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Marie-Claude Vergne

[What is Tangram?] [Playing tangram you understand Mathematics]
[Mathematics arises from problems] [Problems with one or more solutions]
[Insoluble problems] [Classification problems] [Open problems]
[Paradoxes] [Conclusion]

1. What is Tangram?

Tangram is an antique game that originally comes from China. It is formed by dividing a square into seven parts that are called "tan": a square, a parallelogram and five isosceles right-angled triangles, two big ones, a medium and two little ones.
The traditional rules of the game are simple: you have to lay the seven tans on a plane, without overlapping them, trying to form a figure that reproduces, maintaining the proportions, the figures that you have seen earlier in the instruction book.
It may appear very easy to play the game Tangram, especially if you see the pieces already assembled in a square, but normally a beginner has already difficulties to reform the square, after having taken the pieces out of the box.
But Tangram isn't a puzzle as many others. After having played a little bit, you begin to enjoy the subtle elegance with which the square has been divided.
In this way you can understand how Tangram can be made by folding and cutting a square piece of paper.
After we have discovered how to get the tans, it seems obvious that there are many relations between their sides and angles.
Because of that it occurs, as well as for Origami, that, playing the game Tangram, although the material is simple, you can form geometric figures, as the square, in which the characteristics of each single tan vanish, but, on the other hand, you can form any type of figure in which the characteristics of each tan get more evident. Some figures are so expressive that they seem alive and active.
It is also possible to represent the same subject in many positions and therefore you can use Tangram also for illustrate stories and realize cartoons.
A remarkable characteristic of many Tangram figures is that they suggest to ones imagination much more than they effectively represent: in fact they are optical illusions. The Tangram figures offer with their essentiality and effectiveness a variety of perceptions like Zen painting, that is based on the following idea: "the palette of the mind is richer than the possibility that offers the brush".
Tangram figures remind with their expressiveness the silhouettes and the hand shadows plays.
Tangram offers remarkable suggestions for the study of the visual perception and it could be used as base for psychological tests.

In a certain book, that is edited in English up to now, Rouse Ball, sublining the importance of games in teaching and popularising mathematics, wrote about Tangram: "To form figures with these seven pieces of wood....is one of the most antique oriental games. With these pieces it is possible to form hundreds of types of men, women, animals, fishes, houses, ships, everyday objects, geometric figures, and so on, but the kind of offered amusement is not of mathematical nature and so I limit myself just mentioning it".
But now we will see that it isn't like that, because we will discover an unsuspected analogy between certain aspects of playing Tangram and doing mathematics.
We will try to explain, using Tangram as an example and a metaphor, what really is the mathematical activity.
Although mathematics is taught at schools of every level, a lot of people don't recognize what it really is. In popular works there are often biographies or anecdotes that speak about life of famous mathematicians lived in the past. But it's rarely explained what is mathematics and which are its peculiarities. In effect it isn't easy to popularize mathematics. Someone thinks that it can be really understood by doing it.
It rests to say that there are many people, also of a certain culture, that believe that mathematics is an crystallized set of rules to which you can't add anything.
But that isn't true: mathematics always gets on and we can see its many applications every day. But, much more mathematics enters in our daily life, making possible the realization of objects of simple and general use, as the credit cards or the CD’s that we listen or use with the computer, than less we are capable to realize this: this fact was noticed by Chevallard at the sixth international congress of mathematical education at Budapest in 1988.

Mathematics gets on by posing and solving problems.
Einstein said: "The answers are all in front of us: it's enough to find the right questions".
The problems from which mathematical theories arise are sometimes of practical type.
Other times these problems arise from the generalization of results already obtained.
It often occurred that some theories, developed for an internal mathematical aim, get essential after centuries for completely other problems.
For example, the curves that are called conics because you can get them by sectioning a cone with a plane, was maybe originated from the studies about the sun-dials. They've been carried out in the forth century BC by the mathematicians Menaechmus from the Plato's Academy, and have been used for solving the problem of doubling the cube. Then in the third century BC they've been studied by Euclid, Archimedes and Apollonius. The last one obtained them in a more general way and called them: ellipse, hyperbola and parabola. The same curves and their geometrical properties have been used two thousand years later by Kepler for describing the lows of the motion of the planets around the sun.
To give another example, only in recent times, Number's Theory, since ever considered one of the more abstract and pure theories by mathematicians, gave determining contribution in the field of reliability and security of telecommunications: without this the space-travels wouldn't have been possible and the financial transactions on telematical way wouldn't be safe and therefore they couldn't have been developed.

