#### Welcome to Science4All

My name is Lê and I believe that the greatest challenge in education is to make science and math appealing.

This is why I aim at bringing enthusiasm and excitement to the readers’ learning experience.

Science4All is also available in French.

Colors: It’s not just about Wavelengths! Colors: It’s not just about Wavelengths!

By Lê Nguyên Hoang |

Colours… What are they really? Are there the same for all of us? And for other animals? How does color addition or subtraction work? How do they work on computers? And on printers? The mysterious (but not dark) world of colors is actually very colourful!

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 2431Colours… What are they really? Are there the same for all of us? And for other animals? How does color addition or subtraction work? How do they work on computers? And on printers? The mysterious (but not dark) world of colors is actually very colourful!

Construction and Definition of Numbers Construction and Definition of Numbers

By Lê Nguyên Hoang |

Although they have been used for thousands of years, an actual definition of numbers was given less than a century ago! From the most fundamental level of set theory, this article takes you to the journey of the construction of natural, integer, rational, real and complex numbers.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 4092Although they have been used for thousands of years, an actual definition of numbers was given less than a century ago! From the most fundamental level of set theory, this article takes you to the journey of the construction of natural, integer, rational, real and complex numbers.

Type Theory: A Modern Computable Paradigm for Math Type Theory: A Modern Computable Paradigm for Math

By Lê Nguyên Hoang |

In 2013, three dozens of today’s brightest minds have just laid out new foundation of mathematics after a year of collective effort. This new paradigm better fits both informal and computationally-checkable mathematics. There is little doubt that it will fundamentally change our perspective on rigorous knowledge, and it could be that, in a few decades, the book they published turns out to be the bedrock of all mathematics, and, by extension, all human knowledge! Have a primer of this upcoming revolution, with this article on type theory, the theory that the book builds upon!

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 7395In 2013, three dozens of today’s brightest minds have just laid out new foundation of mathematics after a year of collective effort. This new paradigm better fits both informal and computationally-checkable mathematics. There is little doubt that it will fundamentally change our perspective on rigorous knowledge, and it could be that, in a few decades, the book they published turns out to be the bedrock of all mathematics, and, by extension, all human knowledge! Have a primer of this upcoming revolution, with this article on type theory, the theory that the book builds upon!

Dynamics, Chaos, Fractals (pt 2) Dynamics, Chaos, Fractals (pt 2)

By Scott McKinney |

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.

By Scott McKinney |

**Updated:**2015-12 |**Views**: 1125Dynamical 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.

Hypothesis Test with Statistics: Get it Right! Hypothesis Test with Statistics: Get it Right!

By Lê Nguyên Hoang |

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!

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 3466Statistician 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!

P versus NP: A Crucial Open Problem P versus NP: A Crucial Open Problem

By Lê Nguyên Hoang |

P=NP is probably today’s most crucial open problem. Not only is it a very theoretical question in computer science and mathematics, but it also has major implications to real world. Its resolution could revolutionize the world. This article gives the definition and major results of P=NP.

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 5145P=NP is probably today’s most crucial open problem. Not only is it a very theoretical question in computer science and mathematics, but it also has major implications to real world. Its resolution could revolutionize the world. This article gives the definition and major results of P=NP.

The Secretary/Toilet Problem and Online Optimization The Secretary/Toilet Problem and Online Optimization

By Lê Nguyên Hoang |

A large chunk of applied mathematics has focused on optimizing something with respect to all relevant data. However, in practice, especially in the online world, the data is not available to us, and, yet, we’re still expected to make nearly optimal decisions. This problem is exemplified by the famous secretary problem, where a manager needs to decide to hire candidates right after interviews, even though he has not yet met all the candidates. In this article, we review this classic as well as many very recent developments.

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 1933A large chunk of applied mathematics has focused on optimizing something with respect to all relevant data. However, in practice, especially in the online world, the data is not available to us, and, yet, we’re still expected to make nearly optimal decisions. This problem is exemplified by the famous secretary problem, where a manager needs to decide to hire candidates right after interviews, even though he has not yet met all the candidates. In this article, we review this classic as well as many very recent developments.

