## vendredi 30 juin 2017

### Bac

Among the 4 documents that you have prepared, the jury makes you pick 2 subjects out of 4, without looking, then you choose 1 out of 2, after reading them.

For the exercise, they'll make you choose 1 out of 2 subjects, on different themes, among probabilities, sequences, statistics and functions.

1) You'll have 20 min to prepare the exercise that they'll give you, (on probabilities, sequences, functions or statistics) and you'll have 10 min to present it on the white board. (The subjects of the exercises of Baccalauréat DNL of the last 6 years : ).

2) When you present the second part of the oral (the document), introduce your subject make a plan and announce it (1st part, 2nd part), then develop itand as a conclusion give your personal opinion about the document. You'll have 5 min to present your document and then 5 min of questions about it (or about more general questions).

Do not bring any Powerpoint or any documents from home.
Do it all from memory, without learning by heart.
Talk about it freely, but with a structure (introduction, development, conclusion).

Baccalaureat : how are the marks given during the oral exam ?

1) Fractals (T. S or T. ES)
2) Four physics riddles (T. S.)
3) Is the earth flat ? (T. S. or T. ES)
4) Sex and marriage theorems (T. S. or T. ES)
5) Brachistochrone  (T. S.)
(T. S.)
7) Kakeya's needle problem (T. S.)
8) Blob Infinity (T. S.)
9) Hilbert hotel (T. S. or T. ES)
10) Futurama theorem (T. S. or T. ES)
11) Rubik's cube and the Enigma machine (Alan Turing) (T. S.)
12) Magnus effect (Physics) (T. S.)
13) Anti-gravity wheel (Physics) (T. S.)
14) Computer Science Unplugged. (T. S. or T. ES)
15) Conditional probability (T. S. or T. ES)
16) Monty Hall (T. S. or T. ES)
(T. S. or T. ES)

## mercredi 17 mai 2017

### Fractals

1º) Introduce the subject with a summary of what you are going to present.
2º) Present it with a few steps (What is it ? What are they good for ?)
3º) Conclude and give your personal opinion

# Fractal

A Sierpinski triangle, after 7 iterations.
The Mandelbrot set is a famous example of a fractal.
fractal is any pattern, that when seen as an image, produces a picture, which when zoomed into will still make the same picture. It can be cut into parts which look like a smaller version of the picture that was started with. The word fractal was made by Benoît Mandelbrot in 1975 from the Latin word fractus, which means "broken" or "fractured". A simple example is a tree that branches into smaller branches, and those branches into smaller branches and so on. Fractals are not only beautiful, but also have many practical applications.

## Contents

• 1Examples
• 1.1The Koch Curve
• 1.2Similarity dimension
• 1.3Koch snowflake
• 2Uses

## Examples

There are many types of fractals, made in a large variety of ways. One example is the Sierpinski triangle, where there are an infinite number of small triangles inside the large one. Another example is the Mandelbrot set, named for Benoît Mandelbrot. The Sierpinksi triangle is constructed using patterns, but the Mandelbrot set is based on an equation.
There are also many natural examples of fractals in nature including trees, snowflakes, some vegetables and coastlines.

### The Koch Curve

How to make the Koch Curve
The Koch Curve is a simple example of a fractal. First, start with part of a straight line - called a straight line segment. Cut the line into 3 same-sized pieces. Get rid of the middle of those pieces, and put in the top part of a triangle with sides which are the same length as the bit to cut out. We now have 4 line segments which are touching at the ends. We can now do what we just did to the first segment to each of the 4 bits. We can now do the same thing again and again to all the bits we end up with. We now do this forever and look at what we end up with.
The length of the Koch Curve is infinity, and the area of the Koch Curve is zero. This is quite strange. A line segment (with dimension 1) could have a length of 1, but it has an area of 0. A square of length 1 and width 1 (with dimension 2) will have area 1 and length of infinity.

### Similarity dimension

So, the Koch Curve seems to be bigger than something of dimension 1, and smaller than something of dimension 2. The idea of the similarity dimension is to give a dimension which gives a better idea of length or area for fractals. So, for a Koch Curve, we want a dimension between 1 and 2.
The Koch Curve can be cut into 4 pieces, each of which are ${\displaystyle {\frac {1}{3}}}$ of the size of the original.
The Koch Curve is one of the simplest fractal shapes, and so its dimension is easy to work out. Its similarity dimension and Hausdorff dimension are both the same. This is not true for more complex fractals.

### Koch snowflake

The Koch snowflake (or Koch star) is the same as the Koch curve, except it starts with an equilateral triangle instead of a line segment.