What is the kind of the problems studied by mathematicians? Which are their characteristics?
Some people, the most ingenuous, maybe considering what happens if the cashier in the supermarket makes the bill, think that mathematics gives always only one answer. To strengthen their ideas, it seems that they say: "Two plus two is four! Mathematics is not an opinion!".
We must say that it isn't like that. In mathematics, as well as in Tangram, there are problems with only one solution and problems with more solutions and there is also other.
Let's start with a Tangram problem that offers one solution. Reminding of that, we note that the crane can be realized only in one way. In fact the two big triangles can be used only for the body and the wings, the little ones for the paws, after having disposed the other tans in only one possible, determined way.
To show a problem with more solutions, let's have a look to an example of Tangram. To construct a isosceles right-angled triangle like this, we can operate in two different ways.
Also in classical mathematics you find problem with more solutions. Here is one of them: how many are the circles that are tangent to a given straight line and pass by two assigned points?
This problem, with a little knowledge of Euclidean geometry, can be solved with rule and compasses. Let's follow step by step the geometrical construction.
Therefore it's not surprising if mathematicians study problems with two or more solutions: indeed some problems have an infinite number of solutions.

In mathematics as well as in other disciplines you can find problems that don't allow any solution.
But in mathematics, different as it occurs in other sectors, this fact can be proved without any fear of disproof.
Sometimes the impossibility of solving a problem is quite evident.
For example, a theorem of Euclidean geometry establishes that in a triangle every side is shorter than the sum of the other two sides and therefore it may occur that, choosing arbitrarily three segments, it isn't possible to form a triangle with them.
Other situations of impossibility are less evident. Let's begin with a Tangram example. It isn't very obvious that it isn't possible to realize that frame with the seven tans. But it's easy to show.
It's enough to follow an analogous reasoning of the one used to show the uniqueness of the solution for the crane: the big triangles can be arranged only in two opposite angles of the frame. The square tan therefore can be put only in one of the two remaining angles and the parallelogram would have to touch the fourth angle, but in that situation there's no space for the medium triangle and at that point it's no help to consider the little triangle.
Let's speak about mathematics now. One of the most famous classical example is the so-called problem of the trisection of an angle, or the problem to divide any angle into three equal parts. This problem has been studied already in the fifth century BC
By using only rule and compasses, the mathematicians have been able to bisect, that means to divide into two equal parts, any angle, and they have seen that it was possible to construct with the same instruments an angle of thirty degrees, that is the third part of the right angle. Therefore they tried for centuries to solve the problem of the trisection using only rule and compasses, but with these tools nobody was successful. Two thousand years later, in 1837, Pierre Laurent Wantzel showed, by an algebraic process, that there are angles that can't be trisected with rule and compasses.
In any way the impossibility to solve the problem of the frame or the one of trisecting any given angle depends on the tools that have to be used. If, for example, instead of the traditional Tangram we use the tablets of Sei Shanagon, a Japanese writer of the 10th century, the frame can also be constructed.
Already in antique times there have been invented various instruments to solve the problem of trisection. One of these, based on a solution of Archimedes, is called also Pascal's trisector.

Going on with our comparison between Tangram and mathematics, we note that in mathematics there are also studied problems of classification, that will be exemplified now.
The regular polyhedra are solids with regular polygonal and equal faces, that get together always in the same number in each vertex. They are also called Platonic figures because Plato talked about them in "Timaeus". Thinking about the infinite number of regular polygons, you could think that the regular polyedra are also infinite. On the contrary, the antique Greeks found out that there are only five of them: the tetrahedron, that faces are four equilateral triangles, the cube or hexahedron, that’s faces are six squares, the octahedron, that’s faces are eight equilateral triangles, the dodecahedron, that’s faces are twelve regular pentagons, and finally the icosahedron, that’s faces are twenty equilateral triangles.
The problem to classify the regular polyhedra is one of the most antique classification problems that has been solved in mathematics. But there are many others.
A classification problem has been solved recently, around 1980. It's a problem that concerns algebraic structures and regards the classification of the finite simple groups, that in the groups theory represent the same as the prime numbers represent for the integers. In fact by multiplying prime numbers we get all the integers and, analogous to this, by multiplying finite simple groups we get all the finite groups.
There have been involved many generations of mathematicians to solve this problem. To get this result it took more than a century and there have been written more than 15000 pages!
Also with Tangram there are classification problems. One of them is the problem to classify the convex figures that you can get.
Convex figures are mathematically defined in a certain way, but regarding our interests we can say that, if figures would have a certain thickness and we would surround a convex figure with an elastic band, all points of the figure, and only these ones, would be included, whereas for a non-convex figure would rests some empty space.
In 1942 two Chinese mathematicians, Fu Traing Wang and Chuan-Chih Hsiung classified the convex figures that can be realized with Tangram. These are the following 13: a triangle, six quadrilaterals, two pentagons and four hexagons. The quadrilateral are: a square, a rectangle, a parallelogram, an isosceles trapezium, and two right-angled trapezia.
In 1995 a young Italian teacher, Silvio Giordano, showed further that with Tangram you can realize only the quadrilaterals described above, that means only the convex ones.
Regarding the pentagonal Tangram figures, there have been individuated 53, after Martin Gardner had presented the problem to classify them in his rubric "Mathematical games" of "Scientific American" in 1974; this problem was already considered in 1968 by Lindgren. The result was verified by using a computer with especially created programs, but up to now it doesn't exist a complete proof and therefore this Tangram-problem is open.