The Addictive Mathematics of the 2048 Tile Game The Addictive Mathematics of the 2048 Tile Game

By Lê Nguyên Hoang |

2048 is the Internet sensation of the year. This very addictive game has been downloaded hundred of millions of times. Interestingly, this game raises plenty of intriguing mathematical questions. This article unveils some of them!

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 91302048 is the Internet sensation of the year. This very addictive game has been downloaded hundred of millions of times. Interestingly, this game raises plenty of intriguing mathematical questions. This article unveils some of them!

Conditional Probabilities: Know what you Learn Conditional Probabilities: Know what you Learn

By Lê Nguyên Hoang |

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.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 2986Suppose 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.

Game Theory and the Nash Equilibrium Game Theory and the Nash Equilibrium

By Lê Nguyên Hoang |

In the movie “A Beautiful Mind”, the character is John Nash. He is one of the founders of a large and important field of applied mathematics called game theory. Game Theory is the study of human interactions. Its fallouts in economy, politics or biology are countless. This article gives you an introduction to the concepts of this amazing way of thinking.

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 8387In the movie “A Beautiful Mind”, the character is John Nash. He is one of the founders of a large and important field of applied mathematics called game theory. Game Theory is the study of human interactions. Its fallouts in economy, politics or biology are countless. This article gives you an introduction to the concepts of this amazing way of thinking.

Euclidean Geometry and Navigation Euclidean Geometry and Navigation

By Scott McKinney |

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.

By Scott McKinney |

**Updated:**2016-02 |**Views**: 2225This 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.

Regulation of Electricity Markets Regulation of Electricity Markets

By Lê Nguyên Hoang |

Electricity markets are not like any markets. In particular, they cannot be liberalized without regulation. In the article, I list the reasons why this market is specific and I conclude by giving you important features of a good regulation.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 1350Electricity markets are not like any markets. In particular, they cannot be liberalized without regulation. In the article, I list the reasons why this market is specific and I conclude by giving you important features of a good regulation.

High Dynamic Range and Tone Mapping High Dynamic Range and Tone Mapping

By Lê Nguyên Hoang |

Our eyes are amazing! Even today’s cameras are nowhere near competing with them. However, the recent development of high dynamic range (HDR) and tone mapping technologies creates new possibilities to get images nearly as awesome as what our eyes really see!

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 1462Our eyes are amazing! Even today’s cameras are nowhere near competing with them. However, the recent development of high dynamic range (HDR) and tone mapping technologies creates new possibilities to get images nearly as awesome as what our eyes really see!

A Mathematical Guide to Selling A Mathematical Guide to Selling

By Lê Nguyên Hoang |

How to best sell a good? Should we auction it like in movies? Since the 1960s, economists have addressed this question mathematically and found surprising results. Most notably, in 1981, Nobel prize winner Roger Myerson proved that most auctions you could think of would win you just as much as any basic auction, but that, as well, you could do better using his approach. Since, today, billions of dollars are at play in online auctions, you can imagine how hot a topic it has now become!

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 2187How to best sell a good? Should we auction it like in movies? Since the 1960s, economists have addressed this question mathematically and found surprising results. Most notably, in 1981, Nobel prize winner Roger Myerson proved that most auctions you could think of would win you just as much as any basic auction, but that, as well, you could do better using his approach. Since, today, billions of dollars are at play in online auctions, you can imagine how hot a topic it has now become!

The Limitless Vertigo of Cantor’s Infinite The Limitless Vertigo of Cantor’s Infinite

By Lê Nguyên Hoang |

No one believed him. Not even fellow mathematicians. They thought he was wrong. They thought he was crazy. Even he ended up doubting himself and went crazy. And yet, he had mathematically proved it all. Georg Cantor had figured out how to manipulate the infinite. Even more remarkable, he showed that there were actually several infinities; and some are bigger than others!

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 1751No one believed him. Not even fellow mathematicians. They thought he was wrong. They thought he was crazy. Even he ended up doubting himself and went crazy. And yet, he had mathematically proved it all. Georg Cantor had figured out how to manipulate the infinite. Even more remarkable, he showed that there were actually several infinities; and some are bigger than others!

