that do this.i.e

This implies that :

And therefore :

Take a circle, slice it into n equal parts and voila you have your n roots of unity.

Okay, but what does this imply ?

Multiplication by 1 is a 360o / 0o rotation.

When you say that you are multiplying a positive real number(say 1) with 1 , we get  a number(1) that is on the same positive real axis.

Multiplication by (-1) is a 180o rotation.

When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis 

The act of multiplying 1 by (-1) has resulted in a 180o transformation. And doing it again gets us back to 1.

Multiplication by i is a 90o rotation.

Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90o rotation. 

This applies to -i as well.

so on and so forth,

Have a great day!


A Fascinating Calculation

Dividing by 7 yields this fascinating play of the numbers 1,4,2,8,5 and7. I am not going to spoil the fun by letting you on the pattern that emerges and other captivating properties that you might discover along the way.

Although feel free to write to us if you found anything that marveled you and we would definitely share it with the world.

Who knew division could be this much fun. am i right?

Have fun!


Algorithmically generated using circles subjected to constrained randomness

Animated GIFs (600x600)

Mathematica code:

RR[n_, m_] := (SeedRandom[n*m]; RandomReal[])

L[t_, Q_, g_, i_] := Sum[ Exp[-(t - (RR[i, 5*Q] + j))^2/g], {j, {-1, 0, 1}}]

G[c_, F_, P_, s_, o_, g_, A_, a_, w_, t_, r_, Q_, pr_, is_] :=
   {RGBColor[RR[i, Q], RR[i, 2*Q], RR[i, 3*Q], o + A*L[t, 5*Q, g, i]],
     RotationTransform[(k + (-1)^(i*w)*t*a)*2 Pi/F]
      [{r*RR[2 i, 4*Q], r*RR[2 i + 1, 4*Q]}], s*L[t, 5*Q, g, i]]},
  {k, 0, F - 1, 1}],
 {i, 1, P, 1}],
PlotRange -> pr, ImageSize -> is,  Background -> c]

  G[RGBColor[54/255, 70/255, 93/255], 12, 50, .2, 0, .1, .25, 1, 1, t, 1, Q, 1.6, 600],
 {Q, {41, 72, 1, 76}}, {t, 0, .96, .04}],
List of Free Science Books

Here’s an alphabetical list of all available free books. Note that many of the links will bring you to an external page, usually with more info about the book and the download links. Also, the links are updated as frequently as possible, however some of them might be broken. Broken links are constantly being fixed. In case you want to report a broken link, or a link that violates copyrights, use the contact form


  • A Beginner’s Guide to Mathematica
  • A Brief Introduction to Particle Physics
  • A First Course in General Relativity
  • A New Astronomy
  • A No-Nonsense Introduction to General Relativity
  • A Popular History of Astronomy During the Nineteenth Century, Fourth Edition
  • A Review of General Chemistry
  • A Simple Guide to Backyard Astronomy
  • A Text Book for High School Students Studying Physics
  • A Tour of Triangle Geometry
  • About Life: Concepts in Modern Biology
  • Acoustic Emission
  • Adaptive Control
  • Advanced Calculus
  • Advanced Learning
  • Advanced Mathematics for Engineers
  • Advanced Microwave Circuits and Systems
  • Advanced Technologies
  • Advances in Computer Science and IT
  • Advances in Evolutionary Algorithms
  • Advances in Geoscience and Remote Sensing
  • Advances in Haptics
  • Advances in Human Computer Interaction
  • Age of Einstein
  • Aging by Design
  • AMPL:  A Modeling Language for Mathematical Programming
  • An Introduction to Elementary Particles
  • An Introduction to Higher Mathematics
  • An Introduction to Many Worlds in Quantum Computation
  • An Introduction to Mathematical Reasoning
  • An Introduction to Mathematics
  • An Introduction to Proofs and the Mathematical Vernacular
  • An Introduction to Relativistic Quantum Mechanics
  • Analysis 1 (Tao T)
  • Analysis 2 (Tao T)
  • Analytic Functions
  • Astronomical Discovery
  • Astronomy for Amateurs
  • Astronomy Today
  • Astronomy with an Opera-Glass
  • Automation and Robotics


