From Orbits to Pi

The Ptolemaic Cosmos (1660)

For as long as human beings exist, they have looked to the sky and tried to explain life on earth using the motion of stars, planets and the moon. The Greeks were among the first to discover that all celestial objects move on certain paths which can be defined precisely using mathematics. These paths are called orbits.

Greek astronomers like Ptolemy (c. 90 – 168 AD) believed that the Earth is at the centre of the universe (this is called the geocentric model), and that everything else moves around it on the most perfect of all shape: the circle.

Circles are symmetric in every direction, and all points on a circle have the same distance from its centre. The ratio between the circumference and the diameter is the same for circles of any size: it is a mysterious number called Pi.

Pi equals 3.1415926... and its digits goes on forever, without any repeating pattern. Numbers with this property are called irrational numbers. Many Greek mathematicians, including the famous Pythagoras, were shocked and horrified that such a strange and ‘impure’ number was at the very heart of nature:

Flowers Stars and Planets Water Ripples Fruit Storms

Today, mathematicians believe that Pi has an even stranger property: that it is a Normal number. In a Normal number, the digits from 0 to 9 appear completely at random, as if nature had rolled a 10-sided dice, again and again, to find the next digits. The fact the the digits are so random means that if you think of any string of digits, like 12345, it will eventually appear somewhere in the digits of Pi.

We can even convert a whole book, like the complete works of Shakespeare, into a long string of digits (a = 01, b = 02 and so on). And this string must also appear somewhere in the digits of Pi, but even if we used all computers on Earth to calculate more digits, we would probably have to look for longer than the universe exists…

There are many methods to calculate Pi, and many of them use a sequence of numbers. One example is the following sequence discovered by Gottfried Wilhelm Leibniz (1646 – 1716):

π = 41 – 43 + 45 – 47 + 49 – 411 + …

As you add more and more terms to this sequence, and the result will get closer and closer to Pi. Using big computers, Pi has been calculated up to 10 trillion digits (that’s a 1 with 13 zeros)! Because Pi is so easy to understand, yet important in many areas of mathematics, it enjoys a rare popularity in maths, and there is even a “Pi Day”.

The first 10 000 Digits of Pi

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467678374494482553797747268471040

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Conic Sections

The circle is only one of four shapes that can be made when looking at slices through a cone. We can demonstrate this using a torch: if it points vertically downwards, we see a circle of light. If we tilt the cone slightly, we get an ellipse. If we tilt it even further, we first get a parabola and then a hyperbola. These four shapes are called conic sections.

The four Conic Sections

The four conic sections can be created using different orientations of a cone of light: circle, ellipse, parabola and hyperbola.

This animation was programmed in Elica Logo and published with permission of the Elica team. The original version can be seen on YouTube.

The four conic sections look very different on first sight: the circle and ellipse are closed, but parabola and hyperbola are open and go on to infinity. But they can all be described by the same equation:

√x2 + y2  =  l + ex

Here x and y are the usually coordinates in a coordinate system, and e and l are particular values that are different for each conic section. (We call e the “eccentricity” and l the “semi-latus rectum”.)

Let us look at the ellipse in a bit more detail. It is rather similar to the circle, except that we can think of it as having two centres: these are called foci. The further these foci are apart, the more stretched the ellipse will look. If the foci are in exactly the same place, we just get a circle.

Remember that in a circle, the distance from the centre to points on the circle is always the same. In the ellipse, the distance from one focus to a point on the ellipse and back to the second focus is always constant. This gives a very nice method for drawing ellipses, and the other conic sections, using taut thread:

The Circle The Ellipse The Parabola The Hyperbola

In a circle, every point has the same distance from the centre. The circle is just a special case of the ellipse, in which both foci have the same position.

In a circle with radius r (this is the distance from the centre to the line), the circumference is 2 π r and the area is π r2.

The Scientific Revolution

Ptolemy believed that the Earth is at the centre of the universe and that stars, planets, the moon and the sun move around the Earth in circles (the Geocentric Model). Unfortunately the astronomical observations didn’t quite match what was predicted by the model. Therefore scientists added Epicycles to the orbits: all objects in the sky move on a large circle around the Earth, but at the same time they move along a small circle around itself. And, though complicated, this was the most widely accepted model of our universe for more than 1000 years.

But in 1543, Nicolaus Copernicus (1473 – 1534) published the book “On the Revolutions of the Celestial Spheres”, in which he explained what many astronomers had suspected for some time: that the sun is at the centre of the universe and that the Earth moves around the sun on a circle, like all the other planets. This is called the Heliocentric Model.

At this time, the heliocentric model still had many problems and the fact that the Earth is flying though space at a rapid speed was very hard to understand and seemed contradictory to the bible.

The Geocentric (Ptolemaic) Model

Illustration from Cosmographia by Bartolomeu Velho

The Heliocentric Model

Illustration from Harmonia Macrocosmica by Andreas Cellarius

During the following years, the Danish astronomer Tycho Brahe (1546 – 1601) made very accurate observations of the positions of the planets, as well as developing his own model of the solar system. After his (sudden and unexplained) death, his assistant Johannes Kepler (1571 – 1630) succeeded him as imperial mathematician. And many years of research in a variety of areas culminated in Kepler’s three Laws of Planetary Motion, which describe essentially the solar system we know to be true today.

Nicolaus Copernicus1473 – 1534

Tycho Brahe1546 – 1601

Johannes Kepler1571 – 1630

Galileo Galilei1564 – 1642

At the same time in Italy, Galileo Galilei (1564 – 1642) made astronomical observations using newly developed telescopes. He discovered the phases of Venus, four moons of Jupiter, as well as the sunspots.

Galileo was a strong advocate of the new heliocentric model. He was a friend of Pope Urban VIII and tried to connect the new model with the bible. The controversy that arose from his book “Dialogue Concerning the Two Chief World Systems” was founded primarily on politics, not science. Galileo was forced to recant and set under house arrest.

But here are the celebrated laws of Kepler’s that describe the motion of the planets:

1. The planets move on elliptical orbits, not circles, and the sun is in one of the foci.2. A line joining the planet and the Sun sweeps out equal areas during equal times. This

means that the planets speed up as they moves closer to the sun, and slows down as they get further away.

3. The orbital period squared is proportional to the average radius of the orbit cubed. This law allows us to find the time a planet needs for one full rotation, given that we know the size of the orbit.

Isaac Newton (1642 – 1727)

In 1687, Sir Isaac Newton (1642 – 1727) published the “Principia Mathematica”, where he explains the mathematical foundations that give rise to Kepler’s Laws. Between any two masses in the universe, there is an attraction like the one between magnets. This force is called gravity. Gravity is what makes everything fall to the ground and gravity is what makes the planets move around the sun. It is only the great speed at which the planets move that prevents them from falling into the sun.

Using Newton’s laws, one can derive an equation that describes the motion of objects moving under the force of gravity. This equation is the same as the equation that describes conic sections (see above). Planets move along ellipses, but other objects like comets can travel on parabolic or hyperbolic paths: they come close to the sun before turning around and shooting off to infinity, never to come back.

Newton lived and worked at Trinity College Cambridge, where a falling apple is said to have inspired him to think about gravity. His description of classical mechanics, among many other discoveries, makes Newton the greatest and most influential scientist of all times. His ideas shaped our understanding of the world for nearly 300 years, until Albert Einstein discovered the laws of relativity in 1905.

Symmetry and Space

Orbits, Circles, Conic Sections

Symmetry and Groups

Polygons and Polyhedra

Modelling Space

Dimensions and Distortions

Fractals

Numbers and Patterns

Sequences

Functions and Series

Prime Numbers

More on Number Theory

Real, Irrational, Imaginary

Infinity

Combinatorics and Logic

Combinatorics

Graph Theory

Knots, Mazes, Labyrinths

Optimisation

Logic and Paradoxes

Axioms and Proofs

Probability and Games

What is Probability?

The Normal Distribution

Random Walks

Statistics

Game Theory

Codes and Cyphers

Motion and Matter

Forces, Motion, Calculus

Kinematics and Mechanics

Waves

Chaos

Theory of Relativity

Quantum Mechanics

http://world.mathigon.org/Game_Theory.html

http://world.mathigon.org/Conic_Sections.html

Game Theory

Combinatorial Games

Many games involve rolling dice, dealing cards or spinning wheels. In all these cases we can use probability to find how likely certain outcomes are. And we have done so repeatedly in previous chapters. But in this chapter we want to think about games where there is no luck involved and there is no cooperation or conflict between the opponents. These games are called Combinatorial Games.

Chess is an example of a combinatorial game, but there are so many different moves, positions and strategies that it is almost impossible to analyse chess using the methods for other combinatorial games. Here is a much simpler example:

There are two boxes with chocolates, and two players eat them alternatingly. At every turn, a player has to eat at least one chocolate, but he/she can also eat two or more chocolates, as long as they are from the same box. For example, a player could eat 3 chocolates from box A on his/her turn, but not one chocolate from box A and one from box B.

The players continue, alternatingly eating chocolates, until both boxes are empty. Whoever gets the last chocolate wins.

Before reading on, try to play this game against the Computer. You can start simply by clicking on how many chocolates from one box you want to eat.

   

   

   

   

   

   

Click to end your turn

Notice that you lost. In fact it was clear from the beginning that we would always win, unless we had made a stupid mistake. The following sections will explore several different methods to analyse combinatorial games, to find winning strategies and to determine whether it is better to go first or to go second…

Maybe you have already noticed a pattern and worked out a winning strategy. In that case you will find the following paragraphs rather complicated for solving a simple game. However once you have understood these methods you can easily apply them to much more complicated games.

Simple Games and their Solution

One method to think about combinatorial games is to make a list of all possible outcomes. This is best done in a “tree” diagram, where every junction represents different choices the players could make. Here is a tree for a slightly simpler version than above, with only 3 chocolates per box.

Player 2 is destined to win from the beginning, unless he makes a stupid mistake.

P and N-positions

The method above is not very practical is the boxes of chocolates become much bigger. If each contains 5, for example, there would be more than 10,000 possibilities!

You will also notice that we have many different copies of the same position in different XXX of the tree. Instead let us draw a diagram of all the different positions, and connect two positions by an arrow if a player could move from one to the other (remember you can’t put chocolates back or take chocolates from more than one box.

To analyse the game, we will again highlight the positions with different colours, only that they will have a slightly different meaning than above.

Here you can see all possible positions in the game we are playing. Click next to continue…

The pattern is quite obvious: all positions along the diagonal, where there are the same number of chocolates in each box, are P-positions. All the other positions are N-position. And of course it extends to bigger box sizes – including 9 chocolates per box in the game at the beginning of this article. You being the first player, this proves that you had no chance of winning in that game unless we made a mistake.

Think about the actual winning strategy and how it relates to wanting to finish your turns in a position where there are the same number of chocolates in each box.

A P-position is a position in which the previous player will win (who moved to that position) and an N-position is a position where the next player will win. When you play you want to make sure you always end on a P-position.

And remember: from a P-position you can only move to N-position, and from an N-position you can move to at least one P-position.

If we start on a P-position, the next player must lose. Therefore he/she must only be able to move to N-position from where the other player will win (there is no choice).

If we start on a N-position, the next player will win. Therefore there must be at least one P-position to move to, from where the opponent will lose (there is a choice and the game will change if you make the wrong choice).

We always have to start from the end, when we know how would have won. Then we can work backwards using the two rules above to classify all positions in the game.

In any game that can be analysed using this method, the result is determined right from the beginning. If you are unlucky and you are the player destined to lose, there is nothing you can do except hope that your opponent makes a mistake…

The Game of Nim

The game we have been thinking about so far is called Nim. With only two boxes it is quite simple and easy to understand. But it gets much more interesting (and complicated) if we add more boxes and start with different numbers of chocolates in each.

Interactive Game Coming Soon …

To simplify the notation a bit, we can denote any Nim positions by numbers rather than drawing the boxes. For example, (2,5,4) mean there are three boxes with 2, 5 and 4 chocolates respectively. Notice that it doesn’t matter in which order we have to boxes and whether there are additional empty boxes. Therefore (2,5,4) = (5, 0, 4, 2).

Above we have shown that (1,1), (2,2), (3,3) and so on are all P-positions. Positions with two boxes which have different numbers of chocolates are always N-positions. There is a simple method for determining whether positions with three or more boxes are P or N. The method will seem totally random and unrelated to games, but it becomes clear when doing the maths of P and N-positions:

A Nim position (a, b, c, …) is a P-position if the binary digital sum (or Nim sum) of a, b, c, … is 0. If not, than it is a N-position. The Nim sum is often written as a ⊕ b ⊕ c ⊕ … and can be calculates as shown in the following example:

To find the Nim sum of 3 ⊕ 6 ⊕ 7 we proceed as follows:

  124

3110

6011

7111

2010

Above you can see the three numbers to sum, 3, 6 and 7, in rows and the powers of 2 which are 1, 2, 4, 8, … in columns. We first need to write 3, 6 and 7 in binary, which means writing them as a sum of powers of 2.

Try to use the same method to calculate the Nim sum in the following exercises.

Winning Strategies Solutions

Here are a couple Nim positions. Determine which ones are P and which ones are N-positions. For the N-positions assume that it is your turn. What would be your optimal move?

Nim has a couple of useful properties:

Exactly two opponents move alternately. The moves and all options are clearly specified by rules and there are no chance moves. There are only finitely many different positions and the game will always come to an end 

when one player is unable to move. This means that there is no draw and no cycles, which could repeat forever.

The players have perfect information. Card games often don’t have perfect information because one player doesn’t know the opponents cards.

From any one position of the game both players have the same choice of moves. This is not true for chess because from any position, one player can only move white figures and the opponent can only move black ones.

Games with these properties are called Impartial Games. Mathematicians discovered that any impartial game is equivalent to a game of Nim with certain box sizes. This means for example that the P and N-positions match up and there are always the same number of possible moves. A winning strategy for any impartial game can be found by converting it into Nim and the using the Nim sum.

Article on more advanced Combinatorial Game TheoryPresented by Philipp Legner at the "Tomorrow's Mathematician's Today" Conference      

Non-Combinatorial Games

One of the most captivating combinatorial games: Chess.

Impartial games are interesting to analyse from a mathematical point of view, but once you have worked out how they work they are not particularly exciting no play – not only because you know right from the beginning who is going to win.

There are many other combinatorial games, such as chess. These are so complicated that we can’t use the methods we developed for impartial games. Chess computers don’t try hundreds of different possibilities; they play very much like a human being would: analysing the current position and following certain strategies.

What most people associate with Game Theory are situations in daily life where we have a voice of various decisions. The outcome often depends on the decisions other people make – and which we don’t know in advance. This particularly happens in economics were companies “play” against each other in various markets.

Here are a couple of examples of interesting situations which can arise in these cases:

The Prisoners Dilemma Nash Equilibrium Battle of the Sexes

Imagine two prisoners are locked in two separate cells of a prison. They are accused to have committed a crime together and are questioned individually. They are promised to get away if they betray their friend, who gets the full sentence of 10 years.

If both of them stay silent there is not enough evidence so both get a shorter sentence of 1 year. But if both betray each other each is sentenced to 5 years in prison.

The following table shows the four possible outcomes, depending on the actions of Prisoner A and Prisoner B:

 Prisoner A betrays

Prisoner A stays silent

Prisoner B betrays

A: 5 yearsB: 5 years

A: 10 yearsB: 0 years

Prisoner B stays silent

A: 0 yearsB: 10 years

A: 1 yearB: 1 year

Let us suppose we were Prisoner A and thinking about which action to take: betray of stay silent.

If we knew Prisoner B would betray us (row 1), betraying would get us 5 years while staying silent would get us 10 years. Thus we should also betray.

If we know Prisoner B would stay silent (row 2), betraying would get us 0 years while staying silent would get us 1 year. Thus we should betray.

It seems that – no matter what Prisoner B does – betraying will give us a shorter time in jail and hence is the best thing to do.

Of course Prisoner B is thinking exactly the same and will also betray. So both prisoners will betray each other and will both be sentenced to 5 years in jail.

Notice, however, that if they had cooperated and both stayed silent, they would have managed to produce and outcome that would have been better for both of them: just one year each.

http://world.mathigon.org/Game_Theory.html