This is the title and front page of my first book on Tangrams, which I launched at the Spanish club on the 22 of October 2002.
Tangrams and more Tangrams
Tangram is the name given in the western world to a very amusing puzzle-game used today by adults as well as children. To make your own tangram is very simple: It begins with a square, which you then cut into seven pieces called ‘tans’. These ‘tans’ are: a little square, a rhomboid and five triangles (two large triangles, a middle-size one, and two small ones). All seven pieces may be combined to form an infinite variety of shapes. I call this tangram, tangram S (or square).
Other common type of tangram is made by cutting a rectangle and in this case it consists of two triangles, a pentagon and four trapeziums (a large trapezium, a middle-size one, and two small ones). I named this tangram, tangram R (or rectangular).
By combining tangram S and tangram R, I have come up with a new tangram of 14 pieces and called it tangram S-R. The book contains more than 1160 shapes and is divided in three parts of 32 pages each. There are approximately 400 different shapes that can be worked out for each tangram.
DIFFERENT SOLUTIONS TO A TANGRAM
This will be the theme for my next book about tangrams. The book is already completed, and I am seeking to publish it within the next six months. Trying to find as many as possible solutions to a given shape is one of the most challenging ways to play tangrams. In this new book, I consider the tangram-R only, but the same principles could be applied to any of the other tangrams.
There are no known rules to play tangrams. It is only required that all pieces must be used to form any given shape or figure. They must touch, but not overlap. One can put the restriction that not the rhomboid in tangram S, or the large and middle-size quadrilaterals in tangram R should be turned over to form a given shape. Also, once a shape is completed, it is not possible to claim as a different solution, the mirror image of it.
Rotational Symmetry
When a solution to a given tangram is found, it can be observed that in some cases, inside that solution you can see a symmetrical shape formed by two, three or more ‘tans’. For these partial solutions of a given tangram, we can allow the use of Rotational Symmetry around any of the three axes x, y, and z, to assist us in finding some of the remaining possible solutions.
It is only within these partial solutions that we are allowed to turn over any of the ‘tans’ without breaking the restriction stated above, because in this case the ‘tan’ or ‘tans’ involved are part of a group and are not acting as individual pieces.
Virtual Images
In order to find different solutions, I also use virtual images of tans either of individual ‘tans’ or of a group of them. For example the virtual image of the pentagon could be formed using the medium-size trapezium and one of the triangles.
Asymmetric Units
Sometimes it is possible to find inside a solution to a tangram, certain shapes made up of three or more ‘tans’. These same ‘tans’ will provide a new and different solution when positioned differently inside the same shape as shown below. They do not have any symmetrical axis and for that reason I call them asymmetric units.
These are basically the principles comprised in the two books that I am thinking of advertising in the web and their publication depends on the interest they may arise in the readers. If you are interested in any of them, please send me an E-mail and I will include you in the list of prospect costumers. If at the end of this year I have enough requests I will notify you to send the payment for the required book or both.
For more infomation on Luis Fernando Velez, just visit
http://historia-de-un-esfuerzo.wikispaces.com/
for a full biography in spanish.