  • Basic Algebra, Topology and Differential Calculus
  • Basic Concepts of Mathematics
  • Basic Concepts of Thermodynamics
  • Basic Concepts of Thermodynamics Chapter 1
  • Basic Ideas in Chemistry
  • Basic Math: Quick Reference eBook
  • Basic Mathematics for Astronomy
  • Basic Physics
  • Basic Positional Astronomy
  • Basic Principles of Classical and Statistical Thermodynamics
  • Basic Principles of Physics
  • Basics of Physics
  • Beginner’s Botany
  • Biochemistry
  • Biochemistry (practice book)
  • Biology
  • Board Notes for Particle Physics
  • Book of Proof


  • Calculus
  • Calculus Based Physics
  • Celestial Navigation, Elementary Astronomy, Piloting
  • Circuit QED — Lecture Notes
  • Classical Dynamics
  • Classical Geometry
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  • Complex Analysis
  • Computational Geometry
  • Computational Introduction to Number Theory and Algebra
  • Computational Physics with Python
  • Conceptual Physics
  • Consistent Quantum Theory
  • Cook-Book Of Mathematics
  • College Physics
  • Crude Oil Emulsions- Composition Stability and Characterization
  • Curiosities of the Sky


  • Decoherence: Basic Concepts and Their Interpretation
  • Do We Really Understand Quantum Mechanics?
  • Differential Equations
  • Diophantine Analysis
  • Discover Physics
  • Dr. Donald Luttermoser’s Physics Notes
  • Dynamics and Relativity


  • Earthquake Research and Analysis
  • Earthquake-Resistant Structures – Design, Assessment and Rehabilitation
  • Einstein for Everyone
  • Electromagnetic Field Theory
  • Elementary Mathematical Astronomy
  • Elementary Linear Algebra
  • Elementary Particle Physics in a Nutshell
  • Elementary Particles in Physics
  • Elements of Astrophysics
  • Embedded Systems – Theory and Design Methodology
  • Encyclopaedia of Mathematics
  • Encyclopedia of Astrophysics
  • Engineering Mathematics 1
  • Engineering Mathematics with Tables
  • Essential Engineering Mathematics
  • Essential Physics
  • Exoplanet Observing for Amateurs
  • Experimental Particle Physics


  • Fields
  • Foundations of Nonstandard Analysis
  • Frequently Asked Questions about Calendars
  • Fundamental Concepts of Mathematics
  • Fundamentals of Analysis (Chen W.W.L)
  • Further Mathematical Methods
  • Fusion Physics


  • General Chemistry
  • General Relativity
  • General Relativity
  • Geometric Asymptotics
  • Geometry and Group Theory
  • Geometry and Topology
  • Geometry Formulas and Facts
  • Geometry Study Guide
  • Geometry, Topology and Physics
  • Geometry, Topology, Localization and Galois Symmetry
  • Great Astronomers


  • Handbook of Formulae and Physical Constants
  • High School Mathematics Extensions
  • Higher Mathematics for Engineers and Physicists
  • History of Astronomy
  • Homeomorphisms in Analysis
  • How to Use Experimental Data to Compute the Probability of Your Theory


  • Intelligent Systems
  • Intrinsic Geometry of Surfaces
  • Introduction to Astronomy and Cosmology
  • Introduction to Cancer Biology
  • Introduction to Chemistry
  • Introduction to Cosmology
  • Introduction to Elementary Particles
  • Introduction to General Relativity
  • Introduction To Finite Mathematics
  • Introduction to Particle Physics Notes
  • Introduction to PID Controllers
  • Introduction to Quantum Mechanics with Applications to Chemistry
  • Introduction to Quantum Noise, Measurement and Amplification
  • Introduction to Social Network Methods
  • Introduction to String Field Theory
  • Introduction to the Time Evolution of Open Quantum Systems
  • Introduction to Quantum Mechanics
  • Introductory Computational Physics
  • Introductory Physics 1
  • Introductory Physics 2


  • Kinetic Theory


  • Laboratory Manual for Introductory Physics
  • Laws of Physics
  • Learn Physics Today
  • Lecture Notes in Discrete Mathematics
  • Lecture Notes in Quantum Mechanics
  • Lecture Notes in Nuclear and Particle Physics
  • Lecture Notes in Particle Physics
  • Lecture Notes on General Relativity
  • Lectures on Astronomy, Astrophysics, and Cosmology
  • Lectures on Particle Physics
  • Lectures on Riemann Zeta-Function
  • Light and Matter