## Uses

Fractals have many applications e.g. in biology (lung, kidneys, heart rate variability, etc...), in earthquakes, in finance where it is related to the so called heavy tail distributions and in physics. This indicates that fractals should be studied to understand why fractals are so frequent in nature.
Some fractals exist only for artistic reasons, but others are very useful. Fractals are very efficient shapes for radio antennas and are used in computer chips to efficiently connect all the components. Also, coastlines can be thought of as fractals

## samedi 29 avril 2017

### Is the earth flat ?

1) Introduce the subject.
2) Summarize it shortly in 4 minutes (use only a few arguments you could explain).
3) Conclude by explaining in a few sentences why conspiracy theories against science are not coherent, but why a well trained criticism is necessary to understand (and not believe everything in) the news on television and in newspapers.

# Flat Earth

15th century adaptation of a T and O map. This kind of medieval mappa mundi illustrates only the reachable side of a round Earth.
The idea of a flat Earth is that the surface of the Earth is flat (a plane). Belief in a flat Earth is found in the oldest writings. Early Mesopotamian maps showed the world as a flat disk floating in the ocean.
This was a common belief until the Classical Greeks began to discuss the Earth's shape about the 4th century BC. Eratosthenes (276 BC–194 BC) calculated the circumference of the Earth quite well. From then on, few educated people ever believed in its being flat. People first started having the thought that the Earth is round in around 6th century BC. Then, Aristotle proved the Earth was round in around 330 BC.
The large-scale shape of the Earth only matters when considering large distances. Therefore in the Ancient world only sailors, astronomers, philosophers, and theologians would have cared about the Earth's large-scale shape.
The following authors argued for a spherical or ball shaped earth : King Alfred of the Anglo-Saxons, Hildegard von Bingen, Thomas Aquinas, Snorri Sturluson, Marco Polo, Dante Alighieri, Christopher Columbus
Portuguese people explored Africa and Asia, Columbus sailed to the Americas (1492) and Ferdinand Magellan's circumnavigated (sailed all round) the earth (1519-21). This proved finally, in a practical way that the earth is a globe.
During the 19th century, the Romantic ideas about a European "Dark Age" made the Flat Earth model look much more important than it ever had been in history.
The Flammarion woodcut. Flammarion wrote, "A missionary in the Middle Ages tells that he has found the point where heaven and Earth meet..."
The widely circulated woodcut is of a man poking his head through the firmament of a flat Earth to see the machines working the spheres. It was made in 16th century style but cannot be traced to an earlier time than Camille Flammarion's L'Atmosphère: Météorologie Populaire (Paris, 1888, p. 163). The woodcut illustrates the statement in the text that a missionary in the middle Ages claimed that "he reached the horizon where the Earth and the heavens met". That story may be traced back to Voltaire, but not to any known source in the Middle ages. The original woodcut had a decorative border that places it in the 19th century; in later publications, some claimed that the woodcut dated from the 16th century and the border was removed. According to an unproved story Flammarion ordered the woodcut himself; certainly no source of the image earlier than Flammarion's book is known.
An early mention in literature was Ludvig Holberg's comedy Erasmus Montanus (1723). A great many people disagree with Erasmus Montanus when he claims the Earth is round, since all the peasants believe it is flat. He is not allowed to marry his fiancée until he cries "The earth is flat as a pancake". In Rudyard Kipling's The Village that Voted the Earth was Flat, the main characters spread the rumor that a Parish Council meeting had voted in favor of a flat Earth.
An old map of the world made in Amsterdam in 1689
The spherical Earth seen from Apollo 17, disproves the flat Earth model. The Flat Earth Society believes that images like these have been edited by NASA as part of a conspiracy.
Fantasy fiction often imagines a flat Earth. In C. S. Lewis' The Voyage of the Dawn Treader the fictional world of Narnia is "round like a table" (i.e., flat), not "round like a ball", and the characters sail toward the edge of this world. Terry Pratchett's Strata and Discworld novels (1983 onwards) are set on a flat, disc-shaped world resting on the backs of four huge elephants which are in turn standing on the back of an enormous turtle.

## lundi 24 avril 2017

### Four physics riddles

1) Introduce the subject.
2) Summarize it shortly in 4 minutes (only what you can explain).
3) Conclude and give your personal opinion on the subject of open problems.

## lundi 20 mars 2017

### Hilbert hotel

1) Introduce the subject.
2) Summarize it shortly in 4 minutes. (Only what you can explain).

The sequence u(n) = n + 1 proves you can always find space for a new guest.
The sequence v(n) = 2n proves that there are as many even numbers than integers.

3) Conclude for example, by explaining why infinity is a surprising subject full of paradoxes. (Part of an infinite set can have the same number of elements than the whole set).

