Adonis Diaries

Posts Tagged ‘geometry

What the orientation of the streets in Paris do tell us of its history?

La géométrie de la capitale nous raconte les principaux épisodes de son développement

Ceci est une carte visant à révéler l’orientation des rues de Paris. Si elle paraît sophistiquée, sa matière première n’en est pas moins sommaire: les tracés de voies du projet OpenStreetMap.

La couleur d’une rue dépend de son angle sur une échelle de 0 à 90°: deux teintes ont été utilisées, jaune-orangé et magenta, et elles sont d’autant plus claires que l’on se rapproche de l’axe méridien (Nord-Sud) ou parallèle (Est-Ouest).

Cet éventail de couleur est organisé de façon à ce que deux rues perpendiculaires aient la même couleur et à ce qu’une rue qui «perturbe» un quartier bien ordonné ait une couleur différente.

Certaines formes sur la carte, par le jeu des couleurs et des juxtapositions, ont éveillé ma curiosité. Simple géomaticien, peu rompu à l’histoire et à l’urbanisme, je me suis réduit à détailler le procédé de fabrication de la carte sur mon blog.

Plus tard, je me suis lancé dans un travail d’investigation afin de tenter de la comprendre.

De manière générale, Paris s’est développée par à-coups.

Ses différentes enceintes en sont la trace. Un réseau de rues peut se développer progressivement à partir d’un axe de circulation en de multiples ramifications, tel les nervures d’une feuille. Il peut aussi être bousculé par des évènements politiques, historiques.

<a href="https://www.flickr.com/photos/10519370@N04/15654896891/">Voir la carte en grand.</a>

1.La carte, regardée de loin

Convergence

À l’instar d’un tableau, une carte dévoile des choses différentes selon la distance à laquelle on la regarde. Voyons ce que nous réserve une vision globale de la carte.

Cliquez pour agrandir la carte. Source: Bibliothèque en ligne Gallica

Par son jeu de lignes, l’ossature de Paris nous rappelle constamment à son berceau, l’île de la Cité.

Cette île vit la naissance de Lutèce, en 52 av. J.C., après la victoire de Jules César sur Vercingétorix. À mesure que l’on s’en éloigne, les voies semblent régies par d’autres polarités. Le déplacement dans l’espace suit celui du temps.

Parallèles

Les rues épousent souvent des parallèles aux voies navigables: la Seine et ses canaux. Ces derniers ont constitué une épine dorsale à partir de laquelle s’est développée la ville.

L’historien du XVIIIe siècle Jules Michelet qualifiait d’ailleurs la Seine de «grande rue»: ses rives accueillaient, jusqu’au XIXe siècle, moulins, abattoirs, tanneries, établissements de bains, blanchisseries, pompes à eau, activités de pêche.

Jusqu’à l’arrivée du chemin de fer, les deux tiers environ de l’approvisionnement de Paris arrivaient par la Seine.

Cardo Maximus

A l’intérieur de l’enceinte Charles V (XIVe siècle) et de l’enceinte de Philippe-Auguste (XIIIe siècle), dont les périmètres figurent sur l’image ci-dessous, le réseau de rues est largement perpendiculaire.

Cliquez pour agrandir.

Lutèce était construite autour du Cardo Maximus, l’actuelle rue Saint-Jacques de l’île de la Cité, selon un plan quadrillé typique des villes neuves coloniales. Paris s’est souvent reconstruite sur elle-même, conservant en son centre ce schéma romain.

2.Quand on parcourt la carte latéralement

Perspective

Nous discernons sur la carte un axe Est-Ouest.

Cliquez pour agrandir

Préfiguré par l’avenue Victoria, cet axe ne cessera de s’étendre à partir du XVIIIe siècle. Il lie aujourd’hui l’avenue des Champs-Élysées, la rue Saint-Honoré, la rue de Rivoli et la rue du Faubourg Saint-Antoine.

Les hauts lieux de pouvoir et de culture, le Louvre, le Palais Royal, sont mis en scène de façon magistrale par cette perspective monumentale.

