Category Archives: Article

Proof by Mathematical Induction

June 09, 2013Article6674 vuesEdit
This article explores the potency of proofs by induction with 4 different stunning puzzles, from a lock puzzle and a lion issue, to the monk problem and the pencil conundrum!

Spacetime of General Relativity

June 02, 2013Article14199 vuesEdit
Most popular science explanations of the theory of general relativity are very nice-looking. But they are also deeply misleading. This article presents you a more accurate picture of the spacetime envisioned by Albert Einstein.

Does God play dice?

May 28, 2013Article1725 vuesEdit
For Albert Einstein, the answer is no. But what did he mean? Has the greatest theoretical physicist of all time really missed the bandwagon of quantum physics? What are the real issues of the controversy that has opposed him to the Copenhagen School (Bohr, Heisenberg ...)? Back to the physics of the early twentieth century, its history, philosophy and ideas.

Hypothesis Test with Statistics: Get it Right!

May 15, 2013Article4514 vuesEdit
Statistician Johnson recently claimed that up to 25% of published scientific experimental results were just wrong! To see why, let's get to the bottom of the scientific method! And it's probably more complicated than you think. In this article, we apply it rigorously to "prove" $\pi=3$. This will highlight the actually mechanism of the scientific method, its limits, and how much messages of experiments are often deformed!

The Amazing Physics of Water in Trees

April 15, 2013Article64600 vuesEdit
As explained by Derek Muller on Veritasium, the flow of water in trees involves complex physical phenomena including pressure, osmosis, negative pressure, capillarity and evapotranspiration. What seems simple will blow your mind!

Dynamics, Chaos, Fractals (pt 2)

April 13, 2013Article2138 vuesEdit
Dynamical systems such as a system of 3 planetary bodies can exhibit surprisingly complicated behavior. If the initial state of the system is slightly varied, the resulting system behaves in a radically different manner. This "sensitivity to initial conditions" is a key element of what's become (perhaps disproportionately) well-known as chaos. Using the mathematical notion of iterative systems, we can model such systems and understand how chaos arises out of deceptively simple foundations.

A Model of Football Games

April 04, 2013Article7795 vuesEdit
Back then, I simulated the outcome of the 2006 World Cup, based on a modelling of football games. This article explains this model and presents its results.

Dynamics, Chaos, Fractals (pt 1)

March 30, 2013Article4542 vuesEdit
The study of dynamical systems, natural or abstract systems that evolve at each instance in time according to a specific rule, is an active and fruitful area of research in mathematics. Its study has yielded insights into the nature of social networks such as Facebook, the spread of diseases such as influenza, and the behavior of the financial markets. In this series of posts, we'll look in depth at dynamical systems, as well as at the related subjects of chaos theory and fractals, all of which are both interesting and useful for understanding our world.

Poincaré Conjecture and Homotopy

March 25, 2013Article8883 vuesEdit
Poincaré conjecture is the most recent major proven theorem. Posited a century ago by Henri Poincaré, this major conjecture of topology was solved by Gregori Perelman. It has revolutionized our understanding of space and raised intriguing questions regarding the global structure of our Universe.

Shannon’s Information Theory

March 17, 2013Article52328 vuesEdit
Claude Shannon may be considered one of the most influential person of the 20th Century, as he laid out the foundation of the revolutionary information theory. Yet, unfortunately, he is virtually unknown to the public. This article is a tribute to him. And the best way I've found is to explain some of the brilliant ideas he had.

Dynamics of the Wave Function: Heisenberg, Schrödinger, Collapse

March 04, 2013Article10763 vuesEdit
On one hand, the dynamics of the wave function can follow Schrödinger equation and satisfy simple properties like Heisenberg uncertainty principle. But on the other hand, it can be probabilistic. This doesn't mean that it's totally unpredictable, since the unpredictability is amazingly predictable. Find out how these two dynamics work!

Space Deformation and Group Representation

February 22, 2013Article2738 vuesEdit
All along the 20th century, pure algebraists have dug deep into the fundamental structures of mathematics. In this extremely abstract effort, they were greatly help by the possibility of representing these structures by space deformations, which could then be understood much better. This has led to breakthroughs, including the proof of Fermat's las theorem. This article introduces the ideas of group representations.

Conditional Probabilities: Know what you Learn

February 04, 2013Article5015 vuesEdit
Suppose a man has two children, one of them being a boy. What's the probability of the other one being a boy too? This complex question has intrigued thinkers for long until mathematics eventually provided a great framework to better understanding of what's known as conditional probabilities. In this article, we present the ideas through the two-children problem and other fun examples.

Geometry and General Relativity

January 22, 2013Article3791 vuesEdit
From our "intrinsic" point-of-view on the surface of the Earth, it appears to be flat, but if we examine the Earth from the "extrinsic" point of view, somewhere off the Earth's surface, we can see that it is clearly a curved surface. Amazingly, it is possible to determine that the Earth is spherical simply by taking measurements on its surface, and it is possible to generalize these measurements in order to study the shape of the universe. Mathematicians such as Riemann did just this, and Einstein was able to apply these geometric ideas to his "general theory of relativity", which describes the relation between gravitation, space, and time.

The Essence of Quantum Mechanics

January 21, 2013Article11595 vuesEdit
Quantum mechanics is the most accurate and tested scientific theory, Its applications to real life are countless, as all new technologies are based on its principles. Yet, it's also probably the most misunderstood theory, because it constantly contradicts common sense. This article presents the most important features of the theory.

Glaciers: Retreat, Moraines, Valleys, Fjörds

January 09, 2013Article1200 vuesEdit
Glaciers are spectacular phenomenons of nature. The physics they are based on is surprising, while the geological role they have is essential. In this article, we discuss these facts, as well as their retreats and their dangers.

Non-Euclidean Geometry and Map-Making

January 08, 2013Article11298 vuesEdit
Geometry literally means "the measurement of the Earth", and more generally means the study of measurements of different kinds of space. Geometry on a flat surface, and geometry on the surface of a sphere, for example, are fundamentally different. A consequence of this disparity is the fact that it is impossible to create a perfectly accurate (flat) map of the Earth's (spherical) surface. Every map of the Earth necessarily has distortions. In this post we look at a few different methods of map-making and evaluate their distortions as well as their respective advantages.

Euclidean Geometry and Navigation

January 06, 2013Article8573 vuesEdit
This is the first of a series of three posts. In this post we'll see how the Greeks developed a system of geometry - literally "Earth measure" - to assist with planetary navigation. We then will see why their assumption that the Earth is flat means that Euclidean geometry is insufficient for studying the Earth. The Earth's spherical surface looks flat from our perspective, but is actually qualitatively different from a flat surface. In the ensuing posts, we'll see why this implies that it is impossible to make a perfectly accurate map of the Earth, and build on this idea to get a glimpse into Einstein's revolutionary theories regarding the geometry of the space-time universe.

Cryptography and Quantum Physics

December 28, 2012Article2811 vuesEdit
Recent discoveries in the branch of physics known as quantum mechanics have powerful applications in the field of network security - they have the potential to break forms of internet security based on mathematics such as the RSA algorithm, and also present new ways to safely send information. In this article we’ll see how a physics-based method can be used to secure online information.

Topology: from the Basics to Connectedness

December 20, 2012Article8273 vuesEdit
Topology was my favorite course in pure maths. I love it because it's a powerful abstract theory to describe intuitive and visual ideas about space. This article gives you an introduction to this amazing field. We'll introduce graph topology, metric spaces, continuity and connectedness.