David Hilbert (1862-1943)

Imagine an ordinary hotel of 500 rooms, all of them occupied by some clients. You arrive there one evening and the receptionist tells you that there's no vacancy. You walk away sadly. There's no paradox about that.

Now, imagine a hotel in which there are infinitely many rooms, all of them occupied. Although the hotel is full, the receptionist is able to give you a room.How is it possible ? Later, on the same day, infinitely many persons arrive, and the receptionist is able to make room for all of them, (earning a fortune in doing so).

The german mathematician David Hilbert invented these paradoxes in the 1920s to illustrate the mysterious properties of infinity.

Lets see how we can obtain a room in Hilbert's hotel infinity. When only you arrive and the hotel is full, the receptionist is able to give you a room by asking the client who was in room 1 to go to room 2, the one who was in room 2 to go to room 3, and so on forever, each client moving to the next room. Thus, room 1 is now empty for you.

To be able to receive an infinity of new clients, all the previous ones move to the rooms of even numbers, the one who was in room 1 moving to room 2, the one who was in room 2 moving to room 4, the one who was in room 3 moving to room 6, and so on, every client moving to the room having the number double from his previous room. The receptionist is now able to move all the new clients to the rooms of odd numbers.

Hilbert's hotel infinity paradox may be understood with Cantor's theory of transfinite numbers. Thus, while in a normal hotel, the number of odd numbered rooms is inferior to the total number of rooms, in an infinite hotel, the "number" of odd numbered rooms is not inferior to the total "number" of rooms. (The mathematicians use the term of "cardinality" to refer themselves to the size of these sets of rooms).

The receptionist can even receive infinitely many buses containing infinitely many travelers each, by using all the powers of 2 for the previous clients, all the powers of 3 for the first bus, and all the powers of each different prime numbers for each different bus.

See also Zeno's paradox (445 B.C.), Cantor's transfinite numbers (1874), Peano's axioms (1889) and Hilbert's twenty three problems (1900).

### Anti-Gravity Wheel ?

1) Introduce the subject.
2) Summarize it shortly in 4 minutes.
3) Conclude for example, by explaining in a few sentences why physics is sometimes surprising (or not intuitive).

# Gyroscope

A gyroscope
gyroscope is a device used to measure or maintain an angular position. It works using the principles of angular momentum. The gyroscope is made up of a spinning wheel or disc, as well as (in some cases) many other moving parts. It helps with navigation, and plays a part in such things as a gyrocompass and artificial horizon.

### Monty Hall

1) Introduce the subject.
2) Summarize it shortly in 4 minutes (the 1st way, but not the "Mathy" way).
3) Conclude for example, by explaining in a few sentences why conditional probability is sometimes surprising (or not intuitive).

# Monty Hall problem

The Monty Hall problem is a famous problem in probability. The problem is based on a television game show from the United States, Let's Make a Deal. It is named for the host of this show, Monty Hall.
In the problem, there are three doors. A car (prize of high value) is behind one door and goats (booby prizes of low value) behind the other two doors. First, the player chooses a door but does not open it. Then the host, who has knowledge of what is behind every door, opens a different door which they are certain has a goat behind it (opening either door with equal chances if the car is behind the player's door). Last, the host lets the player choose whether to keep what is behind the first door or to change choices to the third door (the one the host did not open). The rules of the problem are that the host has to open a door with a goat behind and has to let the player switch. The question is whether changing choices increases the chances of getting the car.
The chances of the car being behind the two doors that are still closed seem equal, so most people say changing choices does not increase the chances of getting the car. The true answer is that changing choices increases the chances of getting the car from 1/3 (one out of three) to 2/3 (two out of three).
That comes from the fact that the player, by choosing one door out of three, has a one chance in three of selecting the door with the car. The chance of the car being somewhere behind the other two doors is two out of three. So to improve their chance of winning a car, the player if given the choice, should swap their one door for the other two doors right away. But wait! The host then tries to confuse the player by opening one of their own goat doors. That changes nothing, remember that the player is still swapping their one door for the other two doors (even though one of them has been opened).
1.
 Host revealsGoat AorHost revealsGoat B
Player picks carChanging loses.
2.
Host must
reveal Goat B

Player picks Goat AChanging wins.
3.
Host must
reveal Goat A

Player picks Goat BChanging wins.
The player has an equal chance of initially selecting the car, Goat A, or Goat B. Switching results in a win 2/3 of the time.
These are the options:
1. (Lose): If the player picks the car, then the host will show a goat. Then if the player changes their choice, they will get a goat .
2. (Win) : If the player picks a goat, then the host will show the other goat. Then if the player changes their choice, they will get a car.
3. (Win) : If the player picks the other goat, then the host will show the first goat. Then if the player changes their choice, they will get a car.
So, it is true that if the player changes (switches) then the player will win a car two times out of three.