Dichotomie

Voir sur le GéoPortail (OpenStreetMap + Carte État-Major 1820 + Carte Topo 1906 + Photos Aériennes)

Les voies à l’Ouest forment des figures plus alambiquées qu’à l’Est. À partir de la seconde moitié du XVIIIe siècle, l’Ouest est le lieu d’opérations de prestige, prestige que reflètent des places rayonnantes, telles celle de l’Etoile, où se rejoignent pas moins de douze avenues. L’Est, lui, se spécialise dans les activités industrielles et artisanales.

3.Percées

L’avenue de l’Opéra

Sur la carte, des rues se superposent à un réseau préexistant.

Auparavant, les quartiers se développaient en faubourgs le long d’axes de circulation.

Sous Napoléon III, au XIXe siècle, Georges Eugène Haussmann aura pour mission d’assainir et d’embellir la ville. C’est ainsi qu’il détruira, rebâtira sans compter, afin de tracer des voies larges, salubres et somptueuses. On lui doit en grande partie le visage actuel de la capitale.

Boulevard Magenta

Comparaison carte et OpenStreetMap. Voir sur le GéoPortail

La gare de Lyon date de 1855 et celle de l’Est de 1865. Le boulevard Magenta est une traversée importante qui permettra de les relier, ainsi que la place de la République et les boulevards «extérieurs».

Boulevard Raspail

Voir sur le GéoPortail

Le percement du Boulevard Raspail, décidé en 1866, s’étalera sur plus de 40 ans et se fera par tronçons.

Avenue de l’Opéra

Voir sur le GéoPortail

L’Opéra Garnier a été inauguré en 1875. Le percement de l’avenue de l’Opéra, en plus d’offrir un cadre grandiose à ce dernier, connecte le Louvre à la gare Saint-Lazare. Ce chantier prendra dix ans, de 1864 à son année d’inauguration. Il entraînera la destruction d’un quartier ancien, populaire et dégradé.

4.Singularités

Sur la carte, on peut s’étonner de la présence d’îlots de couleur différentes, indiquant un développement a priori singulier.

Un lotissement: le village Orléans

Dans le XIVe arrondissement, la carte comporte une petite tache.

Cliquer pour voir en grand.

À partir de 1820 commencent des opérations de logement très importantes en périphérie de Paris, en réponse à une pression démographique importante. 1830 verra celle du lotissement du village Orléans, visible ci-dessus.

Comme il est possible de le voir sur la carte d’État-Major (1820-1866), le quartier comportait deux rues, qui ont subsisté ajourd’hui: les rues Hallé et Couedic. Le bâti ne suivait pas leur axe mais celui des deux rues environnantes. Désormais, les immeubles suivent l’orientation des deux rues précitées, comme l’indiquent la carte de 1906 et celle d’OpenStreetMap.

Montmartre

Une «anomalie» apparaît au nord de Paris.

CC BY-SA 2.5 par Sam67fr

Il s’agit de la butte Montmartre. L’orientation Nord-Sud de la montée qui mène à la Basilique du Sacré Coeur, édifiée en 1875, et celle de son réseau d’allées, correspond à celle du monument.

Les édifices religieux suivent généralement une orientation Est-Ouest, mais Pie V dira qu’il importe davantage que la façade de l’église soit bien orientée par rapport à la ville.

La basilique offre un promontoire idéal duquel admirer Paris. Réciproquement, son orientation lui permet d’être admirée de face depuis de nombreux endroits de la capitale.

Square du serment de Koufra

En suivant la ceinture verte, on aboutit, vers la porte d’Orléans, à un square dont les allées tracent des obliques.

Photographie Aérienne Géoportail IGN © sur fonds OpenStreetMap MapQuest. Voir sur le GéoPortail

Il s’agit du square du serment de Koufra, créé en 1930. Le général Leclerc prêta ce serment à l’issue de la bataille de Koufra, en 1941. L’emplacement du square est symbolique car c’est par la Porte d’Orléans que ce même général entra le premier à Paris avec les unités alliées. Le parc, en faisant face à la place d’Orléans, rappelle ce moment historique.

5.Renouveau

Parc de Bercy

Voir sur le GéoPortail

Voies d’OpenStreetMap superimposées à une photographie aérienne de 1929 issue du Géoportail IGN ©

Une forme circulaire est de nature à intriguer quelqu’un d’étranger à Paris. Il s’agit du Parc de Bercy et de son dôme.