  • Mag 7 Star Atlas Project
  • Many Particle Physics
  • Math Alive
  • Mathematical Analysis I(Zakon E)
  • Mathematical Biology
  • Mathematical Methods
  • Mathematical Methods 1
  • Mathematical Methods for Physical Sciences
  • Mathematical Methods of Engineering Analysis
  • Mathematics, Basic Math and Algebra
  • Mathematics for Computer Science
  • Mathematics for Computer Science
  • Mathematics for Computer Scientists
  • Mathematics For Engineering Students
  • Mathematics Formulary
  • Motion Mountain
  • Music: A Mathematical Offering
  • Mysteries of the Sun


  • Natural Disasters
  • New Frontiers in Graph Theory
  • Noise Control, Reduction and Cancellation Solutions in Engineering
  • Nondestructive Testing Methods and New Applications
  • Nonlinear Optics
  • Notes on Coarse Geometry
  • Notes on Elementary Particle Physics
  • Notes on Quantum Mechanics


  • Observing the Sky from 30S
  • On Particle Physics
  • Operating Systems: Three Easy Pieces


  • Particle Physics Course Univ. Cape Town
  • Particle Physics Lecture Notes
  • People’s Physics Book
  • Perspectives in Quantum Physics: Epistemological, Ontological and Pedagogical
  • Photons, Schmotons
  • Physics Lectures
  • Physics Tutorials
  • Physics Study Guides
  • Pioneers of Science
  • Practical Astronomy
  • Practical Astronomy for Engineers
  • Preparing for College Physics
  • Primer Of Celestial Navigation
  • Principal Component Analysis – Multidisciplinary Applications
  • Publications of the Astronomical Society of the Pacific Volume 1


  • Quantum Dissipative Systems
  • Quantum Field Theory
  • Quantum Fluctuations
  • Quantum Information Theory
  • Quantum Magnetism
  • Quantum Mechanics
  • Quantum Mechanics
  • Quantum Mechanics: A Graduate Course
  • Quantum Mechanics: An Intermediate Level Course
  • Quantum Notes
  • Quantum Physics Notes
  • Quantum Theory of Many-Particle Systems
  • Quantum Transients


  • Recreations in Astronomy
  • Relativistic Quantum Dynamics
  • Relativity: The Special and General Theory
  • Review of Basic Mathematics
  • Riemann Surfaces, Dynamics and Geometry Course Notes


  • Short History of Astronomy
  • Sintering of Ceramics – New Emerging Techniques
  • Solitons
  • Some Basic Principles from Astronomy
  • Special Relativity
  • Spherical Astronomy
  • Star-Gazer’s Hand-Book
  • Statistical Physics
  • Street-Fighting Mathematics
  • String Theory
  • Structures of Life
  • Supernova Remnants: The X-ray Perspective
  • Superspace: One Thousand and One Lessons in Supersymmetry
  • System of Systems


  • The Astrobiology Primer: An Outline of General Knowledge
  • The Astronomy and the Bible
  • The Astronomy of the Bible: An Elementary Commentary on the Astronomical References of Holy Scripture
  • The Basic Paradoxes of Statistical Classical Physics and Quantum Mechanics
  • The Beginning and the End
  • The Beginning and the End of the Universe
  • The Complete Idiot’s Guide to the Sun
  • The Convenient Setting of Global Analysis
  • The Eightfold Way: The Beauty of Klein’s Quartic Curve
  • The General Theory of Relativity
  • The Geology of Terrestrial Planets
  • The Geometry of the Sphere
  • The Handbook of Essential Mathematics
  • The Moon: A Full Description and Map of its Principal Physical Features
  • The Open Agenda
  • The Origin of Mass in Particle Physics
  • The Particle Detector Brief Book
  • The Physics Hypertextbook
  • The Physics of Quantum Mechanics
  • The Planet Mars
  • The Small n Problem in High Energy Physics
  • The Story of Eclipses
  • The Story of the Heavens
  • The Structure of Life
  • The Wonder Book of Knowledge
  • The World According to the Hubble Space Telescope
  • The Zij as-Sanjari of Gregory Chioniades (June 27, 2009)
  • Three Dimensional Geometry


  • Understanding Physics
  • Unfolding the Labyrinth
  • Utility of Quaternions in Physics
  • Uses of Astronomy