Icosian : A graph theory game

Neutral : for everyone

Level 4 of the game "Icosian"

The Icosian ¹ game is a graph theory game. It was invented in 1857 by Sir W.R.Hamilton (1805-1865), a great mathematician to whom we owe - among other things - a reformulation of mechanics' formalism which now bears his name, and quaternions (which we'll tackle in a future article).


In this post, we will focus on a flash version of this game, written by Neamar. More precisely, we will see the mathematical principles behind the game and a method for solving the last two levels.

ATTENTION SPOILER ALERT !

If you don't know this game, don't read this article right now, but try the game beforehand !  

Play Neamar's Icosian game first.

You liked this game and would like to know a bit about graph theory ? You've been tearing your hairs out on the last two levels and would like to see a detailed solution ? Then here you go...

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What is time ?

Easy : for the curious

Sundial of Saint Rémy de Provence
Source : Wikipedia Commons

« Put your hand on a stove for a minute and it will feel like an hour. Sit with a pretty girl for an hour and it will fly like a minute. That's relativity. »

This famous quote, attributed to Einstein ¹ is a good description of the psychological side of time. Time is undeniably linked to our senses, we perceive it through duration, order and simultaneity. What will be left of it if we take those away ?

The physicist and/or mathematician is used to depicting time rather than asking himself about its nature. For him, time is more of a variable, it's the degree of freedom of bodies that interact or move in space. Concretely, this parameter permits us to measure durations (chronometry), and mark order (chronology). We can mathematically define these properties : duration is the locus in a one-dimensional space, order is represented by the orientation of this straight line.

We will see that this mathematical representation of time as a supplementary dimension of space is not characteristic of relativity theory, and Galilean kinematics can use it as well. We will then see clearly the fundamental differences between the concept of time in the two theories and we will try to understand the geometry behind the notorious change of sign of the mathematician's "ds²". We will also see that during the evolution of physical science, time has progressively lost what we thought were fundamental properties of his...

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About numbers, episode 1 : Natural numbers

Easy : for the curious

« God made the integers, all the rest is the work of man » Léopold Kronecker (1823-1891) ¹

There is no satisfying definition of the general concept of number. However, lots of particular numbers can be rigorously defined. Natural numbers, integers, imaginary, transcendent, algebraic, computable numbers, etc. In this "story", we will see examples of numbers and will try to understand them intuitively and visually. This first episode is about natural numbers.

The "natural number" is a concept that fulfills two needs : that of ordering, and that of comparing sets "in power", i.e. by counting.

For the first need, one defines the ordinal numbers, for the second the cardinal numbers. These two notions seem at first hand to be different faces of the same objects...

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The father of relativity theory : Einstein vs Poincaré

Medium : for amateurs

« We are like dwarfs on the shoulders of giants. »
This famous metaphor, attributed to Bernard de Chartres, a XIIth century philosopher, reused by Newton and Pascal among others, is a tribute to savant predecessors and an acknowledgment of the cumulative nature of scientific knowledge.

In this article, we will pay a tribute to Henri Poincaré, a brilliant mathematician, universal thinker and remarkable physicist. However, we will not try to grant him what is not his, but we will acknowledge some of his numerous contributions to the theory of relativity, which main idea is clearly owed to Albert Einstein.

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The group of special relativity

Medium : for amateurs

In its modern interpretation, the principle of relativity is profoundly linked to the group structure of Lorentz transformations.

We will describe the equivalence relation between the two, and at the same time give a geometrical description of what is an inertial reference frame in special relativity.

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