Ce parc a été réalisé dans les années 90 et son emprise reprend à peu près celle des jardins des demeures du petit château à la propriété des frères Paris. La carte topo IGN de 1906 indique des magasins généraux à cet endroit. Cette prise de vue aérienne de l’IGN de 1929, à laquelle j’ai superposé les données actuelles OpenStreetMap, atteste également de la présence d’entrepôts.

Bassin de La Villette

Voir sur le GéoPortail

Une forme blanche, évoquant un bateau, apparaît dans le quartier de la Villette.

La treemap du MOS indique là une zone d’équipements en 1980 et une zone d’activités en 2000. En allant sur Google Street View, on peut deviner qu’il s’agit là d’entrepôts reconvertis en bureaux.

Le canal Saint-Denis accueillait jadis des activités de fret. Ce secteur est emblématique de la transformation qui a vu, le long des canaux, la disparition progressive des ateliers, usines et entrepôts au profit d’activités de services.

Une fresque trouvée au hasard d’une promenade dans le quartier témoigne effectivement du passé industrieux des bâtiments considérés.

Levallois-Perret

Voir sur le GéoPortail

Dans le sillage de Paris, à Levallois-Perret, on repère des rues bien ordonnées. La commune, classée dixième au niveau de la densité de population, augure d’une nouvelle vision de l’urbanisme. Elle absorbe ses habitants au sein d’un tamis régulier. La fiche communale du MOS nous informe qu’un peu plus de 80% de la surface de la commune est occupée par des espaces construits artificialisés.

Dans la recherche, un bagage scientifique mène à toutes sortes d’expériences. Dans mon cas, l’expérience scientifique, à savoir la conception de cette carte, a amené un travail d’investigation au cours duquel je parcourais l’espace cartographique en même temps que la Toile.

Cet article, bien plus que de vouloir affirmer quelque chose, a pour but d’illustrer que chacun peut mener son enquête à son niveau, grâce aux logiciels et données libres disponibles sur le web.

Ainsi, vous aurez peut-être pu prendre connaissance grâce à lui d’outils géographiques très utiles tels que le GéoPortail, OpenStreetMap ou le MOS île-de-France.

Fractals? Any connection with matters not related with nature?

And why should You care?

Georgemdallas posted this May 2, 2014 (selected as one of the top posts)

Fractal geometry is a field of maths born in the 1970′s and mainly developed by Benoit Mandelbrot. If you’ve already heard of fractals, you’ve probably seen the picture below.

It’s called the Mandelbrot Set and is an example of a fractal shape.

mandelbrot

The geometry that you learnt in school was about how to make shapes; fractal geometry is no different.

While the shapes that you learnt in classical geometry were ‘smooth’, such as a circle or a triangle, the shapes that come out of fractal geometry are ‘rough’ and infinitely complex.

However fractal geometry is still about making shapes, measuring shapes and defining shapes, just like in school.

There are two reasons why you should care about fractal geometry:

1. The process by which shapes are made in fractal geometry is amazingly simple yet completely different to classical geometry. While classical geometry uses formulas to define a shape, fractal geometry uses iteration.

It therefore breaks away from giants such as Pythagoras, Plato and Euclid and heads in another direction. Classical geometry has enjoyed over 2000 years of scrutinisation, Fractal geometry has enjoyed only 40.

2. The shapes that come out of fractal geometry look like nature. This amazing fact that is hard to ignore. As we all know, there are no perfect circles in nature and no perfect squares.

Not only that, but when you look at trees or mountains or river systems they don’t resemble any shapes one is used to in maths.

However with simple formulas iterated multiple times, fractal geometry can model these natural phenomena with alarming accuracy. If you can use simple maths to make things look like the world, you know you’re onto a winner. Fractal geometry does this with ease.

This blog post shall give a quick overview of how to make fractal shapes and show how these shapes can resemble nature.

It shall go on to talk about dimensionality, which is a cool way to measure fractals. It ends by discussing how fractal geometry is also beneficial because randomness can be introduced into the structure of a fractal shape.