Differential Calculus and the Geometry of Derivatives Differential Calculus and the Geometry of Derivatives

By Lê Nguyên Hoang |

Differential calculus is one of the most important concept of mathematics for science and engineering. This article focuses on its fundamental meaning.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 3839Differential calculus is one of the most important concept of mathematics for science and engineering. This article focuses on its fundamental meaning.

Homotopy Type Theory and Higher Inductive Types Homotopy Type Theory and Higher Inductive Types

By Lê Nguyên Hoang |

In this article, we explore the possibilities allowed by higher inductive types. They enable a much more intuitive formalization of integers and new mind-blowing definitions of the (homotopical) circle and sphere.

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 1599In this article, we explore the possibilities allowed by higher inductive types. They enable a much more intuitive formalization of integers and new mind-blowing definitions of the (homotopical) circle and sphere.

Colours and Dimensions Colours and Dimensions

By Lê Nguyên Hoang |

You’ve probably learned early on that there are three primary colours. But why three? And why these three? Surprisingly, the answer lies in the beautiful mathematics of linear algebra and (high) dimension spaces!

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 1489You’ve probably learned early on that there are three primary colours. But why three? And why these three? Surprisingly, the answer lies in the beautiful mathematics of linear algebra and (high) dimension spaces!

The Most Beautiful Equation of Math: Euler’s Identity The Most Beautiful Equation of Math: Euler’s Identity

By Lê Nguyên Hoang |

In 1988, Euler’s identity was elected most beautiful theorem of mathematics. It has been widely taught worldwide. But have you ever stopped to really sense the meaning of this incredible formula? This article does.

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 34165In 1988, Euler’s identity was elected most beautiful theorem of mathematics. It has been widely taught worldwide. But have you ever stopped to really sense the meaning of this incredible formula? This article does.

From Divide and Conquer to Parallelization From Divide and Conquer to Parallelization

By Lê Nguyên Hoang |

Divide and conquer is a extremely powerful concept that is being used a lot in computer science, and that can also be applied in real life. We present its application to sorting algorithms. Then we’ll talk about a major fundamental open mathematical problem, called P=NC.

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 1333Divide and conquer is a extremely powerful concept that is being used a lot in computer science, and that can also be applied in real life. We present its application to sorting algorithms. Then we’ll talk about a major fundamental open mathematical problem, called P=NC.

Self-Reference, Math Foundations and Gödel’s Incompleteness Self-Reference, Math Foundations and Gödel’s Incompleteness

By Lê Nguyên Hoang |

Although highly appreciated by artists, self-reference has been a nightmare for mathematicians. It took one of the greatest, Kurt Gödel, to provide a better understanding of it. This article deals with paradoxes, recursion, fractals and Gödel’s incompleteness theorems.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 2935Although highly appreciated by artists, self-reference has been a nightmare for mathematicians. It took one of the greatest, Kurt Gödel, to provide a better understanding of it. This article deals with paradoxes, recursion, fractals and Gödel’s incompleteness theorems.

Dynamics, Chaos, Fractals (pt 1) Dynamics, Chaos, Fractals (pt 1)

By Scott McKinney |

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.

By Scott McKinney |

**Updated:**2016-02 |**Views**: 2696The 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.

The Amazing Physics of Water in Trees The Amazing Physics of Water in Trees

By Lê Nguyên Hoang |

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!

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 18292As 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!

The Biology Civil War Opposing Kin to Group Selection The Biology Civil War Opposing Kin to Group Selection

By Lê Nguyên Hoang |

In 2010, a controversial article published in Nature violently criticized the last 40 years of developments in evolutionary biology, triggering an ongoing war within the scientific community. This article explains the essence of the controversy!

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 2212In 2010, a controversial article published in Nature violently criticized the last 40 years of developments in evolutionary biology, triggering an ongoing war within the scientific community. This article explains the essence of the controversy!

Imaginary and Complex Numbers Imaginary and Complex Numbers

By Lê Nguyên Hoang |

My first reaction to imaginary numbers was… What the hell is that? Even now, I have trouble getting my head around these mathematical objects. Fortunately, I have a secret weapon: Geometry! This article proposes constructing complex numbers with a very geometrical and intuitive approach, which is probably very different from what you’ve learned (or will learn).