Now we get to the open mathematical problems. Also today there are open problems in mathematics and hopefully there will be always others. Going on in mathematics indeed, as in other sciences, there are opened new horizons. There are problems that are open since centuries and this is important. These ones, even if they may seem of little interest from practical point of views, often determine the birth of new useful and important research fields or suggest unexpected applications.
Some open problems are expressed in such a simple way that they can be understood also by middle-school pupils. One of these, posed in 1742, is the conjecture of Goldbach, who was a German mathematician and friend of Euler. The conjecture says that every even number greater than two can be expressed as a sum of two prime numbers. For instance: 4=2+2, 6=3+3, 8=3+5, 10=5+5, and so on.
But it's obvious that, even using a powerful computer, we can't go on verifying this to the infinite, and up to now nobody has been able to show this conjecture or to find a counterexample. That means that there hasn't been found an even number which can't be expressed as a sum of two prime numbers.

Let's now observe these two figures. At first sight it seems impossible to get both of them with Tangram respecting all the rules, that means using all the seven pieces without overlapping them. They seem two identical men, only that one of them has no foot. Although it appears impossible, both figures can be realized. We aren't surprised any longer if we look how they are composed. Figures like these, completely ignored in the traditional manual Tangram, have been introduced by two famous enigmatographers at the begin of the twentieth century: the American Sam Loyd and the English Ernst Dudney. Logically the area of the figures has to be the same. We ask each other now how it is possible that the first man seems perfectly identical to the second, although there is one part missing. The explanation is the following: in both figures the head, the hat and the arms are formed with the same pieces. The width of the body's base is equal, but the body of the first man is constituted of three pieces, the second of four. The body of the first man differs from the second one exactly from the red strip: this area therefore is equal to that of the foot.
For this reason the second man has a bigger stomach than the first: at first sight we haven't noticed that, and the existence of both figures seemed a contradiction. The existence of these figures produces a paradox.
Also in mathematics there are situations that produce paradoxes, that means that they lead to conclusions contrary to the general opinion. For instance, Galileo himself was impressed by the following mathematical fact: knowing that any integer has a square, you can say that there are as many integers as there are their squares. Nevertheless not all the integers are perfect squares and therefore the squares could seem to be less than the integers. Galileo considered this fact an unsurmountable difficulty and wrote:
"This are difficulties that arise from speaking with our finite intellect about the infinite".
In 1873 the German mathematician Georg Cantor came over the large number of philosophical discussions that existed since Aristotle’s times, with which Galileo himself agreed. Cantor was able to confront the infinite sets. The method he used can be exemplified as follows.
To verify the fact that in this picture of Escher there are as many horses as riders, without counting them, it's enough to observe that each horse has a rider; therefore the elements of this two sets (the one of the horses and the one of the riders) can be coupled.
Analogously you can invent a strategy to couple the points of these two segments and to confront in this way the numerosity of the sets of their points.
Noting that each point P of the segment AB can be coupled with the point P' of the segment CD, showed in the figure, we conclude that the segment AB has as many points as has CD, nevertheless it is shorter.
With this other trick the points of the segment can be coupled with the points of a straight line, and we conclude therefore that a segment has as many points as a whole straight line.
Not all the infinite sets are equal: it's enough to mention that there is no way to couple the points of a straight line only with the integers and neither with the rational numbers, that are the numbers that you write in form of a fraction.
Cantor himself, who was criticized also from eminent mathematicians of his period, as the French Henri Poincaré and his master Leopold Kronecker, wrote in 1899 in a letter to Richard Dedekind: "I see it but I don't believe it".
But today his thesis are fundamental mathematical theories. David Hilbert, the most important mathematician of the first half of the twentieth century, said even: "Nobody could drive us out of the paradise, that has Cantor obtained for us".

To conclude our conversation we have to say that there are mathematical aspects and sectors that we haven't even mentioned. For instance, mathematics doesn't speak only about deterministic problems, and regarding that we want to remind for example that the probability theory, an important part of mathematics, developed in the 16th century starting with the studies about games of chance, speaks about situations in which the outcome is aleatory, that means ensure. The probability theory is applied in the field of insurance.
It wasn't our aim to illustrate the most recent mathematical results or all the various aspects of mathematics, but to subline the idea and the methods that form the base of "doing mathematics".