The post requires almost no maths and includes lot of pretty pictures

How to make a fractal shape

In normal geometry, shapes are defined by a set of rules and definitions. For instance a triangle consists of three straight lines that are connected.

The rules are that if you have the length of all three sides of the triangle it is completely defined. If you have the length of one side and two corresponding angles the triangle is also defined.

Though the rules defining a triangle are simple, huge amounts of useful maths has come out of it, for instance Pythagoras’ Theorum, sin() cos() and tan(), the proof that the shortest distance between two points is a straight line, etc.

Fractal geometry also defines shapes by rules, however these rules are different to the ones in classical geometry.

In fractal geometry a shape is made in two steps:

First by making a rule about how to change a certain (usually classically geometric) shape. This rule is then applied to the shape again and again, until infinity. In maths when you change something it is usually called a function, so what happens is that a function is applied to a shape recursively, like the diagram below.

fractal1

 

After it has repeated an infinite amount of times, the fractal shape is produced. What are these functions then? What do you mean by repeating infinitely? As always, this is best explained by an example…

A good fractal shape is called the von Koch curve. The rules, or function, are extremely simple. First you start with a straight line. This is your ‘initial shape’:

fractal2

The rules are as follows:

1. Split every straight line into 3 equal segments.

2. Replace the middle segment with an equilateral triangle, and remove the side of the triangle corresponding to the initial straight line.

The process is shown in the figure below:

 

 

fractal3

This is what happens to the straight line, our initial shape, when it goes through the function the first time, the first iteration.

Now, the shape it has produced is fed back into the function again for a second iteration:

 

 

fractal4

 

Remember the rule was that any straight line would be split into thirds, so now 4 lines are split up and made into triangles. The shape that is produced after the second iteration is then fed through the function for a third time. This gets hard to draw in MS paint so I’ve used a couple of pictures from this website for the next few stages:

von_Koch_curve (1)

After this has iterated an infinite amount of times the fractal shape is defined.

This may sound bewildering but it is still possible to analyse it mathematically and visually you can see what the shape starts to look like.

The gif below (from Wikipedia) is a good illustration of what the curve looks like by zooming in on it:

Kochsim

 

The von Koch curve is a great example of a fractal: the rule you apply is simple, yet it results in such a complex shape. This kind of shape is impossible to define using conventional maths, yet so easy to define using fractal geometry.

So who cares about the von Koch curve?

Isn’t it just mathematicians wasting time on weird shapes?

I guess that depends on how you look at it, but I’m convinced it’s useful because it looks exactly like a snowflake. This is made more clear if the initial shape you start with is a triangle rather than a straight line:

Von_Koch_curve

There’s a whole debate to be had on the purpose of maths, but as an Engineer I am inclined to say that one of its purposes is to try and replicate the world around us. The shapes that come out of fractal maths are so different to conventional mathematical shapes and so similar to the world around us that I cannot help but be seduced by this topic. Two other shapes that are favorites of mine are the Barnsley Fern:

6hdxhq75-1353976781

And fractal trees:

fractree

These aren’t drawings or pictures, but mathematical shapes. If you look at the shapes you can see what function repeats itself. For instance on the Barsley Fern the function is to draw 30 or so perpendicular lines out of each straight line. The function repeats itself to and looks like a fern. On the tree you can see that each line branches out twice, which will be the function that repeats itself. Another property about these shapes (though strictly not for all fractals) is that they are self-similar. This means that the shape looks like itself however much you zoom in or out. For instance on the tree above, if you snapped a branch off it and stood it up, it would look like the original tree. If you took a twig from the branch and stood it up, it would still look like the original tree. Again, this is a property that occurs in nature, but until fractal geometry there was not a good way to put it into maths.

Not only do these shapes look like natural objects, but the process of iteration sounds intuitive when thinking about nature. When a tree is growing, its trunk will create branches, these branches create further branches, these branches create twigs. It’s as if the function is a genetic code telling the branch how to grow and repeat itself, eventually creating shapes that are ‘natural’. This may sound like pseudo-science (it definitely is) but I think these are concepts worth considering when you are able to imitate nature so closely.

Right enough about nature, time to talk about how fractals have crazy dimensions.