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 4203My first reaction to imaginary numbers was… What the hell is that? Even now, I have trouble getting my head around these mathematical objects. Fortunately, I have a secret weapon: Geometry! This article proposes constructing complex numbers with a very geometrical and intuitive approach, which is probably very different from what you’ve learned (or will learn).

Darwin’s Theory of Evolution Darwin’s Theory of Evolution

By Lê Nguyên Hoang |

Darwin’s theory of evolution is one of the greatest of all time. It’s also one of the most controversial. This article presents the most fundamental concepts of the theory. We’ll also provide results of modern science and discuss to which extend to they confirm Darwin’s postulates.

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 9814Darwin’s theory of evolution is one of the greatest of all time. It’s also one of the most controversial. This article presents the most fundamental concepts of the theory. We’ll also provide results of modern science and discuss to which extend to they confirm Darwin’s postulates.

HDI: a measure of human capabilities HDI: a measure of human capabilities

By Estève Giraud |

By Estève Giraud |

**Updated:**2016-02 |**Views**: 3957Linear Algebra and Higher Dimensions Linear Algebra and Higher Dimensions

By Lê Nguyên Hoang |

Linear algebra is a one of the most useful pieces of mathematics and the gateway to higher dimensions. Using Barney Stinson’s crazy-hot scale, we introduce its key concepts.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 2992Linear algebra is a one of the most useful pieces of mathematics and the gateway to higher dimensions. Using Barney Stinson’s crazy-hot scale, we introduce its key concepts.

Mechanism Design and the Revelation Principle Mechanism Design and the Revelation Principle

By Lê Nguyên Hoang |

Whenever you need to make a group of people interact, you are designing a mechanism. If you want to achieve a good interaction, you need to make sure your mechanism is well designed. In this article, I’ll show you main features of mechanisms through various examples. I’ll also talk about a great mathematical tool for mechanism design: the revelation principle.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 2897Whenever you need to make a group of people interact, you are designing a mechanism. If you want to achieve a good interaction, you need to make sure your mechanism is well designed. In this article, I’ll show you main features of mechanisms through various examples. I’ll also talk about a great mathematical tool for mechanism design: the revelation principle.

Computing Hunger worldwide: the Global Hunger Index (GHI) Computing Hunger worldwide: the Global Hunger Index (GHI)

By Estève Giraud |

The Global Hunger Index was first published in 2006 by the International Food Policy Research Institute and the NGO Welthungerhilfe. In 2007, Concern worldwide joined them. Since then, the Index reports every year the evolution of the hunger situation worldwide and focus on a given topic. How is it calculated? And what is hunger? How are we connected ? What can be done? To learn more, read the article below.

By Estève Giraud |

**Updated:**2015-12 |**Views**: 434The Global Hunger Index was first published in 2006 by the International Food Policy Research Institute and the NGO Welthungerhilfe. In 2007, Concern worldwide joined them. Since then, the Index reports every year the evolution of the hunger situation worldwide and focus on a given topic. How is it calculated? And what is hunger? How are we connected ? What can be done? To learn more, read the article below.

The coach’s dilemma. The coach’s dilemma.

By Herminio López |

By Herminio López |

**Updated:**2016-02 |**Views**: 614Model-Dependent Realism Model-Dependent Realism

By Lê Nguyên Hoang |

Introduced by the two renowned theoretical physicists Stephen Hawking and Leonard Mlodinov in their book The Grand Design in 2010, model-dependent realism is a new controversial understanding of the universe. Based on solid logical reasonings and recent developments in physics, this concept may well be an incredible breakthrough for philosophy and science, as well as metaphysics.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 2757Introduced by the two renowned theoretical physicists Stephen Hawking and Leonard Mlodinov in their book The Grand Design in 2010, model-dependent realism is a new controversial understanding of the universe. Based on solid logical reasonings and recent developments in physics, this concept may well be an incredible breakthrough for philosophy and science, as well as metaphysics.

The New Big Fish Called Mean-Field Game Theory The New Big Fish Called Mean-Field Game Theory

By Lê Nguyên Hoang |

In recent years, at the interface of game theory, control theory and statistical mechanics, a new baby of applied mathematics was given birth. Now named mean-field game theory, this new model represents a new active field of research with a huge range of applications! This is mathematics in the making!

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 6741In recent years, at the interface of game theory, control theory and statistical mechanics, a new baby of applied mathematics was given birth. Now named mean-field game theory, this new model represents a new active field of research with a huge range of applications! This is mathematics in the making!

Euler’s Formula and the Utilities Problem Euler’s Formula and the Utilities Problem

By Lê Nguyên Hoang |

I was a kid when I was first introduced to the deceptively simple utilities problem. It’s only lately that I’ve discovered its solution! And it’s an amazing one! Indeed, it provides a wonderful insight into some fundamental mathematics, including Euler’s formula! This is nothing less than the gateway to the wonderful world of algebraic topology!

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 7344I was a kid when I was first introduced to the deceptively simple utilities problem. It’s only lately that I’ve discovered its solution! And it’s an amazing one! Indeed, it provides a wonderful insight into some fundamental mathematics, including Euler’s formula! This is nothing less than the gateway to the wonderful world of algebraic topology!

Spacetime of Special Relativity Spacetime of Special Relativity

By Lê Nguyên Hoang |

Einstein’s theory of relativity is the best-known breakthrough of the History of science. The reason for that isn’t only the accuracy of the theory, but also and mainly its beauty. As Einstein once said: “Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.” This is what the article aims at showing Einstein’s simple ideas of special relativity and their beauty.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 2188Einstein’s theory of relativity is the best-known breakthrough of the History of science. The reason for that isn’t only the accuracy of the theory, but also and mainly its beauty. As Einstein once said: “Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.” This is what the article aims at showing Einstein’s simple ideas of special relativity and their beauty.

Non-Euclidean Geometry and Map-Making Non-Euclidean Geometry and Map-Making

By Scott McKinney |

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.

By Scott McKinney |

**Updated:**2016-02 |**Views**: 2837Geometry 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.

From Britain’s coast to Julia set: an introduction to fractals From Britain’s coast to Julia set: an introduction to fractals

By Thomas C |

By Thomas C |

**Updated:**2016-02 |**Views**: 2355Proof by Mathematical Induction Proof by Mathematical Induction

By Lê Nguyên Hoang |

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!

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 4675This 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!

Fourier Analysis: Signals and Frequencies Fourier Analysis: Signals and Frequencies

By Lê Nguyên Hoang |

Fourier analysis is a fundamental theory in mathematics with an impressive field of applications. From creating radio to hearing sounds, this concept is a translation between two mathematical worlds: Signals and Frequencies. Here is an introduction to the theory.

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 4929Fourier analysis is a fundamental theory in mathematics with an impressive field of applications. From creating radio to hearing sounds, this concept is a translation between two mathematical worlds: Signals and Frequencies. Here is an introduction to the theory.

Beauty, the Driving Force of our Quest for Truth Beauty, the Driving Force of our Quest for Truth

By Lê Nguyên Hoang |

According to Matthew Colless, the most important aspect of science is beauty. Not only is it what inspires scientists and their quests, I’d even claim that it’s also the compass that guide them in their quests, in a deeper and more surprising way that one can imagine!

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 1042According to Matthew Colless, the most important aspect of science is beauty. Not only is it what inspires scientists and their quests, I’d even claim that it’s also the compass that guide them in their quests, in a deeper and more surprising way that one can imagine!

Geometry and General Relativity Geometry and General Relativity

By Scott McKinney |

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.

By Scott McKinney |

**Updated:**2015-12 |**Views**: 1472From 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 Surprising Flavor of Infinite Series The Surprising Flavor of Infinite Series

By Lê Nguyên Hoang |

1+2+4+8+16+…=-1, as proven by Henry Reich on Minute Physics! Now, as a mathematician, I must say that his proof is far from being rigorous. In fact, anyone familiar with the surprising flavor of infinite series should not find it convincing. Surprisingly though, his proof can be rigorously and naturally justified! Find out how!

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 60301+2+4+8+16+…=-1, as proven by Henry Reich on Minute Physics! Now, as a mathematician, I must say that his proof is far from being rigorous. In fact, anyone familiar with the surprising flavor of infinite series should not find it convincing. Surprisingly though, his proof can be rigorously and naturally justified! Find out how!

Symmetries and Group Theory Symmetries and Group Theory

By Lê Nguyên Hoang |

Beauty is extremely hard to define. Yet, physicists and artists seem to agree on an important feature of beauty, created by mathematicians: Symmetries. This article aims at introducing the beauty and the concepts on symmetries, from the basic geometrical symmetries to the more abstract fundamental automorphisms.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 1755Beauty is extremely hard to define. Yet, physicists and artists seem to agree on an important feature of beauty, created by mathematicians: Symmetries. This article aims at introducing the beauty and the concepts on symmetries, from the basic geometrical symmetries to the more abstract fundamental automorphisms.

Can we measure peace ? The Global Peace Index (GPI) Can we measure peace ? The Global Peace Index (GPI)

By Estève Giraud |

By Estève Giraud |

**Updated:**2016-04 |**Views**: 42The Massive Puzzles of Gravity The Massive Puzzles of Gravity

By Lê Nguyên Hoang |

This article follows the footsteps of the giants of physics that have moulded our current understanding of gravity. It is a series of brilliant inspirations, usually accompanied by deceiving misconceptions. After all, even today, gravity is still a slippery concept.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 2346This article follows the footsteps of the giants of physics that have moulded our current understanding of gravity. It is a series of brilliant inspirations, usually accompanied by deceiving misconceptions. After all, even today, gravity is still a slippery concept.

A Model of Football Games A Model of Football Games

By Lê Nguyên Hoang |

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.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 4503Back 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.

Logarithms and Age Counting Logarithms and Age Counting

By Lê Nguyên Hoang |

Amusingly, the age difference between a 45-year-old man and a 25-year-old woman doesn’t seem as big as the age difference between them 20 years earlier, when the woman was a little 5-year-old girl. This remark was the insight the late science popularizer Albert Jacquart liked to give to his readers to explain logarithms. This article pays tribute to the great scientist by introducing age difference as he liked to tell it.

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 2255Amusingly, the age difference between a 45-year-old man and a 25-year-old woman doesn’t seem as big as the age difference between them 20 years earlier, when the woman was a little 5-year-old girl. This remark was the insight the late science popularizer Albert Jacquart liked to give to his readers to explain logarithms. This article pays tribute to the great scientist by introducing age difference as he liked to tell it.

Entropy and the Second Law of Thermodynamics Entropy and the Second Law of Thermodynamics

By Lê Nguyên Hoang |

The second law of thermodynamics is my favorite law in physics, mainly because of the troubling puzzles it raises! Indeed, what your professors may have forgotten to tell you is that this law connects today’s world to its first instant, the Big Bang! Find out why!

By Lê Nguyên Hoang |

**Updated:**2016-01 |**Views**: 18779The second law of thermodynamics is my favorite law in physics, mainly because of the troubling puzzles it raises! Indeed, what your professors may have forgotten to tell you is that this law connects today’s world to its first instant, the Big Bang! Find out why!

Glaciers: Retreat, Moraines, Valleys, Fjörds Glaciers: Retreat, Moraines, Valleys, Fjörds

By Lê Nguyên Hoang |

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.

By Lê Nguyên Hoang |

**Updated:**2016-02 |**Views**: 865Glaciers 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.

The Tortuous Geometry of the Flat Torus The Tortuous Geometry of the Flat Torus

By Lê Nguyên Hoang |

Take a square sheet of paper. Can you glue opposite sides without ever folding the paper? This is a conundrum that many of the greatest modern mathematicians, like Gauss, Riemann, and Mandelbrot, couldn’t figure out. While John Nash did answer yes, he couldn’t say how. After 160 years of research, Vincent Borrelli and his collaborators have finally provided a revolutionary and breathtaking example of a bending of a square sheet of paper! And it is spectacularly beautiful!

By Lê Nguyên Hoang |

**Updated:**2015-12 |**Views**: 8742Take a square sheet of paper. Can you glue opposite sides without ever folding the paper? This is a conundrum that many of the greatest modern mathematicians, like Gauss, Riemann, and Mandelbrot, couldn’t figure out. While John Nash did answer yes, he couldn’t say how. After 160 years of research, Vincent Borrelli and his collaborators have finally provided a revolutionary and breathtaking example of a bending of a square sheet of paper! And it is spectacularly beautiful!