Dimensions

So now we know what fractal shapes are and how to make them, we would like to know a few things about them. One of them first things to try and figure out is the length of some of these shapes. Let’s go back to the von Koch curve.

In order to figure out how long the full von Koch curve is (after being iterated an infinite amount of times), it is useful to consider what happens at the first stage again:

fractal6

The line is split into three, then the middle section is replaced by two lines that are as long as it (as it’s an equal triangle). So if the original straight line had a length of 1, the length of the curve after the first iteration is 4/3. It turns out that every time you iterate the shape, it gets 4/3 longer. So the length of the curve after the second iteration is 4/3 x 4/3 = 16/9:

fractal5

As 4/3 is greater than 1, the line gets longer every time it is iterated through the function. As you iterate the function an infinite amount of times, the full von Koch curve has a perimeter that is infinitely long! This is the case for all fractal shapes: they have infinitely long perimeters. That isn’t useful for mathematicians so they don’t measure the perimeter of the shape. Now the next few paragraphs require a bit of abstract thought, but if you think a bit outside the box it does make sense.

The perimeter measures the length around something. Length is a 1 dimensional measure of space. Length is 1D because it only measures a straight line. A 2D measure of space is area, 3D is volume. Now we’ve shown that it isn’t useful to measure fractal patterns in 1 dimension as they are infinitely long, but what is odd is that fractal shapes are not 1D, 2D, or 3D. Each fractal shape has it’s own unique dimension, which is usually a number with a decimal place.

The dimension of a fractal shape is a measure of how quickly the shape becomes complicated when you are iterating it. What do we mean by becoming complicated? Well in the von Koch curve you can see that the first few iterations produce quite simple shapes, however at about iteration 4 it starts to become quite small and complex.

The way to measure how fast a shape becomes complicated, and hence its dimension, is to measure how much longer the perimeter gets after each iteration. This makes sense intuitively, as if the line gets much longer after each iteration it is probably becoming very complicated very fast, whereas if the line stays pretty much the same length after each iteration then it probably isn’t getting very complex.

As we’ve already shown, the von Koch curve gets 4/3 longer each iteration. This means that the von Koch curve is 4/3 D, or 1.3333…D. Pretty crazy right? It exists somewhere between 1D and 2D. But this measure is really useful to mathematicians as it gives information about the shape (whereas perimeter doesn’t, it’s always infinite). For instance if there was another fractal shape which was 1.93D, you could say with confidence that that shape gets complex quicker than the von Koch curve, as the perimeter gets 1.93 times longer after each iteration rather than 1.3333, implying it gets complex more quickly. When studying a fractal shape, knowing its dimension is of integral importance.

Randomness

The last thing I’m going to talk about is the fact that randomness can be inserted into fractal shapes. Random (or seemingly random) events occur in nature all the time and affect different things in a variety of different ways, for instance a large part of Information Engineering is dealing with noise, which randomly fluctuates an electronic signal. When trying to replicate this, you usually add randomness on top of a signal. For instance in electronics you would create a nice sine wave and then add noise on top of it (borrowed from this website):

noise

The bottom image is the ‘pure’ wave, and the top image is the wave with noise added on. An inherent assumption when doing this is that there is an underlying ‘pure’ signal which is randomly altered. While this may be true for a lot of electronics, the same cannot be said for nature. Often there isn’t a ‘pure’ shape that is randomly altered around the edges (for instance there are not many fuzzy squares in nature), but rather randomness effects the structure of the shape itself at each stage of its evolution. Classical geometry is not good at incorporating randomness into shapes, whereas fractal geometry can do it easily. For the last time lets turn to the von Koch curve. However this time we will insert randomness into it.

We know the rule is that for each iteration a triangle is created in the middle third of a line. However every time the triangles always faced ‘outwards’. We could insert randomness by saying that for each triangle created, it goes either above the line or below the line depending on a coin toss:

fractal7

Now the shape will develop at random according to the coin toss. For instance after multiple iterations the von Koch curve can look like this:

random

Or it can look completely different. What is cool about this is that you can insert randomness into the shape itself rather than adding it on top of an existing shape. This has exciting potential, for instance (going back to nature) this may be a good way to model random genetic mutations.

 

This blog post has provided a brief introduction to fractal geometry. I hope you’ve found it interesting!

Learning paradigm for our survival; (Nov. 9, 2009)

Einstein, the great theoretical physicist, confessed that most theoretical scientists are constantly uneasy until they discover, from their personal experiences, natural correspondences with their abstract models.  I am not sure if this uneasiness is alive before or after a mathematician is an expert professional. 

For example, mathematicians learn Riemann’s metrics in four-dimensional spaces and solve the corresponding problems. How many of them were briefed that this abstract construct, which was invented two decades before relativity, was to be used as foundation for modern science? Would these kinds of knowledge make a difference in the long run for professional mathematicians?

During the construction of theoretical (mathematical) models, experimental data contribute to revising models to taking into account real facts that do not match previous paradigms. I got into thinking: If mathematicians receive scientific experimental training at the university and are exposed to various scientific fields, they might become better mathematicians by getting aware of the scientific problems and be capable of interpreting purely mathematical models to corresponding natural or social phenomenon that are defying comprehension.

By the way, I am interested to know if there are special search engines for mathematical concepts and models that can be matched to those used in fields of sciences.  By now, it would be absurd if no projects have worked on sorting out the purely mathematical models and theories that are currently applied in sciences.

I got this revelation.  Schools use different methods for comprehending languages and natural sciences.  Kids are taught the alphabet, words, syntax, grammars, spelling and then much later are asked to compose essays.  Why this process is not applied in learning natural sciences?  

Why students learning math are not asked to write essays on how formulas and equations they had learned apply to natural or social realities?

I have strong disagreement on the pedagogy of learning languages:

First, we know that children learn to talk years before they can read. Why then kids are not encouraged to tell verbal stories before they can read?  Why kids’ stories are not recorded and translated into the written words to encourage the kids into realizing that what they read is indeed another story telling medium?

Second, we know that kids have excellent capabilities to memorize verbally and visually entire short sentences before they understand the fundamentals. Why don’t we develop their cognitive abilities before we force upon them the traditional malignant methodology?  The proven outcomes are that kids are devoid of verbal intelligence, hate to read, and would not attempt to write, even after they graduate from universities.

Arithmetic, geometry, algebra, and math are used as the foundations for learning natural sciences. The Moslem scientist and mathematician Ibn Al Haitham set the foundation for required math learning, in the year 850, if we are to study physics and sciences. Al  Haitham said that it is almost impossible to do science without strong math background. 

Ibn Al Haitham wrote math equations to describe the cosmos and its movement over 9 centuries before Kepler emulated Ibn Al Haitham’s analysis. Currently, Kepler is taunted as the discoverer of modern astronomy science.

We learn to manipulate equations; we then are asked to solve examples and problems by finding the proper equations that correspond to the natural problem (actually, we are trained to memorize the appropriate equations that apply to the problem given!).  Why we are not trained to compose a story that corresponds to an equation, or set of equations (model)?

If kids are asked to compose essays as the final outcome of learning languages, why students are not trained to compose the natural phenomena from given set of equations? Would not that be the proper meaning for comprehending the physical world or even the world connected with human behavior? 

Would not the skill of modeling a system be more meaningful and straightforward after we learn to compose a world from a model or set of equations?  Consequently, scientists and engineers, by researching natural phenomena and man-made systems that correspond to the mathematical models, would be challenged to learn about natural phenomena. Thus, their modeling abilities would be enhanced, more valid, and more instructive!

If mathematicians are trained to compose or view the appropriate natural phenomenon and human behavior from equations and mathematical models, then the scientific communities in natural and human sciences would be far richer in quality and quantity.

Our survival needs mathematicians to be members of scientific teams.  This required inclusion would be the best pragmatic means into reforming math and sciences teaching programs.

Note: This post is a revised version of “Oh, and I hate math: Alternative teaching methods (February 8, 2009)”.


adonis49

adonis49

adonis49

October 2020
M T W T F S S
 1234
567891011
12131415161718
19202122232425
262728293031  

Blog Stats

  • 1,426,861 hits

Enter your email address to subscribe to this blog and receive notifications of new posts by email.adonisbouh@gmail.com

Join 774 other followers

%d bloggers like this: